程式開發中常用的10種演演算法，你用過幾種？

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

1. 氣泡排序 (Bubble Sort):

氣泡排序是一種簡單的比較排序演演算法，它多次遍歷陣列，將較大的元素逐漸浮動到陣列的末尾。

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演算法，它通過選擇一個基準元素並將陣列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演算法，它將陣列分成兩半，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. 二分查詢 (Binary Search):

二分查詢是一種高效的查詢演演算法，它要求在有序陣列中查詢特定元素。

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. 深度優先搜尋 (Depth-First Search, DFS):

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

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. 廣度優先搜尋 (Breadth-First Search, BFS):

BFS 是一種圖遍歷演演算法，它從起始節點開始，逐層遍歷，先存取所有相鄰的節點，然後再逐層擴充套件。

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演演算法:

Dijkstra演演算法是一種用於查詢圖中最短路徑的演演算法。

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演演算法:

Prim演演算法用於找到圖的最小生成樹，它從一個初始頂點開始，逐漸擴充套件生成樹。

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演演算法:

Kruskal演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演演算法是一種用於解決所有點對最短路徑的動態規劃演演算法。

下面是C#中的Floyd-Warshall演演算法的實現範例：

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);

        // 建立一個二維陣列dist，用於儲存最短路徑的長度
        int[,] dist = new int[V, V];

        // 初始化dist陣列
        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陣列
        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);
    }
}

在這個範例中，我們使用Floyd-Warshall演演算法來計算給定圖的最短路徑矩陣。該演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。