程式開發中常用的10種演算法，你用過幾種？

當編寫程式時，瞭解和使用不同的演算法對解決問題至關重要。以下是C#中常用的10種演算法，每個演算法都伴隨著示例代碼和詳細說明。

1. 冒泡排序 (Bubble Sort):

冒泡排序是一種簡單的比較排序演算法，它多次遍曆數組，將較大的元素逐漸浮動到數組的末尾。

public static void BubbleSort(int[] arr)
{
    int n = arr.Length;
    for (int i = 0; i < n - 1; i++)
    {
        for (int j = 0; j < n - i - 1; j++)
        {
            if (arr[j] > arr[j + 1])
            {
                int temp = arr[j];
                arr[j] = arr[j + 1];
                arr[j + 1] = temp;
            }
        }
    }
}

2. 快速排序 (Quick Sort):

快速排序是一種高效的分治排序演算法，它通過選擇一個基準元素並將數組分為較小和較大的兩部分來進行排序。

public static void QuickSort(int[] arr, int low, int high)
{
    if (low < high)
    {
        int partitionIndex = Partition(arr, low, high);
        QuickSort(arr, low, partitionIndex - 1);
        QuickSort(arr, partitionIndex + 1, high);
    }
}

public static int Partition(int[] arr, int low, int high)
{
    int pivot = arr[high];
    int i = low - 1;

    for (int j = low; j < high; j++)
    {
        if (arr[j] < pivot)
        {
            i++;
            int temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
        }
    }

    int swap = arr[i + 1];
    arr[i + 1] = arr[high];
    arr[high] = swap;

    return i + 1;
}

3. 合併排序 (Merge Sort):

合併排序是一種穩定的分治排序演算法，它將數組分成兩半，分別排序後再合併。

public static void MergeSort(int[] arr)
{
    int n = arr.Length;
    if (n > 1)
    {
        int mid = n / 2;
        int[] left = new int[mid];
        int[] right = new int[n - mid];

        for (int i = 0; i < mid; i++)
            left[i] = arr[i];
        for (int i = mid; i < n; i++)
            right[i - mid] = arr[i];

        MergeSort(left);
        MergeSort(right);

        int i = 0, j = 0, k = 0;
        while (i < mid && j < (n - mid))
        {
            if (left[i] < right[j])
                arr[k++] = left[i++];
            else
                arr[k++] = right[j++];
        }
        while (i < mid)
            arr[k++] = left[i++];
        while (j < (n - mid))
            arr[k++] = right[j++];
    }
}

4. 二分查找 (Binary Search):

二分查找是一種高效的查找演算法，它要求在有序數組中查找特定元素。

public static int BinarySearch(int[] arr, int target)
{
    int low = 0, high = arr.Length - 1;
    while (low <= high)
    {
        int mid = (low + high) / 2;
        if (arr[mid] == target)
            return mid;
        else if (arr[mid] < target)
            low = mid + 1;
        else
            high = mid - 1;
    }
    return -1;
}

5. 深度優先搜索 (Depth-First Search, DFS):

DFS 是一種圖遍歷演算法，它從起始節點開始，沿著路徑儘可能深入，然後返回並繼續搜索。

using System;
using System.Collections.Generic;

public class Graph
{
    private int V;
    private List<int>[] adj;

    public Graph(int v)
    {
        V = v;
        adj = new List<int>[v];
        for (int i = 0; i < v; i++)
            adj[i] = new List<int>();
    }

    public void AddEdge(int v, int w)
    {
        adj[v].Add(w);
    }

    public void DFS(int v)
    {
        bool[] visited = new bool[V];
        DFSUtil(v, visited);
    }

    private void DFSUtil(int v, bool[] visited)
    {
        visited[v] = true;
        Console.Write(v + " ");

        foreach (var n in adj[v])
        {
            if (!visited[n])
                DFSUtil(n, visited);
        }
    }
}

6. 廣度優先搜索 (Breadth-First Search, BFS):

BFS 是一種圖遍歷演算法，它從起始節點開始，逐層遍歷，先訪問所有相鄰的節點，然後再逐層擴展。

using System;
using System.Collections.Generic;

public class Graph
{
    private int V;
    private List<int>[] adj;

    public Graph(int v)
    {
        V = v;
        adj = new List<int>[v];
        for (int i = 0; i < v; i++)
            adj[i] = new List<int>();
    }

    public void AddEdge(int v, int w)
    {
        adj[v].Add(w);
    }

    public void BFS(int s)
    {
        bool[] visited = new bool[V];

        Queue<int> queue = new Queue<int>();
        visited[s] = true;
        queue.Enqueue(s);

        while (queue.Count != 0)
        {
            s = queue.Dequeue();
            Console.Write(s + " ");

            foreach (var n in adj[s])
            {
                if (!visited[n])
                {
                    visited[n] = true;
                    queue.Enqueue(n);
                }
            }
        }
    }
}

7. Dijkstra演算法:

Dijkstra演算法是一種用於查找圖中最短路徑的演算法。

public class Dijkstra
{
    private static int V = 9;

    private int MinDistance(int[] dist, bool[] sptSet)
    {
        int min = int.MaxValue;
        int minIndex = 0;

        for (int v = 0; v < V; v++)
        {
            if (!sptSet[v] && dist

[v] <= min)
            {
                min = dist[v];
                minIndex = v;
            }
        }

        return minIndex;
    }

    private void PrintSolution(int[] dist)
    {
        Console.WriteLine("Vertex \t Distance from Source");
        for (int i = 0; i < V; i++)
        {
            Console.WriteLine(i + " \t " + dist[i]);
        }
    }

    public void FindShortestPath(int[,] graph, int src)
    {
        int[] dist = new int[V];
        bool[] sptSet = new bool[V];

        for (int i = 0; i < V; i++)
        {
            dist[i] = int.MaxValue;
            sptSet[i] = false;
        }

        dist[src] = 0;

        for (int count = 0; count < V - 1; count++)
        {
            int u = MinDistance(dist, sptSet);

            sptSet[u] = true;

            for (int v = 0; v < V; v++)
            {
                if (!sptSet[v] && graph[u, v] != 0 && dist[u] != int.MaxValue && dist[u] + graph[u, v] < dist[v])
                {
                    dist[v] = dist[u] + graph[u, v];
                }
            }
        }

        PrintSolution(dist);
    }
}

8. 最小生成樹 (Minimum Spanning Tree, MST) - Prim演算法:

Prim演算法用於找到圖的最小生成樹，它從一個初始頂點開始，逐漸擴展生成樹。

public class PrimMST
{
    private static int V = 5;

    private int MinKey(int[] key, bool[] mstSet)
    {
        int min = int.MaxValue;
        int minIndex = 0;

        for (int v = 0; v < V; v++)
        {
            if (!mstSet[v] && key[v] < min)
            {
                min = key[v];
                minIndex = v;
            }
        }

        return minIndex;
    }

    private void PrintMST(int[] parent, int[,] graph)
    {
        Console.WriteLine("Edge \t Weight");
        for (int i = 1; i < V; i++)
        {
            Console.WriteLine(parent[i] + " - " + i + " \t " + graph[i, parent[i]]);
        }
    }

    public void FindMST(int[,] graph)
    {
        int[] parent = new int[V];
        int[] key = new int[V];
        bool[] mstSet = new bool[V];

        for (int i = 0; i < V; i++)
        {
            key[i] = int.MaxValue;
            mstSet[i] = false;
        }

        key[0] = 0;
        parent[0] = -1;

        for (int count = 0; count < V - 1; count++)
        {
            int u = MinKey(key, mstSet);

            mstSet[u] = true;

            for (int v = 0; v < V; v++)
            {
                if (graph[u, v] != 0 && !mstSet[v] && graph[u, v] < key[v])
                {
                    parent[v] = u;
                    key[v] = graph[u, v];
                }
            }
        }

        PrintMST(parent, graph);
    }
}

9. 最小生成樹 (Minimum Spanning Tree, MST) - Kruskal演算法:

Kruskal演算法也用於找到圖的最小生成樹，它基於邊的權重排序。

using System;
using System.Collections.Generic;

public class Graph
{
    private int V, E;
    private List<Edge> edges;

    public Graph(int v, int e)
    {
        V = v;
        E = e;
        edges = new List<Edge>(e);
    }

    public void AddEdge(int src, int dest, int weight)
    {
        edges.Add(new Edge(src, dest, weight));
    }

    public void KruskalMST()
    {
        edges.Sort();

        int[] parent = new int[V];
        int[] rank = new int[V];

        for (int i = 0; i < V; i++)
        {
            parent[i] = i;
            rank[i] = 0;
        }

        int i = 0;
        int e = 0;

        List<Edge> mst = new List<Edge>();

        while (e < V - 1)
        {
            Edge nextEdge = edges[i++];
            int x = Find(parent, nextEdge.src);
            int y = Find(parent, nextEdge.dest);

            if (x != y)
            {
                mst.Add(nextEdge);
                Union(parent, rank, x, y);
                e++;
            }
        }

        Console.WriteLine("Edges in Minimum Spanning Tree:");
        foreach (var edge in mst)
        {
            Console.WriteLine($"{edge.src} - {edge.dest} with weight {edge.weight}");
        }
    }

    private int Find(int[] parent, int i)
    {
        if (parent[i] == i)
            return i;
        return Find(parent, parent[i]);
    }

    private void Union(int[] parent, int[] rank, int x, int y)
    {
        int xRoot = Find(parent, x);
        int yRoot = Find(parent, y);

        if (rank[xRoot] < rank[yRoot])
            parent[xRoot] = yRoot;
        else if (rank[xRoot] > rank[yRoot])
            parent[yRoot] = xRoot;
        else
        {
            parent[yRoot] = xRoot;
            rank[xRoot]++;
        }
    }
}

public class Edge : IComparable<Edge>
{
    public int src, dest, weight;

    public Edge(int src, int dest, int weight)
    {
        this.src = src;
        this.dest = dest;
        this.weight = weight;
    }

    public int CompareTo(Edge other)
    {
        return weight - other.weight;
    }
}

10.Floyd-Warshall演算法是一種用於解決所有點對最短路徑的動態規划算法。

下麵是C#中的Floyd-Warshall演算法的實現示例：

using System;

class FloydWarshall
{
    private static int INF = int.MaxValue; // 代表無窮大的值

    public static void FindShortestPath(int[,] graph)
    {
        int V = graph.GetLength(0);

        // 創建一個二維數組dist，用於保存最短路徑的長度
        int[,] dist = new int[V, V];

        // 初始化dist數組
        for (int i = 0; i < V; i++)
        {
            for (int j = 0; j < V; j++)
            {
                dist[i, j] = graph[i, j];
            }
        }

        // 逐個頂點考慮，如果經過k頂點路徑比原路徑短，就更新dist數組
        for (int k = 0; k < V; k++)
        {
            for (int i = 0; i < V; i++)
            {
                for (int j = 0; j < V; j++)
                {
                    if (dist[i, k] != INF && dist[k, j] != INF
                        && dist[i, k] + dist[k, j] < dist[i, j])
                    {
                        dist[i, j] = dist[i, k] + dist[k, j];
                    }
                }
            }
        }

        // 輸出最短路徑矩陣
        Console.WriteLine("最短路徑矩陣：");
        for (int i = 0; i < V; i++)
        {
            for (int j = 0; j < V; j++)
            {
                if (dist[i, j] == INF)
                    Console.Write("INF\t");
                else
                    Console.Write(dist[i, j] + "\t");
            }
            Console.WriteLine();
        }
    }

    static void Main(string[] args)
    {
        int V = 4; // 頂點數
        int[,] graph = {
            {0, 5, INF, 10},
            {INF, 0, 3, INF},
            {INF, INF, 0, 1},
            {INF, INF, INF, 0}
        };

        FindShortestPath(graph);
    }
}

在這個示例中，我們使用Floyd-Warshall演算法來計算給定圖的最短路徑矩陣。該演算法通過考慮逐個中間頂點k，不斷更新最短路徑矩陣dist。最終，我們可以獲得所有點對之間的最短路徑長度。