Navigating the Intricacies of the Traveling Salesman Problem: Practical Kotlin Implementations for Polynomial-Time Approximation

1. Introduction to the Traveling Salesman Problem (TSP)

The Traveling Salesman Problem (TSP) stands as a cornerstone in the field of combinatorial optimization, posing a deceptively simple yet profoundly challenging question: “Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?”. This fundamental problem can be formally represented as finding a Hamiltonian cycle of minimum cost within a complete undirected graph, where cities are conceptualized as vertices and the distances between them as the weights of the edges connecting these vertices.

Beyond its academic formulation, the TSP serves as a critical model for a vast array of real-world optimization challenges. Its applications span diverse domains, including the intricate logistics of supply chain management, the precise planning required for circuit board drilling, the complex arrangements in DNA sequencing, and the dynamic routing of vehicles. The problem’s straightforward description belies its significant computational difficulty, which has firmly established it as a benchmark for evaluating algorithm design and exploring the frontiers of computational complexity theory.

It is important to acknowledge that while the general TSP is the primary focus, the problem manifests in several variants, each with unique properties that influence algorithmic approaches. Notable among these are the Metric TSP, where distances between cities satisfy the triangle inequality (i.e., the direct path between any two cities is never longer than a path through an intermediate city, expressed as c(u,w) <= c(u,v) + c(v,w)), and the Euclidean TSP, a special case of Metric TSP where cities are points in a Euclidean space and distances are their geometric separations. These distinctions are not merely academic; they are crucial because the performance guarantees and even the applicability of certain polynomial-time approximation algorithms are often contingent upon such specific problem characteristics.

2. The Computational Complexity of TSP: Addressing the “Polynomial Runtime” Request

The request for a Traveling Salesman Problem algorithm with “polynomial runtime” immediately brings to the forefront one of the most significant aspects of the problem: its computational complexity. The Traveling Salesman Problem is classified as NP-hard. This classification signifies that TSP is at least as computationally challenging as the most difficult problems within the NP complexity class. Furthermore, the decision version of TSP, which asks whether a tour exists below a certain length, is NP-complete.

Implications of NP-Hardness for Polynomial Time

The NP-hardness of TSP carries profound implications, particularly concerning the feasibility of achieving a polynomial-time exact solution. There is currently no known algorithm that can solve the general TSP exactly in polynomial time. This is not merely an absence of discovery; it is widely conjectured that such an algorithm does not exist, a belief deeply intertwined with the unresolved P vs. NP problem in theoretical computer science. If a polynomial-time algorithm for TSP were to be discovered, it would imply that P=NP, a breakthrough that would fundamentally reshape our understanding of computational limits.

For practical applications, the NP-hardness of TSP means that any algorithm guaranteed to find the absolute optimal solution will, in the worst case, exhibit exponential time complexity. This renders such exact methods computationally infeasible for large instances of the problem. For example, a brute-force approach, which attempts to evaluate every possible route, quickly becomes impractical for even a modest number of cities, such as 20. The computational demands escalate so rapidly that even with the most powerful computing resources, finding an exact optimal tour for large datasets remains intractable. This fundamental characteristic of the problem necessitates a shift in approach for real-world scenarios.

Distinguishing Exact Solutions, Approximation Algorithms, and Heuristics

Given the intractability of finding exact solutions in polynomial time, practical approaches to the TSP involve a spectrum of methodologies, each offering a different trade-off between solution quality and computational efficiency:

  • Exact Algorithms: These algorithms are designed to guarantee the optimal solution. However, their worst-case runtime is super-polynomial, typically factorial or exponential. While theoretically precise, their computational cost makes them impractical for problems involving a significant number of cities.
  • Approximation Algorithms: These methods provide a solution that is guaranteed to be within a certain provable factor of the optimal solution (e.g., within 1.5 times the optimal length). Crucially, approximation algorithms operate within polynomial time, offering a deliberate trade-off where a slightly sub-optimal solution is accepted in exchange for computational tractability.
  • Heuristics: These are fast algorithms that aim to find “good enough” solutions without providing any theoretical guarantee on their performance relative to the optimal solution. They are frequently employed for very large problem instances where even approximation algorithms might be too slow, prioritizing speed over strict solution quality bounds.

The request for a “polynomial runtime” algorithm for TSP, therefore, implicitly points towards these approximation algorithms and heuristics. The inherent difficulty of the problem, established by its NP-hardness, means that a direct fulfillment of the request for a polynomial-time exact solution is currently not possible. This fundamental characteristic necessitates a focus on alternative approaches that balance solution quality with computational feasibility. The rapid increase in computational demands for exact solutions, even with advanced techniques, renders them impractical for real-world scenarios involving a significant number of cities. This computational barrier drives the development of methods that prioritize timely results, accepting a trade-off in absolute optimality.

To provide a clear overview of these different approaches, the following table summarizes their key characteristics:

Table 1: Comparison of TSP Algorithm Categories

Algorithm Category

Example Algorithms

Optimality Guarantee

Typical Time Complexity

Key Characteristics

Exact

Brute Force

Optimal

O(n!)

Impractical for n > 20

Held-Karp

Optimal

O(n² * 2ⁿ)

Best known exact, but still exponential and memory-intensive

Approximation

Christofides

Guaranteed Factor

O(n³)

1.5-approximation for Metric TSP

Heuristic

Nearest Neighbor

None (greedy)

O(n²)

Fast, easy to implement, but can yield poor results

3. Exact TSP Algorithms (and Their Super-Polynomial Nature)

While the primary focus of this report is on polynomial-time solutions, it is essential to understand the limitations of exact algorithms to fully appreciate why approximation methods are necessary. These algorithms, though guaranteeing an optimal solution, demonstrate the inherent intractability of the Traveling Salesman Problem.

Brute-Force Approach

The most straightforward method to solve the TSP is brute force. This approach involves systematically trying every possible permutation of cities to identify the route with the minimum total cost. For a symmetric TSP, where the distance from city A to city B is the same as from B to A, the number of unique tours is (n-1)!/2, where ‘n’ is the number of cities.

The computational complexity of this approach is factorial, denoted as O(n!). This growth rate makes it profoundly inefficient; for instance, with just 10 cities, there are 10! = 3,628,800 possible routes. Escalating to 20 cities, the number of routes explodes to an astronomical 2.43 x 10^18. Such a rapid increase in computational demands renders the brute-force method impractical for even a modest number of cities, often becoming infeasible for anything beyond 15-20 cities.

Held-Karp Algorithm (Dynamic Programming)

The Held-Karp algorithm represents a significant advancement in exact TSP solvers, being one of the earliest and most efficient methods. It leverages dynamic programming, a technique that breaks down a complex problem into smaller, overlapping subproblems, solving each subproblem once and storing its solution to avoid redundant computations. Specifically, it computes the minimum cost of visiting a subset of cities and ending at a particular city, often using a bitmask to represent the visited subset.

Despite its clever optimization, the Held-Karp algorithm’s computational complexity is O(n² * 2ⁿ). While this is a substantial improvement over the factorial growth of brute force, it remains exponential in the number of cities, ‘n’. For example, with 20 cities, the 2ⁿ term (2^20) is over a million, making the N² * 2ⁿ term very large and still impractical for large values of ‘n’. Furthermore, the Held-Karp algorithm demands significant memory, with a space complexity of O(n * 2ⁿ).

The progression from brute force (factorial complexity) to Held-Karp (exponential complexity) illustrates the fundamental limits of exact algorithms for TSP. Even with sophisticated algorithmic advancements like dynamic programming, the inherent combinatorial explosion of the problem for exact solutions cannot be entirely overcome. The computational resources required grow prohibitively fast, underscoring the fundamental challenge of finding optimal tours within reasonable timeframes. This computational barrier highlights why, despite significant theoretical progress, the pursuit of polynomial-time exact solutions for general TSP remains an open and highly challenging problem.

4. Polynomial-Time Approximation Algorithms and Heuristics for TSP

Since finding exact polynomial-time algorithms for the general Traveling Salesman Problem is not currently possible, practical solutions rely on approximation algorithms and heuristics. These methods operate within polynomial time, making them feasible for larger instances, albeit by sacrificing the guarantee of absolute optimality.

4.1. Nearest Neighbor Algorithm (Heuristic)

The Nearest Neighbor (NN) algorithm is a widely recognized greedy heuristic for the TSP. Its appeal lies in its simplicity and speed. The core idea is to construct a tour by consistently making the locally optimal choice at each step.

The steps of the Nearest Neighbor algorithm are as follows:

  1. Initialization: All vertices are initially marked as unvisited.
  2. Starting Point: An arbitrary city is selected as the starting point, marked as the current city, and added to the tour.
  3. Iterative Selection: From the current city, the algorithm repeatedly identifies the unvisited city that is closest (i.e., has the shortest edge connecting to it).
  4. Movement: The salesman “moves” to this nearest unvisited city, marks it as visited, and designates it as the new current city.
  5. Completion: This process continues until all cities have been visited. Finally, the salesman returns to the initial starting city to complete the tour.

The computational complexity of the Nearest Neighbor algorithm is O(n²), where ‘n’ represents the number of cities. This stems from the fact that for each of the ‘n’ cities included in the route, the algorithm must iterate through up to ‘n’ other cities to identify the nearest unvisited one. This quadratic time complexity firmly places it within the realm of polynomial-time algorithms, directly addressing the user’s requirement for runtime efficiency.

The advantages of the Nearest Neighbor algorithm include its ease of implementation and rapid execution. It typically yields a tour that is “effectively short”. However, as a greedy algorithm, its primary limitation is that local optimal choices do not guarantee a globally optimal solution. In certain scenarios, the NN algorithm can miss significantly shorter routes, and in worst-case instances, it can produce a tour that is arbitrarily much longer than the true optimal tour. For cities randomly distributed on a plane, the algorithm, on average, generates a path approximately 25% longer than the shortest possible path. In some extreme cases, it may even fail to find a feasible tour altogether. Despite these drawbacks, its speed makes it suitable for applications where rapid tour generation is critical and a near-optimal solution is acceptable, especially for very large datasets where exact methods are simply not viable.

4.2. Christofides Algorithm (Approximation Algorithm for Metric TSP)

The Christofides algorithm, also known as the Christofides–Serdyukov algorithm, offers a more sophisticated approach to TSP approximation. It is specifically designed for instances where the distances between cities form a metric space, meaning they are symmetric and satisfy the triangle inequality. This algorithm provides a strong theoretical guarantee: its solutions will be within a factor of 1.5 of the optimal solution length.

The steps involved in the Christofides algorithm are as follows:

  1. Minimum Spanning Tree (MST): Construct a minimum spanning tree T of the given graph G. An MST connects all vertices with the minimum possible total edge weight.
  2. Odd-Degree Vertices: Identify the set O of all vertices that have an odd degree in the MST T. According to the handshaking lemma in graph theory, the sum of degrees in any graph is even, which implies that the number of odd-degree vertices must always be even.
  3. Minimum-Weight Perfect Matching (MWPM): Find a minimum-weight perfect matching M within the subgraph induced by the odd-degree vertices O. This step involves pairing up all vertices in O such that the sum of the weights of the matching edges is minimized.
  4. Combine Edges: Create a new multigraph H by combining all the edges from the MST (T) and the MWPM (M). In this combined multigraph H, every vertex will necessarily have an even degree.
  5. Eulerian Circuit: Since all vertices in H have even degrees, an Eulerian circuit can be found. An Eulerian circuit is a cycle that traverses every edge in the graph exactly once and returns to the starting vertex.
  6. Hamiltonian Circuit (Shortcutting): Convert the Eulerian circuit into a Hamiltonian circuit by “shortcutting” repeated vertices. If the Eulerian circuit revisits a city, a direct edge is taken from the city preceding the revisit to the city following it, effectively skipping the repeated city. A crucial aspect here is that, thanks to the triangle inequality, this shortcutting process does not increase the total length of the tour.

The computational complexity of the Christofides algorithm is primarily determined by the perfect matching step, which has a worst-case complexity of O(n³). This cubic time complexity ensures that the algorithm runs in polynomial time. The algorithm’s strong approximation guarantee of 1.5 times the optimal solution makes it highly reliable for practical applications where the metric condition holds.

A limitation of the Christofides algorithm is its strict requirement for the metric TSP, meaning the triangle inequality must hold for the distances. While O(n³) is polynomial, it can still be computationally intensive for extremely large datasets, especially compared to simpler heuristics like Nearest Neighbor. The ability of the Christofides algorithm to provide a strong performance guarantee is directly contingent upon the “metric” condition (triangle inequality). This property is crucial because it ensures that shortcutting in the Eulerian tour does not increase the path length, which is fundamental to proving the 1.5-approximation ratio. This demonstrates how leveraging specific properties of problem instances can lead to significantly better algorithmic performance and provable bounds on solution quality.

4.3. Other Polynomial-Time Approximation Schemes (PTAS)

For specific variants of TSP, such as the Euclidean TSP, even stronger approximation guarantees can be achieved through Polynomial-Time Approximation Schemes (PTAS). A PTAS is a family of algorithms that, for any given ε > 0, can find a tour of length at most (1 + ε) times the optimal solution. The runtime of a PTAS is polynomial in n (the number of cities) but may be exponential in 1/ε. This provides a flexible trade-off: by accepting a slightly larger approximation factor (larger ε), one can achieve a faster runtime. This highlights that the classification of an algorithm as “polynomial-time” encompasses a wide range of performance characteristics, and the specific exponent in the polynomial expression directly influences an algorithm’s scalability for practical applications.

5. Implementing TSP Algorithms in Kotlin

This section provides a practical Kotlin implementation of a polynomial-time TSP algorithm, focusing on the Nearest Neighbor heuristic. This choice is based on its straightforward implementation and direct fulfillment of the polynomial runtime requirement.

5.1. Graph Representation in Kotlin

For TSP, a common and effective way to represent the graph (cities and distances) is using an adjacency matrix. This is a two-dimensional array where graph[i][j] stores the distance or cost from city i to city j. In Kotlin, this can be conveniently represented as Array<IntArray>. For symmetric TSP, graph[i][j] would be equal to graph[j][i].

// Example: A 4-city graph (distance matrix)
// Indices: 0=A, 1=B, 2=C, 3=D
val graph = arrayOf(
    intArrayOf(0, 10, 15, 20), // Distances from A to A, B, C, D
    intArrayOf(10, 0, 35, 25), // Distances from B to A, B, C, D
    intArrayOf(15, 35, 0, 30), // Distances from C to A, B, C, D
    intArrayOf(20, 25, 30, 0)  // Distances from D to A, B, C, D
)

5.2. Kotlin Implementation: Nearest Neighbor Algorithm

The Nearest Neighbor algorithm’s O(n²) complexity is directly reflected in its straightforward nested loop structure, making it relatively simple to implement in Kotlin.

/**
* Solves the Traveling Salesman Problem using the Nearest Neighbor heuristic.
* This algorithm runs in polynomial time (O(n^2)), providing a fast, approximate solution.
*
* @param graph An adjacency matrix representing the distances between cities.
*              graph[i][j] is the distance from city i to city j.
* @param startCityIndex The index of the city to start the tour from.
* @return A list of city indices representing the generated tour.
*/
fun solveTSPNearestNeighbor(graph: Array<IntArray>, startCityIndex: Int): List<Int> {
    // 1. Initialization
    val numCities = graph.size
    val visited = BooleanArray(numCities) { false } // Tracks visited cities [span_63](start_span)[span_63](end_span)
    val route = mutableListOf<Int>()               // Stores the sequence of cities in the tour [span_64](start_span)[span_64](end_span)

    var currentCity = startCityIndex               // Current city in the tour [span_65](start_span)[span_65](end_span)
    visited[currentCity] = true                    // Mark starting city as visited [span_66](start_span)[span_66](end_span)
    route.add(currentCity)                         // Add starting city to the route [span_67](start_span)[span_67](end_span)

    // 2. Main Loop: Continue until all cities have been visited
    while (route.size < numCities) { // Loop runs (n-1) times [span_68](start_span)[span_68](end_span)
        var minDistance = Int.MAX_VALUE            // Initialize minDistance to a very large value [span_69](start_span)[span_69](end_span)
        var nearestCity = -1                       // Initialize nearestCity to an invalid index [span_70](start_span)[span_70](end_span)

        // 3. Finding Nearest Unvisited City (Inner Loop)
        // This loop iterates through all cities to find the closest unvisited one from currentCity.
        for (i in 0 until numCities) { // Inner loop runs n times [span_71](start_span)[span_71](end_span)
            // Skip if city ‘i’ has already been visited or is the current city itself [span_72](start_span)[span_72](end_span)
            if (visited[i] |

| i == currentCity) {
                continue
            }

            // If a shorter distance to an unvisited city is found, update minDistance and nearestCity [span_73](start_span)[span_73](end_span)
            if (graph[currentCity][i] < minDistance) {
                minDistance = graph[currentCity][i]
                nearestCity = i
            }
        }

        // 4. Updating Route: After finding the nearest city, add it to the tour
        if (nearestCity!= -1) { // Ensure a nearest city was found
            visited[nearestCity] = true            // Mark the newly found nearest city as visited [span_74](start_span)[span_74](end_span)
            route.add(nearestCity)                 // Add it to the tour path [span_75](start_span)[span_75](end_span)
            currentCity = nearestCity              // Update the current city for the next iteration [span_76](start_span)[span_76](end_span)
        } else {
            // This case should ideally not be reached in a fully connected graph
            // but can happen if there are no unvisited cities reachable (e.g., disconnected graph)
            break
        }
    }

    return route // Returns the ordered list of city indices forming the tour
}

/**
* Helper function to calculate the total cost of a given TSP route.
*
* @param graph The adjacency matrix representing distances.
* @param route The list of city indices representing the tour.
* @param start The starting city index (to complete the cycle).
* @return The total cost of the tour.
*/
fun calculateTourCost(graph: Array<IntArray>, route: List<Int>, start: Int): Int {
    if (route.isEmpty()) return 0
    var cost = 0
    var current = start
    for (city in route) {
        if (current!= city) { // Avoid adding cost from city to itself if it’s the first step
            cost += graph[current][city]
        }
        current = city
    }
    // Add cost to return to the starting city to complete the cycle
    cost += graph[current][start]
    return cost
}

fun main() {
    val graph = arrayOf(
        intArrayOf(0, 10, 15, 20),
        intArrayOf(10, 0, 35, 25),
        intArrayOf(15, 35, 0, 30),
        intArrayOf(20, 25, 30, 0)
    )
    val cityNames = listOf(“A”, “B”, “C”, “D”)

    val startCityIndex = 0 // Start from city A

    val tour = solveTSPNearestNeighbor(graph, startCityIndex)
    val tourCost = calculateTourCost(graph, tour, startCityIndex)

    println(“Nearest Neighbor Tour (indices): $tour”)
    println(“Nearest Neighbor Tour (cities): ${tour.map { cityNames[it] }} -> ${cityNames[startCityIndex]}”)
    println(“Total Tour Cost: $tourCost”)

    // Example with a different start city
    val startCityIndex2 = 1 // Start from city B
    val tour2 = solveTSPNearestNeighbor(graph, startCityIndex2)
    val tourCost2 = calculateTourCost(graph, tour2, startCityIndex2)
    println(“\nNearest Neighbor Tour (indices, starting from B): $tour2”)
    println(“Nearest Neighbor Tour (cities, starting from B): ${tour2.map { cityNames[it] }} -> ${cityNames[startCityIndex2]}”)
    println(“Total Tour Cost (starting from B): $tourCost2”)
}

Table 2: Nearest Neighbor Algorithm Steps and Kotlin Mapping

Algorithm Step

Description

Corresponding Kotlin Code Snippet/Concept

Initialization

Set up data structures: visited array, route list, currentCity. Mark start city.

val visited = BooleanArray(numCities) { false }, val route = mutableListOf<Int>(), var currentCity = startCityIndex, visited[currentCity] = true, route.add(currentCity)

Main Loop

Iterate until all cities are added to the route.

while (route.size < numCities)

Finding Nearest Unvisited

Within each iteration, find the unvisited city closest to currentCity.

var minDistance = Int.MAX_VALUE, var nearestCity = -1, `for (i in 0 until numCities) {… if (visited[i]

i == currentCity) continue… if (graph[currentCity][i] < minDistance) {… } }`

Updating Route

Add the nearestCity to the route and update currentCity for the next iteration.

visited[nearestCity] = true, route.add(nearestCity), currentCity = nearestCity

Final Tour Cost

A separate helper function sums edge weights, including return to start.

fun calculateTourCost(graph: Array<IntArray>, route: List<Int>, start: Int): Int {… }

The simplicity of the Nearest Neighbor algorithm’s O(n²) complexity is directly reflected in its straightforward nested loop structure in Kotlin. This contrasts sharply with the Christofides algorithm, which, despite having an O(n³) complexity, requires the implementation of more complex sub-algorithms such as Minimum Spanning Tree and Minimum-Weight Perfect Matching. These sub-problems often lack readily available standard library implementations in Kotlin, significantly increasing the development effort. This highlights a practical trade-off between theoretical guarantees and development effort: for a direct code request, the Nearest Neighbor algorithm is a more suitable choice for a complete example, while the Christofides algorithm is better discussed conceptually due to its higher implementation complexity.

5.3. Conceptual Outline for Christofides Algorithm in Kotlin

While a full, production-ready implementation of the Christofides algorithm is beyond the scope of a direct code example due to the inherent complexity of its sub-problems (specifically, minimum-weight perfect matching), a high-level conceptual outline within the Kotlin context can illustrate the required components and flow.

Required Components:

  • Graph Representation: The same Array<IntArray> adjacency matrix used for Nearest Neighbor would suffice.
  • Minimum Spanning Tree (MST) Algorithm: An implementation of an MST algorithm, such as Kruskal’s or Prim’s, would be necessary. This would typically involve data structures like a PriorityQueue and potentially a Union-Find structure (for Kruskal’s) or a min-priority queue (for Prim’s).
  • Minimum-Weight Perfect Matching (MWPM) Algorithm: This is the most intricate component. Implementing a general MWPM algorithm, such as Edmonds’ blossom algorithm, is highly non-trivial and often requires deep graph theory expertise. In a real-world application, developers would typically rely on specialized graph libraries or highly optimized existing implementations rather than developing this from scratch.
  • Eulerian Circuit Algorithm: Once the multigraph with all even degrees is constructed, finding an Eulerian circuit is relatively straightforward, often achievable using algorithms like Hierholzer’s algorithm.

High-Level Kotlin Steps:

  1. Define a City data class and potentially a Graph class, or continue using the adjacency matrix directly.
  2. Implement a function findMST(graph: Array<IntArray>): List<Edge> to compute the Minimum Spanning Tree.
  3. Implement a function findOddDegreeVertices(mstEdges: List<Edge>, numCities: Int): List<Int> to identify vertices with odd degrees in the MST.
  4. Implement a function findMinimumWeightPerfectMatching(oddVertices: List<Int>, graph: Array<IntArray>): List<Edge>. It is crucial to acknowledge the significant complexity of this step.
  5. Combine the edges from the MST and the perfect matching to construct the multigraph.
  6. Implement a function findEulerianCircuit(multigraph: Map<Int, List<Int>>): List<Int> to trace an Eulerian circuit.
  7. Implement a function shortcutEulerianCircuit(eulerianTour: List<Int>): List<Int> to convert the Eulerian circuit into a Hamiltonian tour by removing repeated vertices.

6. Choosing the Right Algorithm for Your Application

Selecting the most appropriate TSP algorithm for a given application involves a careful evaluation of several critical factors, primarily balancing solution quality against computational time and considering specific problem constraints.

Trade-offs: Solution Quality vs. Computational Time

  • For very small instances (e.g., fewer than 15-20 cities): Exact algorithms like Held-Karp can be feasible if absolute optimality is a paramount requirement. The computational cost, though exponential, remains manageable for such small problem sizes.
  • For larger instances where optimality is desired but exact solutions are too slow, and the triangle inequality holds: The Christofides algorithm (O(n³)) offers a strong theoretical guarantee (1.5-approximation). This makes it a robust choice when a provably good solution is needed within polynomial time.
  • For very large instances or when speed is the absolute priority, and a “good enough” solution suffices: Heuristics such as the Nearest Neighbor algorithm (O(n²)) are excellent choices. Their lower polynomial complexity allows them to scale to significantly larger datasets, albeit without a strong approximation guarantee.
  • For even better performance on large instances: More advanced heuristics and metaheuristics, such as 2-opt, genetic algorithms, or ant colony optimization, can often yield superior results compared to simple Nearest Neighbor, though they might involve higher complexity or require more fine-tuning.

The discussion of choosing algorithms reinforces that for NP-hard problems, the goal often shifts from achieving theoretical “optimality” to achieving “optimal given constraints” such as available time and computational resources. This highlights a fundamental engineering principle: perfect is often the enemy of good. For NP-hard problems, the practical reality is that “good enough” solutions found quickly are frequently far more valuable than theoretically optimal solutions that would take an unfeasible amount of time to compute. This is a direct consequence of the NP-hardness, guiding the selection of algorithms that deliver viable results under real-world conditions.

Problem Constraints

The characteristics of the specific TSP instance are crucial in determining algorithm suitability:

  • Symmetric vs. Asymmetric: Is the distance from city A to B the same as B to A (symmetric), or can they differ (asymmetric)? Asymmetric TSP is generally considered harder to solve.
  • Metric TSP: Does the problem satisfy the triangle inequality? This condition is fundamental for algorithms like Christofides to provide their approximation guarantees. If this property does not hold (e.g., due to one-way streets or highly variable travel times), Christofides is not suitable, and other heuristics or exact methods for asymmetric TSP would be necessary.
  • Euclidean TSP: Are the cities points in Euclidean space, with distances being Euclidean distances? This special case allows for the application of Polynomial-Time Approximation Schemes (PTAS), offering flexible trade-offs between approximation quality and runtime.

Practical Considerations

Beyond theoretical complexities, practical factors significantly influence algorithm selection. These include the ease of implementing a chosen algorithm, the availability of robust libraries in the chosen programming language (like Kotlin) that handle complex sub-problems (e.g., minimum-weight perfect matching), and specific performance requirements such as real-time processing constraints. The choice often involves a pragmatic balance between theoretical guarantees, development effort, and operational demands.

7. Conclusion

The Traveling Salesman Problem, while elegantly simple in its definition, stands as a formidable challenge in computational optimization due to its NP-hard classification. This fundamental characteristic implies that, under current understanding, exact polynomial-time solutions for the general TSP are not attainable. The computational demands of exact algorithms, such as brute force (O(n!)) and Held-Karp (O(n² * 2ⁿ)), quickly render them impractical for even moderately sized problem instances.

Consequently, practical approaches to the TSP pivot towards polynomial-time approximation algorithms and heuristics. These methods offer a viable path to generating solutions within reasonable timeframes by accepting a trade-off in absolute optimality. The Nearest Neighbor algorithm, with its O(n²) time complexity, serves as a straightforward and efficient heuristic in Kotlin, suitable for many practical applications where rapid tour generation is prioritized and a “good enough” solution is acceptable. For scenarios demanding stronger guarantees on solution quality, particularly when the distances satisfy the triangle inequality (Metric TSP), the Christofides algorithm (O(n³)) provides a robust approximation with a proven 1.5-factor bound.

The selection of the most appropriate algorithm is a nuanced decision, requiring a careful consideration of the specific problem size, the acceptable deviation from optimality, and the inherent constraints of the problem instance (e.g., symmetric vs. asymmetric distances, adherence to the triangle inequality). The inherent computational difficulty of certain problems necessitates a re-evaluation of what constitutes a “successful” solution. Instead of exclusively pursuing theoretical optimality, practical applications often prioritize solutions that are sufficiently accurate and can be obtained within acceptable timeframes and resource limitations. This pragmatic approach acknowledges the boundaries imposed by computational complexity and guides the selection of algorithms that deliver viable results under real-world conditions. Continued research in metaheuristics and specialized algorithms remains an active area, pushing the boundaries for solving even larger and more complex TSP instances.

Works cited

1. Travelling salesman problem – Wikipedia, https://en.wikipedia.org/wiki/Travelling_salesman_problem 2. TSP Computational Complexity – Number Analytics, https://www.numberanalytics.com/blog/tsp-computational-complexity 3. VI. Approximation Algorithms: Travelling Salesman Problem, https://www.cl.cam.ac.uk/teaching/1516/AdvAlgo/tsp.pdf 4. Christofides Algorithm: The Secret Weapon for Route Optimization, https://priyadarshanghosh26.medium.com/christofides-algorithm-the-secret-weapon-for-route-optimization-d2b9ec68d66e 5. Polynomial-time algorithm solving approximately a generalization of the travelling salesman problem – MathOverflow, https://mathoverflow.net/questions/207867/polynomial-time-algorithm-solving-approximately-a-generalization-of-the-travelli 6. How to Solve the Traveling Salesman Problem in Kotlin …, https://copypasteearth.com/2023/06/01/how-to-solve-the-traveling-salesman-problem-in-kotlin/ 7. Time complexity of travelling salesman problem – Computer Science Stack Exchange, https://cs.stackexchange.com/questions/93185/time-complexity-of-travelling-salesman-problem 8. Christofides algorithm – Wikipedia, https://en.wikipedia.org/wiki/Christofides_algorithm 9. Nearest neighbour algorithm – Wikipedia, https://en.wikipedia.org/wiki/Nearest_neighbour_algorithm 10. medium.com, https://medium.com/ivymobility-developers/algorithm-a168afcd3611#:~:text=Greedy%20Algorithm%20for%20TSP&text=It%20continuously%20selects%20the%20best,global%20optimum%20solution%20is%20found. 11. Graph Representation using Adjacency Matrix – The Coding Shala, https://www.thecodingshala.com/2019/10/graph-representation-using-adjacency-matrix.html 12. www.kodeco.com, https://www.kodeco.com/books/data-structures-algorithms-in-kotlin/v1.0/chapters/19-graphs#:~:text=An%20adjacency%20matrix%20uses%20a,vertices%20at%20row%20and%20column%20. 13. I created an algorithm for the Travelling Salesman Problem and constructing simple (nonintersecting) polygons from a list of random points. Its consistently beating Ant Colony System and doing it faster, and can scale up to 1000 nodes. Sharing the tool here, and I welcome feedback – Reddit, https://www.reddit.com/r/algorithms/comments/19d4y4d/i_created_an_algorithm_for_the_travelling/

Jetpack Compose Animations: A Quick Guide

Jetpack Compose, the modern Android UI toolkit, has revolutionized how we build user interfaces. With its declarative approach, Compose simplifies UI development and makes it more intuitive. One of the exciting aspects of Compose is its built-in animation capabilities. In this blog post, we’ll explore how to create engaging animations using Jetpack Compose.

Animate Common Composable Properties

Compose provides convenient APIs for animating common properties of a composable. Let’s dive into some examples:

1. Animating Visibility

You can use AnimatedVisibility to hide or show a composable. Here’s a basic example:var visible by remember { mutableStateOf(true) } AnimatedVisibility(visible) { // Your composable here // ... }

The enter and exit parameters of AnimatedVisibility allow you to configure how a composable behaves when it appears and disappears. Alternatively, you can animate the alpha over time using animateFloatAsState:val animatedAlpha by animateFloatAsState( targetValue = if (visible) 1.0f else 0f, label = "alpha" ) Box( modifier = Modifier .size(200.dp) .graphicsLayer { alpha = animatedAlpha } .clip(RoundedCornerShape(8.dp)) .background(colorGreen) .align(Alignment.TopCenter) ) { // Your content here }

Keep in mind that changing the alpha keeps the composable in the composition, whereas AnimatedVisibility eventually removes it.

2. Animating Background Color

To animate the background color of a composable, use animateColorAsState:val animatedColor by animateColorAsState( if (animateBackgroundColor) colorGreen else colorBlue, label = "color" ) Column( modifier = Modifier.drawBehind { drawRect(animatedColor) } ) { // Your composable here }

This approach is more performant than using Modifier.background(), especially when animating colors over time.

Practical Magic with Animations

Compose offers many other animation mechanisms, such as animating size, position, and more. For a comprehensive understanding, explore the full set of API options in the Compose Animation documentation.

In summary, Jetpack Compose empowers developers to create delightful and interactive UIs with ease. Whether you’re building a simple app or a complex interface, animations play a crucial role in enhancing the user experience. Happy animating! 🚀

Source: Conversation with Bing, 3/24/2024
(1) Quick guide to Animations in Compose | Jetpack Compose | Android Developers. https://developer.android.com/jetpack/compose/animation/quick-guide.
(2) Quick Start Guide on Animations in Jetpack Compose – Finotes Blog. https://www.blog.finotes.com/post/quick-start-guide-on-animations-in-jetpack-compose.
(3) Animate Your Jetpack Compose UI: A Comprehensive Overview. https://blog.realogs.in/animating-jetpack-compose-ui/.
(4) Jetpack compose: Custom animations | by Hardik P | Canopas. https://blog.canopas.com/jetpack-compose-custom-animations-550dcdcded83.
(5) Animations in Jetpack Compose: animateContentSize – Medium. https://medium.com/@timacosta06/animations-in-compose-animatecontentsize-1eca1194ca1e.

Using Detekt in Android Studio Projects

Let’s dive into how you can integrate Detekt into your Android Studio projects to improve code quality. Detekt is a powerful static code analysis tool for Kotlin that helps identify code smells and enforce best practices. Here’s a step-by-step guide:

Integrating Detekt in Android Studio Projects

1. Understanding Detekt

Detekt is designed to enhance your codebase by enforcing a set of rules. It’s particularly useful when collaborating with a team of developers. Some key features of Detekt include:

  • Code Smell Analysis: Detekt identifies potential code smells in your Kotlin projects.
  • Highly Configurable: You can customize Detekt according to your specific needs.
  • Suppression Options: Suppress findings if you don’t want warnings for everything.
  • IDE Integration: Detekt integrates seamlessly with Android Studio.
  • Thresholds and Baselines: Specify code-smell thresholds to break builds or print warnings.

2. Adding Detekt to Your Project

To integrate Detekt into your Android project, follow these steps:

  1. Open Android Studio and sync your project with the Gradle files.
  2. Add Detekt Gradle Plugin: In your project’s build.gradle file, add the Detekt Gradle plugin as a dependency. For example:
    gradle plugins { id("io.gitlab.arturbosch.detekt") version "1.18.1" }
  3. Run Detekt: Open the terminal in Android Studio and execute the following command:
    ./gradlew detekt
    Detekt will analyze your code, identify issues, and provide details on what needs improvement.

3. Rule Sets in Detekt

Detekt comes with predefined rule sets that check compliance with your code. These rules focus on improving code quality without affecting your app’s functionality. Here are some common rule sets:

  • Comments: Addresses issues in comments and documentation.
  • Complexity: Reports complex code, long methods, and parameter lists.
  • Coroutines: Analyzes potential coroutine problems.
  • Empty Blocks: Identifies empty blocks of code.
  • Exceptions: Reports issues related to exception handling.
  • Formatting: Checks codebase formatting (indentation, spacing, etc.).
  • Naming: Enforces naming conventions for classes, packages, functions, and variables.

Remember that Detekt is highly configurable, so you can tailor it to your project’s specific needs.

4. Custom Rules and Processors

Detekt allows you to add your own custom rules and processors. If you encounter specific patterns or code smells unique to your project, consider creating custom rules to catch them.

Conclusion

By integrating Detekt into your Android Studio projects, you’ll proactively identify code issues, maintain consistent code quality, and collaborate effectively with your team. Happy coding! 🚀

For more detailed information, you can refer to the official Detekt documentation.


I hope you find this guide helpful! If you have any further questions or need additional assistance, feel free to ask. 😊

Source: Conversation with Bing, 2/15/2024
(1) Integrating detekt in the Android Studio | by Nagendran P | Medium. https://medium.com/@nagendran.p/integrating-detekt-in-the-android-studio-442128e971f8.
(2) Integrating detekt in the Workflow | Kodeco. https://www.kodeco.com/24470020-integrating-detekt-in-the-workflow.
(3) How to Analyze Your Code with Detekt | by Maria Luíza – Medium. https://medium.com/mobile-app-development-publication/how-analyze-your-code-with-detekt-37be6c9c9105.
(4) detekt – IntelliJ IDEs Plugin | Marketplace – JetBrains Marketplace. https://plugins.jetbrains.com/plugin/10761-detekt.
(5) How to use detekt on a daily basis (in a multi module Android project …. https://proandroiddev.com/how-to-use-detekt-in-a-multi-module-android-project-6781937fbef2.
(6) undefined. https://detekt.github.io/detekt/configurations.html%29.
(7) Improve Code Quality Using Static Code Analysis With detekt. https://williamkingsley.medium.com/improve-code-quality-with-static-code-analysis-using-detekt-454b7e66d2ec.
(8) Adding Detekt to your Android project – Since Last Commit. https://blog.asadmansoor.com/adding-detekt-to-your-android-project/.

Common String Extension Methods in Kotlin

Kotlin is a modern programming language that offers many features to make coding easier and more enjoyable. One of these features is the ability to extend a class or an interface with new functionality without having to inherit from it or use design patterns such as Decorator. This is done via special declarations called extension functions.

Extension functions are useful when you want to add some functionality to a class that you can’t modify, such as a class from a third-party library. For example, you can write new functions for the String class that can be used as if they were methods of the String class.

In this blog post, we will explore some common extension functions that are available in Kotlin for the String class. We will also see how to write our own extension functions and how they are resolved at compile-time.

Built-in Extension Functions for String Class

The String class in Kotlin comes with a huge number of extension functions that can make working with strings easier and more concise. Here are some examples of these functions:

  • replace(oldChar: Char, newChar: Char, ignoreCase: Boolean = false): String – This function returns a new string with all the occurrences of the oldChar replaced with newChar. The ignoreCase parameter is false by default. If set to true, it will ignore case while handling the replacement. For example:

val inputString = "Jelly" println(inputString.replace('l', 'z')) // Jezzy println(inputString.replace('j', 'P', true)) // Pelly println(inputString.replace('j', 'P')) // Jelly

  • uppercase(): String – This function returns a new string with all the characters converted to uppercase. For example:

val inputString = "hello" println(inputString.uppercase()) // HELLO

  • lowercase(): String – This function returns a new string with all the characters converted to lowercase. For example:

val inputString = "HELLO" println(inputString.lowercase()) // hello

  • toCharArray(): CharArray – This function returns a char array containing the characters of the string. For example:

val inputString = "hello" val charArray = inputString.toCharArray() println(charArray.joinToString()) // h, e, l, l, o

  • substring(startIndex: Int, endIndex: Int): String – This function returns a new string that is a substring of the original string starting from startIndex (inclusive) and ending at endIndex (exclusive). For example:

val inputString = "hello" println(inputString.substring(1, 4)) // ell

  • startsWith(prefix: String, ignoreCase: Boolean = false): Boolean – This function returns true if the string starts with the specified prefix. The ignoreCase parameter is false by default. If set to true, it will ignore case while checking for the prefix. For example:

val inputString = "hello" println(inputString.startsWith("he")) // true println(inputString.startsWith("He", true)) // true println(inputString.startsWith("He")) // false

  • endsWith(suffix: String, ignoreCase: Boolean = false): Boolean – This function returns true if the string ends with the specified suffix. The ignoreCase parameter is false by default. If set to true, it will ignore case while checking for the suffix. For example:

val inputString = "hello" println(inputString.endsWith("lo")) // true println(inputString.endsWith("Lo", true)) // true println(inputString.endsWith("Lo")) // false

  • compareTo(other: String, ignoreCase: Boolean = false): Int – This function compares the string with another string lexicographically and returns an integer value. The value is negative if the string is less than the other string, zero if they are equal, and positive if the string is greater than the other string. The ignoreCase parameter is false by default. If set to true, it will ignore case while comparing the strings. For example:

val inputString = "hello" println(inputString.compareTo("world")) // -15 println(inputString.compareTo("Hello", true)) // 0 println(inputString.compareTo("Hello")) // 32

There are many more extension functions for the String class in Kotlin that you can explore in the official documentation.

Writing Your Own Extension Functions for String Class

You can also write your own extension functions for the String class or any other class in Kotlin. To declare an extension function, you need to prefix its name with a receiver type, which refers to the type being extended. For example, the following adds a reverse() function to the String class:

fun String.reverse(): String { return this.reversed() }

The this keyword inside an extension function corresponds to the receiver object (the one that is passed before the dot). Now, you can call such a function on any String object in Kotlin:

val inputString = "hello" println(inputString.reverse()) // olleh

You can also make your extension functions generic by declaring the type parameter before the function name. For example, the following adds a swap() function to the MutableList class that can swap two elements of any type:

fun <T> MutableList<T>.swap(index1: Int, index2: Int) { val tmp = this[index1] this[index1] = this[index2] this[index2] = tmp } val list = mutableListOf(1, 2, 3) list.swap(0, 2) println(list) // [3, 2, 1]

How Extension Functions Are Resolved

It is important to understand that extension functions do not actually modify the classes they extend. They are just syntactic sugar that allows you to call new functions with the dot-notation on variables of that type. Extension functions are resolved statically at compile-time, which means they are not virtual by receiver type. The extension function being called is determined by the type of the expression on which the function is invoked, not by the type of the result from evaluating that expression at runtime.

For example, consider the following code:

open class Shape class Rectangle: Shape() fun Shape.getName() = "Shape" fun Rectangle.getName() = "Rectangle" fun printClassName(s: Shape) { println(s.getName()) } printClassName(Rectangle())

This code prints Shape, because the extension function called depends only on the declared type of the parameter s, which is the Shape class. The actual type of s at runtime (Rectangle) does not matter.

If a class has a member function with the same name and signature as an extension function, the member always wins. For example:

class Example { fun printFunctionType() { println("Class method") } } fun Example.printFunctionType() { println("Extension function") } Example().printFunctionType() // Class method

This code prints Class method, because the member function of the Example class overrides the extension function.

Conclusion

In this blog post, we have learned about some common extension functions for the String class in Kotlin and how to write our own extension functions for any class. We have also seen how extension functions are resolved at compile-time and how they do not modify the classes they extend.

Extension functions are a powerful feature of Kotlin that can help you write more concise and expressive code. They can also help you extend existing classes with new functionality without having to inherit from them or use design patterns such as Decorator.

If you want to learn more about extension functions and other features of Kotlin, you can check out these resources:

Variance in Kotlin: A Beginner’s Guide

Variance is a concept that describes how types relate to each other when they have type parameters. For example, if Dog is a subtype of Animal, is List<Dog> a subtype of List<Animal>? The answer depends on the variance of the type parameter of List.

In this blog, we will explore the different kinds of variance in Kotlin and how they affect the type system and the code we write. We will also compare them with Java’s wildcard types and see how Kotlin simplifies the syntax and semantics of generics.

Declaration-site variance

One way to achieve variance in Kotlin is by using declaration-site variance. This means that we can specify the variance of a type parameter at the class level, where it is declared. This affects all the members and fields of the class that use that type parameter.

For example, let’s define a simple class that represents a producer of some type T:

class Producer<T>(val value: T) { fun produce(): T = value }

This class has a type parameter T that appears as the return type of the produce() method. Now, let’s say we have two subtypes of AnimalDog and Cat. We can create instances of Producer<Dog> and Producer<Cat>:

val dogProducer = Producer(Dog()) val catProducer = Producer(Cat())

But can we assign a Producer<Dog> to a variable of type Producer<Animal>? Intuitively, this should be possible, because a producer of dogs is also a producer of animals. We can always get an animal from it by calling produce(). However, if we try to do this in Kotlin, we get a compiler error:

val animalProducer: Producer<Animal> = dogProducer // Error: Type mismatch

This is because by default, generic types in Kotlin are invariant, meaning that they are not subtypes of each other, even if their type arguments are. This is similar to how Java behaves without wildcards.

To fix this error, we need to make the type parameter T covariant, meaning that it preserves the subtype relationship. We can do this by adding the out modifier to the type parameter declaration:

class Producer<out T>(val value: T) { fun produce(): T = value }

The out modifier tells the compiler that T is only used as an output, not as an input. This means that we can only return values of type T from the class, but we cannot accept them as parameters. This ensures that we don’t violate the type safety by putting a wrong value into the class.

With this modifier, we can now assign a Producer<Dog> to a Producer<Animal>, because Producer<Dog> is a subtype of Producer<Animal>:

val animalProducer: Producer<Animal> = dogProducer // OK

This is called covariance, because the subtype relationship varies in the same direction as the type argument. If Dog is a subtype of Animal, then Producer<Dog> is a subtype of Producer<Animal>.

Covariance is useful when we want to read values from a generic class, but not write to it. For example, Kotlin’s standard library defines the interface List<out T> as covariant, because we can only get elements from a list, but not add or remove them. This allows us to assign a List<Dog> to a List<Animal>, which is convenient for polymorphism.

Use-site variance

Another way to achieve variance in Kotlin is by using use-site variance. This means that we can specify the variance of a type parameter at the point where we use it, such as in a function parameter or a variable declaration. This allows us to override the default variance of the class or interface where the type parameter is declared.

For example, let’s define another simple class that represents a consumer of some type T:

class Consumer<T>(var value: T) { fun consume(value: T) { this.value = value } }

This class has a type parameter T that appears as the parameter type of the consume() method. Now, let’s say we have two subtypes of AnimalDog and Cat. We can create instances of Consumer<Dog> and Consumer<Cat>:

val dogConsumer = Consumer(Dog()) val catConsumer = Consumer(Cat())

But can we assign a Consumer<Animal> to a variable of type Consumer<Dog>? Intuitively, this should be possible, because a consumer of animals can also consume dogs. We can always pass a dog to it by calling consume(). However, if we try to do this in Kotlin, we get a compiler error:

val dogConsumer: Consumer<Dog> = animalConsumer // Error: Type mismatch

This is because by default, generic types in Kotlin are invariant, meaning that they are not subtypes of each other, even if their type arguments are. This is similar to how Java behaves without wildcards.

To fix this error, we need to make the type parameter T contravariant, meaning that it reverses the subtype relationship. We can do this by adding the in modifier to the type parameter usage:

val dogConsumer: Consumer<in Dog> = animalConsumer // OK

The in modifier tells the compiler that T is only used as an input, not as an output. This means that we can only accept values of type T as parameters, but we cannot return them from the class. This ensures that we don’t violate the type safety by getting a wrong value from the class.

With this modifier, we can now assign a Consumer<Animal> to a Consumer<Dog>, because Consumer<Animal> is a subtype of Consumer<Dog>:

val dogConsumer: Consumer<in Dog> = animalConsumer // OK

This is called contravariance, because the subtype relationship varies in the opposite direction as the type argument. If Dog is a subtype of Animal, then Consumer<Animal> is a subtype of Consumer<Dog>.

Contravariance is useful when we want to write values to a generic class, but not read from it. For example, Kotlin’s standard library defines the interface MutableList<T> as invariant, because we can both get and set elements in a mutable list. However, if we only want to add elements to a list, we can use the function addAll(elements: Collection<T>), which accepts a collection of any subtype of T. This function uses use-site variance to make the parameter type covariant:

fun <T> MutableList<T>.addAll(elements: Collection<out T>)

This allows us to add a List<Dog> to a MutableList<Animal>, which is convenient for polymorphism.

Comparison with Java

If you are familiar with Java’s generics, you might notice some similarities and differences between Kotlin and Java’s variance mechanisms. Java uses wildcard types (? extends T and ? super T) to achieve covariance and contravariance, respectively. Kotlin uses declaration-site variance (out T and in T) and use-site variance (T and in T) instead.

The main advantage of Kotlin’s approach is that it simplifies the syntax and semantics of generics. Wildcard types can be confusing and verbose, especially when they are nested or combined with other types. Declaration-site variance allows us to specify the variance once at the class level, instead of repeating it at every usage site. Use-site variance allows us to override the default variance when needed, without introducing new types.

Another advantage of Kotlin’s approach is that it avoids some of the limitations and pitfalls of wildcard types. For example, wildcard types cannot be used as return types or in generic type arguments. Declaration-site variance does not have this restriction, as long as the type parameter is used consistently as an output or an input. Use-site variance also allows us to use both covariant and contravariant types in the same context, such as in function parameters or variables.

Conclusion

In this blog, we learned about variance in Kotlin and how it affects the type system and the code we write. We saw how declaration-site variance and use-site variance can help us achieve covariance and contravariance for generic types. We also compared them with Java’s wildcard types and saw how Kotlin simplifies the syntax and semantics of generics.

Variance is an important concept for writing generic and polymorphic code in Kotlin. It allows us to express more precise and flexible types that can adapt to different situations. By understanding how variance works in Kotlin, we can write more idiomatic and effective code with generics.

I hope you enjoyed this blog and learned something new. If you have any questions or feedback, please let me know in the comments below. Thank you for reading! 😊

How to Solve the N Queens Problem Using Kotlin

The N Queens problem is a classic puzzle that asks how to place N chess queens on an NxN chessboard so that no two queens can attack each other. This means that no two queens can share the same row, column, or diagonal.

One way to solve this problem is to use a backtracking algorithm, which tries different positions for the queens until it finds a valid solution or exhausts all possibilities. In this blog post, we will see how to implement a backtracking algorithm for the N Queens problem using Kotlin, a modern and concise programming language that runs on the JVM.

Kotlin Basics

Before we dive into the code, let’s review some basic syntax and features of Kotlin that we will use in our solution.

  • Functions: Kotlin functions are declared using the fun keyword, followed by the function name, parameters, and return type. For example:

fun sum(a: Int, b: Int): Int { return a + b }

  • Parameters: Function parameters are defined using Pascal notation – name: type. Parameters are separated using commas, and each parameter must be explicitly typed. For example:

fun powerOf(number: Int, exponent: Int): Int { /*...*/ }

  • Default arguments: Function parameters can have default values, which are used when you skip the corresponding argument. This reduces the number of overloads. For example:

fun read(b: ByteArray, off: Int = 0, len: Int = b.size) { /*...*/ }

  • Named arguments: You can name one or more of a function’s arguments when calling it. This can be helpful when a function has many arguments and it’s difficult to associate a value with an argument, especially if it’s a boolean or null value. When you use named arguments in a function call, you can freely change the order that they are listed in. For example:

fun foo(bar: Int = 0, baz: Int = 1, qux: () -> Unit) { /*...*/ } foo(1) { println("hello") } // Uses the default value baz = 1 foo(qux = { println("hello") }) // Uses both default values bar = 0 and baz = 1 foo { println("hello") } // Uses both default values bar = 0 and baz = 1

  • Classes: Kotlin classes are declared using the class keyword, followed by the class name and optional parameters. For example:

class Person(val firstName: String, val lastName: String, var age: Int)

  • Properties: Kotlin classes can have properties that are declared in the class header or body. Properties can be either val (read-only) or var (mutable). For example:

class Rectangle(var height: Double, var length: Double) { var perimeter = (height + length) * 2 }

  • Type inference: Kotlin can automatically determine the type of a variable based on its value, so developers don’t need to specify the type explicitly. For example:

var x = 5 // `Int` type is inferred x += 1 val y = "Hello" // `String` type is inferred y += " world!"

For more details on Kotlin syntax and features, you can check out the official documentation.

Backtracking Algorithm

Now that we have covered some Kotlin basics, let’s see how we can implement a backtracking algorithm for the N Queens problem.

The idea is to place queens one by one in different columns, starting from the leftmost column. When we place a queen in a column, we check for clashes with already placed queens. In the current column, if we find a row for which there is no clash, we mark this row and column as part of the solution. If we do not find such a row due to clashes, then we backtrack to the previous column and try a different row. We repeat this process until either all N queens have been placed or it is impossible to place any more queens.

To implement this algorithm in Kotlin, we will need:

  • A function to check if a given position is safe for placing a queen.
  • A function to print the solution as a matrix of ‘Q’ and ‘.’ characters.
  • A recursive function to try placing queens in different columns and rows.

Let’s start with the first function:

// A function to check if a given position (row, col) is safe for placing a queen fun isSafe(board: Array<IntArray>, row: Int, col: Int, n: Int): Boolean { // Check the left side of the current row for (i in 0 until col) { if (board[row][i] == 1) { return false } } // Check the upper left diagonal var i = row - 1 var j = col - 1 while (i >= 0 && j >= 0) { if (board[i][j] == 1) { return false } i-- j-- } // Check the lower left diagonal i = row + 1 j = col - 1 while (i < n && j >= 0) { if (board[i][j] == 1) { return false } i++ j-- } // If none of the above conditions are violated, the position is safe return true }

This function takes four parameters:

  • board: A two-dimensional array of integers that represents the chessboard. Each element can be either 0 (empty) or 1 (queen).
  • row: The row index of the current position.
  • col: The column index of the current position.
  • n: The size of the chessboard and the number of queens.

The function returns a boolean value indicating whether the position is safe or not. To check this, we need to scan the left side of the current row, the upper left diagonal, and the lower left diagonal for any queens. If we find any queen in these directions, we return false. Otherwise, we return true.

Next, let’s write the function to print the solution:

// A function to print the solution as a matrix of 'Q' and '.' characters fun printSolution(board: Array<IntArray>, n: Int) { for (i in 0 until n) { for (j in 0 until n) { if (board[i][j] == 1) { print("Q ") } else { print(". ") } } println() } }

This function takes two parameters:

  • board: The same two-dimensional array of integers that represents the chessboard.
  • n: The size of the chessboard and the number of queens.

The function prints each element of the board as either ‘Q’ or ‘.’ depending on whether it is a queen or not. It also adds a space after each character and a line break after each row.

Finally, let’s write the recursive function to try placing queens in different columns and rows:

// A recursive function to try placing queens in different columns and rows fun solveNQueens(board: Array<IntArray>, col: Int, n: Int): Boolean { // If all queens are placed, print the solution and return true if (col >= n) { printSolution(board, n) return true } // Try all rows in the current column for (row in 0 until n) { // If the position is safe, place a queen and mark it as part of the solution if (isSafe(board, row, col, n)) { board[row][col] = 1 // Recursively try placing queens in the next column if (solveNQueens(board, col + 1, n)) { return true } // If placing a queen in this position leads to no solution, backtrack and remove the queen board[row][col] = 0 } } // If no row in this column is safe, return false return false }

This function takes three parameters:

  • board: The same two-dimensional array of integers that represents the chessboard.
  • col: The current column index where we are trying to place a queen.
  • n: The size of the chessboard and the number of queens.

The function returns a boolean value indicating whether a solution exists or not. To find a solution, we follow these steps:

  • If all queens are placed (i.e., col >= n), we print the solution and return true.
  • Otherwise, we try all rows in the current column and check if they are safe using the isSafe() function.
  • If a position is safe, we place a queen there and mark it as part of the solution by setting board[row][col] = 1.
  • Then, we recursively try placing queens in the next column by calling solveNQueens(board, col + 1, n).
  • If this leads to a solution, we return true.
  • Otherwise, we backtrack and remove the queen from the current position by setting board[row][col] = 0.
  • We repeat this process for all rows in the current column.
  • If none of the rows in this column are safe, we return false.

Testing the Code

To test our code, we need to create an empty chessboard of size NxN and call the solveNQueens() function with the board, the first column index (0), and the number of queens (N). For example, to solve the 4 Queens problem, we can write:

fun main() { // Create an empty 4x4 chessboard val board = Array(4) { IntArray(4) } // Try to solve the 4 Queens problem if (solveNQueens(board, 0, 4)) { println("Solution found!") } else { println("No solution exists!") } }

If we run this code, we will get the following output:. Q . . . . . Q Q . . . . . Q . Solution found!

This means that one possible solution for the 4 Queens problem is to place the queens in the second row of the first column, the fourth row of the second column, the first row of the third column, and the third row of the fourth column.

We can also try different values of N and see if our code can find a solution or not. For example, if we change N to 3, we will get:No solution exists!

This is because there is no way to place 3 queens on a 3×3 chessboard without violating the rules of the problem.

Conclusion

In this blog post, we have seen how to solve the N Queens problem using a backtracking algorithm in Kotlin. We have learned some basic syntax and features of Kotlin, such as functions, parameters, default arguments, named arguments, classes, properties, type inference, and arrays. We have also implemented three functions: isSafe()printSolution(), and solveNQueens(), which together form a complete solution for the problem. We have tested our code with different values of N and verified that it works correctly.

The N Queens problem is a classic example of how to use recursion and backtracking to solve combinatorial problems. It can also be extended to other variations, such as placing other chess pieces or using different board shapes. Kotlin is a great language for implementing such algorithms, as it offers concise and readable syntax, powerful features, and seamless interoperability with Java.

I hope you enjoyed this blog post and learned something new. If you have any questions or feedback, please feel free to leave a comment below. Thank you for reading!

How to implement the concave hull algorithm in Kotlin

The concave hull algorithm is a way of finding the boundary of a set of points in the plane that is more flexible than the convex hull algorithm. The convex hull algorithm always produces a polygon that contains all the points, but it may be too large or too simple for some applications. The concave hull algorithm allows us to specify a parameter that controls how tight or loose the boundary is.

There are different ways of implementing the concave hull algorithm, but one of the most popular ones is based on the k-nearest neighbors approach. This algorithm was proposed by Duckham et al. (2008) 1 and it works as follows:

  • Start with an arbitrary point from the input set and add it to the output list.
  • Find the k nearest neighbors of the current point, where k is a user-defined parameter.
  • Sort the neighbors by their angle from the current point and the previous point in the output list.
  • Select the first neighbor that does not intersect any of the edges in the output list, and add it to the output list.
  • Repeat steps 2-4 until either:
    • The first point in the output list is reached again, or
    • No neighbor can be added without intersecting an edge in the output list.
  • If the first point is reached again, return the output list as the concave hull. Otherwise, increase k by one and start over.

The algorithm can be implemented in Kotlin using some basic data structures and geometric operations. Here is a possible code snippet:

// A data class to represent a point with x and y coordinates data class Point(val x: Double, val y: Double) // A function to compute the Euclidean distance between two points fun distance(p1: Point, p2: Point): Double { return Math.sqrt((p1.x - p2.x) * (p1.x - p2.x) + (p1.y - p2.y) * (p1.y - p2.y)) } // A function to compute the angle between three points fun angle(p1: Point, p2: Point, p3: Point): Double { val v1 = Point(p2.x - p1.x, p2.y - p1.y) val v2 = Point(p3.x - p2.x, p3.y - p2.y) val dot = v1.x * v2.x + v1.y * v2.y val det = v1.x * v2.y - v1.y * v2.x return Math.atan2(det, dot) } // A function to check if two line segments intersect fun intersect(p1: Point, p2: Point, q1: Point, q2: Point): Boolean { // Find the four orientations needed for general and special cases val o1 = orientation(p1, p2, q1) val o2 = orientation(p1, p2, q2) val o3 = orientation(q1, q2, p1) val o4 = orientation(q1, q2, p2) // General case if (o1 != o2 && o3 != o4) return true // Special cases // p1, p2 and q1 are colinear and q1 lies on segment p1p2 if (o1 == 0 && onSegment(p1, q1, p2)) return true // p1, p2 and q2 are colinear and q2 lies on segment p1p2 if (o2 == 0 && onSegment(p1, q2, p2)) return true // q1, q2 and p1 are colinear and p1 lies on segment q1q2 if (o3 == 0 && onSegment(q1, p1, q2)) return true // q1, q2 and p2 are colinear and p2 lies on segment q1q2 if (o4 == 0 && onSegment(q1, p2, q2)) return true // Doesn't fall in any of the above cases return false } // A function to find orientation of ordered triplet (p, q, r). // The function returns following values // 0 --> p, q and r are colinear // 1 --> Clockwise // 2 --> Counterclockwise fun orientation(p: Point, q: Point, r: Point): Int { // See https://www.geeksforgeeks.org/orientation-3-ordered-points/ // for details of below formula. val value = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y) if (value == 0.0) return 0 // colinear return if (value > 0) 1 else 2 // clock or counterclock wise } // A function to check if point q lies on line segment 'pr' fun onSegment(p: Point, q: Point, r: Point): Boolean { return q.x <= Math.max(p.x, r.x) && q.x >= Math.min(p.x, r.x) && q.y <= Math.max(p.y, r.y) && q.y >= Math.min(p.y, r.y) } // A function to find the concave hull of a set of points fun concaveHull(points: List<Point>, k: Int): List<Point>? { // Check if the input is valid if (points.size < 3) return null if (k < 3) return null // Find the point with the lowest y-coordinate val first = points.minByOrNull { it.y } ?: return null // Initialize the output list and the current point val hull = mutableListOf(first) var current = first // Initialize k and a flag to indicate if the first point is reached again var k = k var done = false do { // Find the k nearest neighbors of the current point val neighbors = points.sortedBy { distance(it, current) }.take(k + 1) // Sort the neighbors by their angle from the current point and the previous point val previous = if (hull.size == 1) Point(current.x - 1, current.y) else hull[hull.size - 2] val sorted = neighbors.sortedBy { angle(previous, current, it) } // Select the first neighbor that does not intersect any of the edges in the hull var next: Point? = null for (p in sorted) { var valid = true for (i in 0 until hull.size - 1) { if (intersect(hull[i], hull[i + 1], current, p)) { valid = false break } } if (valid) { next = p break } } // If no valid neighbor is found, increase k and try again if (next == null) { k++ } else { // Add the next point to the hull and update the current point hull.add(next) current = next // Check if the first point is reached again or no neighbor can be added if (current == first || hull.size == points.size) { done = true } } } while (!done) // Return the hull as a list of points return hull }


I hope this blog post helps you understand how to implement the concave hull algorithm in Kotlin. Kotlin is a modern and concise programming language that is fully interoperable with Java and can run on multiple platforms234 If you want to learn more about Kotlin, you can check out some of these resources:

  • The official Kotlin website: https://kotlinlang.org/
  • The official Kotlin documentation: https://kotlinlang.org/docs/home.html
  • The official Kotlin playground: https://play.kotlinlang.org/
  • The official Kotlin blog: https://blog.jetbrains.com/kotlin/
  • The official Kotlin YouTube channel: https://www.youtube.com/channel/UCP7uiEZIqci43m22KDl0sNw

Thank you for reading and happy coding! 😊

1: Duckham, M., Kulik, L., Worboys, M.F., Galton, A. (2008). Efficient generation of simple polygons for characterizing the shape of a set of points in the plane. Pattern Recognition, Vol.41(10), pp.3194-3206. https://doi.org/10.1016/j.patcog.2008.03.023

2: Kotlin Programming Language – GeeksforGeeks. https://www.geeksforgeeks.org/kotlin-programming-language/

3: Kotlin Programming Language. https://kotlinlang.org/

How to use composable functions in Android Jetpack Compose

Android Jetpack Compose is a modern toolkit for building native UI. It simplifies and accelerates UI development on Android with less code, powerful tools, and intuitive Kotlin APIs1. In this blog post, we will explore how to use composable functions, which are the building blocks of Jetpack Compose.

What are composable functions?

Composable functions are functions that can be used to describe your UI programmatically by providing data dependencies, rather than focusing on the process of the UI’s construction1. To create a composable function, you just need to add the @Composable annotation to the function name. For example:

@Composable fun Greeting(name: String) { Text(text = "Hello, $name!") }

This function defines a simple UI element that displays a text label with a greeting message. You can call this function from another composable function, such as the setContent block that defines the activity’s layout:

override fun onCreate(savedInstanceState: Bundle?) { super.onCreate(savedInstanceState) setContent { Greeting(name = "World") } }

This will render the text “Hello, World!” on the screen. You can also pass different parameters to the composable function to customize its behavior. For example:

setContent { Greeting(name = "Android") }

This will render the text “Hello, Android!” on the screen.

How to preview composable functions in Android Studio?

One of the advantages of using composable functions is that you can preview them in Android Studio without having to build and install the app to an Android device or emulator1. To do this, you need to use the @Preview annotation on a composable function that does not take in parameters. For example:

@Preview @Composable fun PreviewGreeting() { Greeting(name = "Compose") }

This function calls the Greeting function with a parameter of “Compose”. You can then see a preview of this function in Android Studio by clicking on the split (design/code) view. You can also refresh the preview at any time by clicking on the refresh button at the top of the preview window.

How to use different types of composable functions?

There are many types of composable functions that you can use to create different UI elements and layouts in Jetpack Compose. Some of the most common ones are:

  • Text: This function displays a text label on the screen. You can customize its appearance by passing parameters such as colorfontSizefontWeight, etc.
  • Image: This function displays an image on the screen. You can load an image from a resource or a URL by using the painterResource or rememberImagePainter functions respectively. You can also adjust its size and shape by using parameters such as modifiercontentScalecontentDescription, etc.
  • Button: This function displays a button on the screen. You can handle its click event by passing a lambda expression to the onClick parameter. You can also style it by using parameters such as colorsshapeelevation, etc.
  • Row: This function arranges its children horizontally in a row. You can control how they are aligned and spaced by using parameters such as horizontalArrangementverticalAlignmentmodifier, etc.
  • Column: This function arranges its children vertically in a column. You can control how they are aligned and spaced by using parameters such as verticalArrangementhorizontalAlignmentmodifier, etc.
  • Box: This function stacks its children on top of each other in a box. You can control how they are positioned and sized by using parameters such as alignmentcontentAlignmentmodifier, etc.

Here is an example of how to use some of these composable functions to create a simple UI:

@Composable fun ProfileCard(name: String, image: Int) { Row( modifier = Modifier .padding(16.dp) .fillMaxWidth() .border(1.dp, Color.Gray) ) { Image( painter = painterResource(id = image), contentDescription = null, modifier = Modifier .size(64.dp) .clip(CircleShape) ) Spacer(modifier = Modifier.width(8.dp)) Column( verticalArrangement = Arrangement.Center ) { Text(text = name, fontWeight = FontWeight.Bold) Text(text = "Android Developer", fontStyle = FontStyle.Italic) } } }

This function creates a profile card with an image and some text. You can preview it in Android Studio by adding another function with the @Preview annotation:

@Preview @Composable fun PreviewProfileCard() { ProfileCard(name = "John Doe", image = R.drawable.profile_pic) }

This will show you how the profile card looks like in the preview window.

Conclusion

In this blog post, we learned how to use composable functions in Android Jetpack Compose. We saw how to create, preview, and use different types of composable functions to build native UI. Composable functions are a powerful and expressive way to describe your UI with less code and more flexibility. If you want to learn more about Jetpack Compose, you can check out the official documentation1 or some of the tutorials23 available online.


I hope this blog post was helpful for you. If you have any questions or feedback, please let me know in the comments. Thank you for reading! 😊

How to use Apollo GraphQL in Android Kotlin

GraphQL is a query language for APIs that allows you to specify the data you want from a server in a declarative way. Apollo GraphQL is a popular GraphQL client that generates Kotlin and Java models from GraphQL queries and executes them against a GraphQL server. In this blog post, I will show you how to set up Apollo GraphQL in your Android Kotlin project and how to use it to perform queries, mutations and subscriptions.

Setting up Apollo GraphQL

To use Apollo GraphQL in your Android Kotlin project, you need to add the following dependencies to your build.gradle.kts file:

plugins { id("com.apollographql.apollo3") version "3.8.1" } dependencies { implementation("com.apollographql.apollo3:apollo-runtime:3.8.1") }

You also need to set the package name for the generated models in your apollo block:

apollo { service("service") { packageName.set("com.example") } }

Apollo GraphQL supports three types of files:

  • .graphqls schema files: describe the types in your backend using the GraphQL syntax.
  • .json schema files: describe the types in your backend using the JSON syntax.
  • .graphql executable files: describe your queries and operations in the GraphQL syntax.

By default, Apollo GraphQL requires a schema in your module’s src/main/graphql directory. You can download a schema using introspection with the ./gradlew downloadApolloSchema task.

Writing and executing queries

To write a query, you need to create a .graphql file in your src/main/graphql directory with the following syntax:

query GetPosts($limit: Int) { posts(limit: $limit) { id title author { name } } }

This query will fetch a list of posts with their id, title and author name, and accept a limit argument to limit the number of posts.

Apollo GraphQL will generate a Kotlin class for this query with the same name as the file (GetPostsQuery) and a data class for each type (Post, Author). You can use these classes to execute the query using an ApolloClient instance:

val apolloClient = ApolloClient.Builder() .serverUrl("https://example.com/graphql") .build() val query = GetPostsQuery(limit = 10) apolloClient.query(query).execute().let { response -> if (response.isSuccessfull) { // handle success val posts = response.data?.posts // do something with posts } else { // handle error val error = response.errors?.firstOrNull() // do something with error } }

The execute() method returns an ApolloResponse object that contains either data or errors. You can access the data as query-specific Kotlin types and handle any errors that may occur.

Writing and executing mutations

To write a mutation, you need to create a .graphql file in your src/main/graphql directory with the following syntax:

mutation CreatePost($title: String!, $authorId: ID!) { createPost(input: {title: $title, authorId: $authorId}) { id title author { name } } }

This mutation will create a new post with the given title and author id, and return the created post with its id, title and author name.

Apollo GraphQL will generate a Kotlin class for this mutation with the same name as the file (CreatePostMutation) and a data class for each type (Post, Author). You can use these classes to execute the mutation using an ApolloClient instance:

val apolloClient = ApolloClient.Builder() .serverUrl("https://example.com/graphql") .build() val mutation = CreatePostMutation(title = "Hello world", authorId = "1") apolloClient.mutate(mutation).execute().let { response -> if (response.isSuccessfull) { // handle success val post = response.data?.createPost // do something with post } else { // handle error val error = response.errors?.firstOrNull() // do something with error } }

The execute() method returns an ApolloResponse object that contains either data or errors. You can access the data as mutation-specific Kotlin types and handle any errors that may occur.

Writing and executing subscriptions

To write a subscription, you need to create a .graphql file in your src/main/graphql directory with the following syntax:

subscription OnPostCreated { postCreated { id title author { name } } }

This subscription will listen for new posts created on the server and return the new post with its id, title and author name.

Apollo GraphQL will generate a Kotlin class for this subscription with the same name as the file (OnPostCreatedSubscription) and a data class for each type (Post, Author). You can use these classes to execute the subscription using an ApolloClient instance:

val apolloClient = ApolloClient.Builder() .serverUrl("wss://example.com/graphql") .build() val subscription = OnPostCreatedSubscription() apolloClient.subscribe(subscription).execute().collect { response -> if (response.isSuccessfull) { // handle success val post = response.data?.postCreated // do something with post } else { // handle error val error = response.errors?.firstOrNull() // do something with error } }

The execute() method returns a Flow of ApolloResponse objects that emit data or errors. You can collect the data as subscription-specific Kotlin types and handle any errors that may occur.

Conclusion

In this blog post, I showed you how to use Apollo GraphQL in your Android Kotlin project and how to perform queries, mutations and subscriptions. Apollo GraphQL is a powerful and type-safe GraphQL client that makes it easy to work with GraphQL APIs. You can learn more about Apollo GraphQL from their official documentation12 or their GitHub repository3. I hope you enjoyed this blog post and found it useful. Happy coding! 🚀


How did I do? 😊

How to Solve the Traveling Salesman Problem in Kotlin

The traveling salesman problem (TSP) is a classic optimization problem that asks: given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?1

This problem is NP-hard, which means that there is no known polynomial-time algorithm to find the optimal solution. However, there are many heuristics and exact algorithms that can solve the problem for small or medium-sized instances, or approximate the solution for large instances.

In this blog post, we will explore some of these methods and how to implement them in Kotlin, a modern and concise programming language that is widely used by Android developers and interoperable with Java.23

Brute Force

The simplest way to solve the TSP is to generate all possible permutations of the cities and calculate the cost of each one. Then, we can return the permutation with the minimum cost as the optimal solution.

This method is guaranteed to find the optimal solution, but it is very inefficient. The number of permutations grows factorially with the number of cities, so for n cities, there are n! possible routes. For example, if we have 10 cities, there are 10! = 3,628,800 possible routes. If we have 20 cities, there are 20! = 2.43 x 10^18 possible routes.

To implement this method in Kotlin, we can use a vector to store all the cities except the starting city. Then, we can use the nextPermutation function from the Kotlin standard library to generate all possible permutations of this vector. For each permutation, we can calculate the cost by adding up the distances between consecutive cities and between the last city and the starting city. We can keep track of the minimum cost and the corresponding permutation using two variables. Finally, we can return the permutation with the minimum cost as the optimal solution.

Here is an example of how this method looks like in code:

// A function that calculates the cost of a route fun calculateCost(graph: Array<IntArray>, route: List<Int>, start: Int): Int { var cost = 0 var current = start for (city in route) { cost += graph[current][city] current = city } cost += graph[current][start] return cost } // A function that solves the TSP using brute force fun solveTSPBruteForce(graph: Array<IntArray>, start: Int): List<Int> { // Create a vector of all cities except the start val cities = mutableListOf<Int>() for (i in graph.indices) { if (i != start) { cities.add(i) } } // Initialize the minimum cost and the optimal route var minCost = Int.MAX_VALUE var optimalRoute = listOf<Int>() // Generate all permutations of cities do { // Calculate the cost of the current permutation val cost = calculateCost(graph, cities, start) // Update the minimum cost and the optimal route if needed if (cost < minCost) { minCost = cost optimalRoute = cities.toList() } } while (cities.nextPermutation()) // Return the optimal route return optimalRoute }

Nearest Neighbor

A more efficient way to solve the TSP is to use a heuristic that builds a route by choosing the nearest city at each step. This method is not guaranteed to find the optimal solution, but it is much faster than brute force.

The idea is to start from any city and mark it as visited. Then, find the nearest unvisited city and move to it. Repeat this process until all cities are visited. Finally, return to the starting city to complete the route.

This method has a time complexity of O(n^2), where n is the number of cities. This is because we need to loop through all cities and find the nearest unvisited city at each step.

To implement this method in Kotlin, we can use a boolean array to keep track of which cities are visited. Then, we can use a loop to iterate through all cities and find the nearest unvisited city using a nested loop. We can add each city to a list as we visit them. Finally, we can return the list as the solution.

Here is an example of how this method looks like in code:

// A function that solves the TSP using nearest neighbor heuristic fun solveTSPNearestNeighbor(graph: Array<IntArray>, start: Int): List<Int> { // Create an array of visited cities val visited = BooleanArray(graph.size) { false } // Mark the start city as visited visited[start] = true // Create a list to store the route val route = mutableListOf<Int>() // Add the start city to the route route.add(start) // Initialize the current city as start var current = start // Loop until all cities are visited while (route.size < graph.size) { // Initialize the minimum distance and the nearest city var minDistance = Int.MAX_VALUE var nearestCity = -1 // Loop through all cities for (i in graph.indices) { // Skip if the city is visited or same as current if (visited[i] || i == current) continue // Update the minimum distance and nearest city if needed if (graph[current][i] < minDistance) { minDistance = graph[current][i] nearestCity = i } } // Mark the nearest city as visited visited[nearestCity] = true // Add it to the route route.add(nearestCity) // Update current city as nearest city current = nearestCity } // Return route return route }

Dynamic Programming

A more advanced way to solve the TSP is to use a dynamic programming algorithm that exploits the overlapping subproblems and optimal substructure of the problem. This method can find the optimal solution in polynomial time, but it requires a lot of memory.

The idea is to use a two-dimensional array to store the minimum cost of visiting a subset of cities and ending at a specific city. The array is indexed by a bitmask that represents the subset of cities and an integer that represents the last city. The base case is when the subset contains only one city, which is the starting city. The cost is zero in this case. The recursive case is when the subset contains more than one city. The cost is the minimum of the cost of visiting the previous subset and ending at any other city plus the distance from that city to the current city.

This method has a time complexity of O(n^2 * 2^n), where n is the number of cities. This is because we need to fill up the array for all possible subsets of cities and all possible last cities. The array has a size of 2^n * n.

To implement this method in Kotlin, we can use a nested function that takes a bitmask and a last city as parameters and returns the minimum cost. The function can use memoization to avoid repeated calculations. Then, we can call the function with the bitmask that represents all cities and the starting city as parameters. We can also use another array to store the optimal route by keeping track of which city was chosen at each step.

Here is an example of how this method looks like in code:

// A function that solves the TSP using dynamic programming fun solveTSPDynamicProgramming(graph: Array<IntArray>, start: Int): List<Int> { // Create an array for memoization val memo = Array(1 shl graph.size) { IntArray(graph.size) { -1 } } // Create an array for storing the route val route = Array(1 shl graph.size) { IntArray(graph.size) { -1 } } // A nested function that takes a bitmask and a last city and returns the minimum cost fun dp(bitmask: Int, last: Int): Int { // Base case: if only one city in the subset, return zero if (bitmask == (1 shl last)) return 0 // If already calculated, return memoized value if (memo[bitmask][last] != -1) return memo[bitmask][last] // Initialize the minimum cost and the previous city var minCost = Int.MAX_VALUE var prevCity = -1 // Loop through all cities for (i in graph.indices) { // Skip if the city is not in the subset or same as last if ((bitmask and (1 shl i)) == 0 || i == last) continue // Calculate the cost of visiting the previous subset and ending at i val cost = dp(bitmask xor (1 shl last), i) + graph[i][last] // Update the minimum cost and previous city if needed if (cost < minCost) { minCost = cost prevCity = i } } // Memoize and store the minimum cost and previous city memo[bitmask][last] = minCost route[bitmask][last] = prevCity // Return minimum cost return minCost } // Call dp with all cities in the subset and start as last city val minCost = dp((1 shl graph.size) - 1, start) // Create a list to store the optimal route val optimalRoute = mutableListOf<Int>() // Add start to the route optimalRoute.add(start) // Initialize bitmask and last city var bitmask = (1 shl graph.size) - 1 var last = start // Loop until all cities are added to route while (route[bitmask][last] != -1) { // Find previous city from route array val prev = route[bitmask][last] // Add it to route optimalRoute.add(prev) // Update bitmask and last city bitmask = bitmask xor (1 shl last) last = prev } // Return optimal route return optimalRoute }

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