廣度優先搜尋 (BFS) 演算法

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廣度優先搜尋 (BFS) 演算法

廣度優先搜尋 (BFS) 演算法以廣度優先的方式遍歷圖資料結構，以搜尋滿足一組條件的節點。它使用佇列來記住下次搜尋的起始頂點，當任何迭代中出現死鎖時。

廣度優先搜尋 (BFS) 演算法從樹根開始，在移動到下一深度級別的節點之前，探索當前深度級別上的所有節點。

Breadth First Traversal

如上例所示，BFS演算法首先從A到B到E到F遍歷，然後到C和G，最後到D。它採用以下規則。

  • 規則1 - 訪問相鄰的未訪問頂點。將其標記為已訪問。顯示它。將其插入佇列。

  • 規則2 - 如果找不到相鄰頂點，則從佇列中移除第一個頂點。

  • 規則3 - 重複規則1和規則2，直到佇列為空。

步驟 遍歷 描述
1 Breadth First Search Step One 初始化佇列。
2 Breadth First Search Step Two 我們從訪問S（起始節點）開始，並將其標記為已訪問。
3 Breadth First Search Step Three 然後我們看到S的未訪問相鄰節點。在這個例子中，我們有三個節點，但按字母順序我們選擇A，將其標記為已訪問並將其入隊。
4 Breadth First Search Step Four 接下來，S的未訪問相鄰節點是B。我們將其標記為已訪問並將其入隊。
5 Breadth First Search Step Five 接下來，S的未訪問相鄰節點是C。我們將其標記為已訪問並將其入隊。
6 Breadth First Search Step Six 現在，S沒有未訪問的相鄰節點了。因此，我們出隊並找到A。
7 Breadth First Search Step Seven 從A我們有D作為未訪問的相鄰節點。我們將其標記為已訪問並將其入隊。

在這個階段，我們沒有未標記（未訪問）的節點了。但是根據演算法，我們繼續出隊以獲得所有未訪問的節點。當佇列為空時，程式結束。

示例

以下是各種程式語言中廣度優先搜尋 (BFS) 演算法的實現：

C C++ Java Python
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#define MAX 5
struct Vertex {
   char label;
   bool visited;
};
//queue variables
int queue[MAX];
int rear = -1;
int front = 0;
int queueItemCount = 0;
//graph variables
//array of vertices
struct Vertex* lstVertices[MAX];
//adjacency matrix
int adjMatrix[MAX][MAX];
//vertex count
int vertexCount = 0;
//queue functions
void insert(int data) {
   queue[++rear] = data;
   queueItemCount++;
}
int removeData() {
   queueItemCount--;
   return queue[front++]; 
}
bool isQueueEmpty() {
   return queueItemCount == 0;
}
//graph functions
//add vertex to the vertex list
void addVertex(char label) {
   struct Vertex* vertex = (struct Vertex*) malloc(sizeof(struct Vertex));
   vertex->label = label;  
   vertex->visited = false;     
   lstVertices[vertexCount++] = vertex;
}
//add edge to edge array
void addEdge(int start,int end) {
   adjMatrix[start][end] = 1;
   adjMatrix[end][start] = 1;
}
//display the vertex
void displayVertex(int vertexIndex) {
   printf("%c ",lstVertices[vertexIndex]->label);
}       
//get the adjacent unvisited vertex
int getAdjUnvisitedVertex(int vertexIndex) {
   int i;
	
   for(i = 0; i<vertexCount; i++) {
      if(adjMatrix[vertexIndex][i] == 1 && lstVertices[i]->visited == false)
         return i;
   }
   return -1;
}
void breadthFirstSearch() {
   int i;
   //mark first node as visited
   lstVertices[0]->visited = true;
   //display the vertex
   displayVertex(0);   
   //insert vertex index in queue
   insert(0);
   int unvisitedVertex;
   while(!isQueueEmpty()) {
      //get the unvisited vertex of vertex which is at front of the queue
      int tempVertex = removeData();   
      //no adjacent vertex found
      while((unvisitedVertex = getAdjUnvisitedVertex(tempVertex)) != -1) {    
         lstVertices[unvisitedVertex]->visited = true;
         displayVertex(unvisitedVertex);
         insert(unvisitedVertex);               
      }	
   }   
   //queue is empty, search is complete, reset the visited flag        
   for(i = 0;i<vertexCount;i++) {
      lstVertices[i]->visited = false;
   }    
}
int main() {
   int i, j;

   for(i = 0; i<MAX; i++) { // set adjacency 
      for(j = 0; j<MAX; j++) // matrix to 0
         adjMatrix[i][j] = 0;
   }
   addVertex('S');   // 0
   addVertex('A');   // 1
   addVertex('B');   // 2
   addVertex('C');   // 3
   addVertex('D');   // 4
   addEdge(0, 1);    // S - A
   addEdge(0, 2);    // S - B
   addEdge(0, 3);    // S - C
   addEdge(1, 4);    // A - D
   addEdge(2, 4);    // B - D
   addEdge(3, 4);    // C - D
   printf("\nBreadth First Search: ");
   breadthFirstSearch();
   return 0;
}

輸出

Breadth First Search: S A B C D

//C++ code for Breadth First Traversal
#include <iostream>
#include <stdlib.h>
#include <stdbool.h>
#define MAX 5
struct Vertex {
   char label;
   bool visited;
};
//queue variables
int queue[MAX];
int rear = -1;
int front = 0;
int queueItemCount = 0;
//graph variables
//array of vertices
struct Vertex* lstVertices[MAX];
//adjacency matrix
int adjMatrix[MAX][MAX];
//vertex count
int vertexCount = 0;
//queue functions
void insert(int data) {
   queue[++rear] = data;
   queueItemCount++;
}
int removeData() {
   queueItemCount--;
   return queue[front++]; 
}
bool isQueueEmpty() {
   return queueItemCount == 0;
}
//graph functions
//add vertex to the vertex list
void addVertex(char label) {
   struct Vertex* vertex = (struct Vertex*) malloc(sizeof(struct Vertex));
   vertex->label = label;  
   vertex->visited = false;     
   lstVertices[vertexCount++] = vertex;
}
//add edge to edge array
void addEdge(int start,int end) {
   adjMatrix[start][end] = 1;
   adjMatrix[end][start] = 1;
}
//display the vertex
void displayVertex(int vertexIndex) {
   std::cout << lstVertices[vertexIndex]->label << " ";
}       
//get the adjacent unvisited vertex
int getAdjUnvisitedVertex(int vertexIndex) {
   int i;
   for(i = 0; i<vertexCount; i++) {
      if(adjMatrix[vertexIndex][i] == 1 && lstVertices[i]->visited == false)
         return i;
   }
   return -1;
}
void breadthFirstSearch() {
   int i;
   //mark first node as visited
   lstVertices[0]->visited = true;
  //display the vertex
   displayVertex(0);   
   //insert vertex index in queue
   insert(0);
   int unvisitedVertex;
   while(!isQueueEmpty()) {
      //get the unvisited vertex of vertex which is at front of the queue
      int tempVertex = removeData();   
      //no adjacent vertex found
      while((unvisitedVertex = getAdjUnvisitedVertex(tempVertex)) != -1) {    
         lstVertices[unvisitedVertex]->visited = true;
         displayVertex(unvisitedVertex);
         insert(unvisitedVertex);               
      }
		
   }   
   //queue is empty, search is complete, reset the visited flag        
   for(i = 0;i<vertexCount;i++) {
      lstVertices[i]->visited = false;
   }    
}
int main() {
   int i, j;
   for(i = 0; i<MAX; i++) { // set adjacency 
      for(j = 0; j<MAX; j++) // matrix to 0
         adjMatrix[i][j] = 0;
   }
   addVertex('S');   // 0
   addVertex('A');   // 1
   addVertex('B');   // 2
   addVertex('C');   // 3
   addVertex('D');   // 4
   addEdge(0, 1);    // S - A
   addEdge(0, 2);    // S - B
   addEdge(0, 3);    // S - C
   addEdge(1, 4);    // A - D
   addEdge(2, 4);    // B - D
   addEdge(3, 4);    // C - D
   std::cout << "Breadth First Search: ";
   breadthFirstSearch();
   return 0;
}

輸出

Breadth First Search: S A B C D

//Java code for Breadth First Traversal
import java.util.LinkedList;
import java.util.Queue;
class Vertex {
    char label;
    boolean visited;
    public Vertex(char label) {
        this.label = label;
        visited = false;
    }
}
public class Graph {
    private static final int MAX = 5;
    private Vertex[] lstVertices;
    private int[][] adjMatrix;
    private int vertexCount;
    public Graph() {
        lstVertices = new Vertex[MAX];
        adjMatrix = new int[MAX][MAX];
        vertexCount = 0;
    }
    private void addVertex(char label) {
        Vertex vertex = new Vertex(label);
        lstVertices[vertexCount++] = vertex;
    }
    private void addEdge(int start, int end) {
        adjMatrix[start][end] = 1;
        adjMatrix[end][start] = 1;
    }
    private void displayVertex(int vertexIndex) {
        System.out.print(lstVertices[vertexIndex].label + " ");
    }
    private int getAdjUnvisitedVertex(int vertexIndex) {
        for (int i = 0; i < vertexCount; i++) {
            if (adjMatrix[vertexIndex][i] == 1 && !lstVertices[i].visited)
                return i;
        }
        return -1;
    }
    private void breadthFirstSearch() {
        lstVertices[0].visited = true;
        displayVertex(0);
        Queue<Integer> queue = new LinkedList<>();
        queue.add(0);
        while (!queue.isEmpty()) {
            int tempVertex = queue.poll();
            int unvisitedVertex;
            while ((unvisitedVertex = getAdjUnvisitedVertex(tempVertex)) != -1) {
                lstVertices[unvisitedVertex].visited = true;
                displayVertex(unvisitedVertex);
                queue.add(unvisitedVertex);
            }
        }
        // Reset the visited flag
        for (int i = 0; i < vertexCount; i++) {
            lstVertices[i].visited = false;
        }
    }
    public static void main(String[] args) {
        Graph graph = new Graph();
        for (int i = 0; i < MAX; i++) {
            for (int j = 0; j < MAX; j++)
                graph.adjMatrix[i][j] = 0;
        }
        graph.addVertex('S');   // 0
        graph.addVertex('A');   // 1
        graph.addVertex('B');   // 2
        graph.addVertex('C');   // 3
        graph.addVertex('D');   // 4
        graph.addEdge(0, 1);    // S - A
        graph.addEdge(0, 2);    // S - B
        graph.addEdge(0, 3);    // S - C
        graph.addEdge(1, 4);    // A - D
        graph.addEdge(2, 4);    // B - D
        graph.addEdge(3, 4);    // C - D
        System.out.print("Breadth First Search: ");
        graph.breadthFirstSearch();
    }
}

輸出

Breadth First Search: S A B C D

#Python program for Breadth First Search
# defining MAX 5
MAX = 5
class Vertex:
   def __init__(self, label):
      self.label = label
      self.visited = False
# queue variables
queue = [0] * MAX
rear = -1
front = 0
queueItemCount = 0
# graph variables
#array of vertices
lstVertices = [None] * MAX
#adjacency matrix
adjMatrix = [[0] * MAX for _ in range(MAX)]
#vertex count
vertexCount = 0
# queue functions
def insert(data):
   global rear, queueItemCount
   rear += 1
   queue[rear] = data
   queueItemCount += 1
def removeData():
   global front, queueItemCount
   queueItemCount -= 1
   data = queue[front]
   front += 1
   return data
def isQueueEmpty():
   return queueItemCount == 0
# graph functions
#add vertex to the vertex list
def addVertex(label):
   global vertexCount
   vertex = Vertex(label)
   lstVertices[vertexCount] = vertex
   vertexCount += 1
#add edge to edge array
def addEdge(start, end):
   adjMatrix[start][end] = 1
   adjMatrix[end][start] = 1
#Display the vertex
def displayVertex(vertexIndex):
   print(lstVertices[vertexIndex].label, end=" ")
#Get the adjacent unvisited vertex
def getAdjUnvisitedVertex(vertexIndex):
   for i in range(vertexCount):
      if adjMatrix[vertexIndex][i] == 1 and not lstVertices[i].visited:
         return i
   return -1
def breadthFirstSearch():
    #mark first node as visited
   lstVertices[0].visited = True
   #Display the vertex
   displayVertex(0)
   #insert vertex index in queue
   insert(0)
   while not isQueueEmpty():
    #get the unvisited vertex of vertex which is at front of the queue
      tempVertex = removeData()     
      #no adjacent vertex found
      unvisitedVertex = getAdjUnvisitedVertex(tempVertex)
      while unvisitedVertex != -1:
         lstVertices[unvisitedVertex].visited = True
         displayVertex(unvisitedVertex)
         insert(unvisitedVertex)
         unvisitedVertex = getAdjUnvisitedVertex(tempVertex)     
    #queue is empty, search is complete, reset the visited flag 
   for i in range(vertexCount):
      lstVertices[i].visited = False
# main function
if __name__ == "__main__":
    #set adjacency
   for i in range(MAX):
       #matrix to 0
       for j in range(MAX):
         adjMatrix[i][j] = 0
   addVertex('S')
   addVertex('A')
   addVertex('B')
   addVertex('C')
   addVertex('D')
   addEdge(0, 1)
   addEdge(0, 2)
   addEdge(0, 3)
   addEdge(1, 4)
   addEdge(2, 4)
   addEdge(3, 4)
   print("Breadth First Search: ", end="")
   breadthFirstSearch()

輸出

Breadth First Search: S A B C D

點選檢視廣度優先搜尋 (BFS) 演算法的C語言實現

BFS演算法的複雜度

時間複雜度

BFS演算法的時間複雜度表示為O(V + E)，其中V是節點數，E是邊數。

空間複雜度

BFS演算法的空間複雜度為O(V)。

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