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Preface
The single-source shortest path problem is an important and common application problem in graph theory. In this problem, we need to find the shortest path from a given source node to other nodes. The breadth-first search (BFS) algorithm is a commonly used and effective algorithm to solve this problem.
This blog will focus on the single-source shortest path problem and its practical applications, while briefly introducing the basic idea of the BFS algorithm.
1. Introduction to the single-source shortest path problem
The single-source shortest path problem means that in a weighted graph, given a source node (starting point), it is necessary to find the shortest path from the source node to all other nodes. Among them, the shortest path may refer to the sum of edge weights or the number of edges between two nodes.
Specific examples can be finding the shortest route from one location to another in a city map; or determining the shortest route from one bus stop to other stops in a transportation network, etc. This type of problem has many application scenarios in real life, so effective algorithms to solve this problem are very important.
2. BFS algorithm idea
The breadth-first search algorithm is similar to the level-order traversal of a tree and is implemented based on the queue data structure. The algorithm idea is as follows:
First visit the starting vertex, then starting from the starting vertex, visit each unvisited adjacent vertex of the starting vertex in sequence, and then visit the unvisited adjacent vertices of these adjacent vertices in sequence until all vertices in the graph have been visited. Until it passes.
If there are still unvisited vertices in the graph at this time, select another unvisited vertex in the graph as the starting vertex, and repeat the above process until all vertices are visited.
For more detailed information about the BFS algorithm, you can check out my other article!
Breadth First Search (BFS) algorithm idea, algorithm implementation and its application scenarios
3. BFS algorithm to solve single source shortest path (pseudocode)
int d[Maxsize]; //d数组记录从起始顶点v到各个顶点最短路径
int path[Maxsize]; //path数组记录最短路径从哪个顶点过来
int visited[Maxsize]; //访问标记数组
void BFS_MIN_Distance(Graph G,int v)
{
for(int i=0;i<G.vexnum;i++)
{
d[i]=∞; //初始化路径长度
path[i]=-1; //初始化路径
}
d[v]=0; //起始顶点路径长度为0
visited[v]=1; //标记起始顶点已访问
EnQueue(Q,v); //起始顶点入队
while(!isEmpty(Q))
{
DelQueue(Q,v); //队头元素出队
for(int w=FirstNeighbor(G,v);w>=0;w=NextNeighbor(G,v,w))
if(!visited[w])
{
d[w]=d[v]+1; //路径长度+1
path[w]=v; //最短路径从顶点v到顶点w
visited[w]=1; //标记已访问
EnQueue(Q,w); //将顶点w入队
}
}
}
Breadth-first search finds the shortest path for unweighted graphs. BFS can be used to solve the single-source shortest path problem, which is determined by the property that breadth-first search always traverses each vertex in the graph according to the distance from near to far.
Summarize
The single-source shortest path problem is an important application problem, and the BFS algorithm is an effective algorithm to solve this problem. The BFS algorithm uses the ideas of queues and layer-by-layer expanded search to find the shortest path from the source node to other nodes.
In real life, the BFS algorithm is widely used in traffic navigation systems, social network analysis, network routing, game development and other fields. Learning and understanding the BFS algorithm is of great significance for solving similar shortest path problems.