Breadth-first search
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In graph theory, breadth-first search (BFS) is a graph search algorithm that begins at the root node and explores all the neighboring nodes. Then for each of those nearest nodes, it explores their unexplored neighbor nodes, and so on, until it finds the goal.
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[edit] How it works
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BFS is an uninformed search method that aims to expand and examine all nodes of a graph systematically in search of a solution. In other words, it exhaustively searches the entire graph without considering the goal until it finds it. It does not use a heuristic.
From the standpoint of the algorithm, all child nodes obtained by expanding a node are added to a FIFO queue. In typical implementations, nodes that have not yet been examined for their neighbors are placed in some container (such as a queue or linked list) called "open" and then once examined are placed in the container "closed".
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[edit] Algorithm (informal)
- Put the ending node (the root node) in the queue.
- Pull a node from the beginning of the queue and examine it.
- If the searched element is found in this node, quit the search and return a result.
- Otherwise push all the (so-far-unexamined) successors of this node into the end of the queue, if there are any.
- If the queue is empty, every node on the graph has been examined -- quit the search and return "not found".
- repeat from step 2.
[edit] C++ implementation
This is the implementation of the above informal algorithm, where the "so-far-unexamined" is handled by the parent array.
- Suppose we have a struct:
struct Vertex {
...
vector<int> out;
...
};
- and an array of vertices: (the algorithm will use the indexes of this array, to handle the vertices)
Vertex* graph = new Vertex[1 + vertices]; // 1 based indexing
- the algorithm starts from start and returns true if there is a directed path from start to end:
bool BFS(Vertex* graph, int num_of_vertices, int start, int end) {
queue<int> next;
int parent[1 + num_of_vertices]; // 1 based indexing, don't care about 0th
memset(parent + 1, 0, sizeof(int) * num_of_vertices); // fill with zeros
parent[start] = -1;
next.push(start);
while(!next.empty()) {
int u = next.front(); next.pop();
// Here is the point where you can examine the u th vertex of graph
// For example:
if(u == end) return true;
for(size_t j = 0; j < graph[u].out.size(); ++j) {
int v = graph[u].out[j];
if(parent[v] == 0) {
parent[v] = u;
next.push(v);
}
}
}
return false;
}
- it also stores the parents of each node, from which you can get the path.
[edit] Features
[edit] Space Complexity
Since all nodes discovered so far have to be saved, the space complexity of breadth-first search is O(|V| + |E|) where |V| is the number of nodes and |E| the number of edges in the graph. Note: another way of saying this is that it is O(B ^ M) where B is the maximum branching factor and M is the maximum path length of the tree. This immense demand for space is the reason why breadth-first search is impractical for larger problems.
[edit] Time Complexity
Since in the worst case breadth-first search has to consider all paths to all possible nodes the time complexity of breadth-first search is O(|V| + |E|) where |V| is the number of nodes and |E| the number of edges in the graph. The best case of this search is o(1).It occurs when the node is found at first time.
[edit] Completeness
Breadth-first search is complete. This means that if there is a solution breadth-first search will find it regardless of the kind of graph. However, if the graph is infinite and there is no solution breadth-first search will diverge.
[edit] Optimality
In general breadth-first search is not optimal since it always returns the result with the fewest edges between the start node and the goal node. If the graph is a weighted graph, and therefore has costs associated with each step, a goal next to the start does not have to be the cheapest goal available. This problem is solved by improving breadth-first search to uniform-cost search which considers the path costs. Nevertheless, if the graph is not weighted, and therefore all step costs are equal, breadth-first search will find the nearest and the best solution.
[edit] Applications of BFS
Breadth-first search can be used to solve many problems in graph theory, for example:
- Finding all connected components in a graph.
- Finding all nodes within one connected component
- Finding the shortest path between two nodes u and v (in an unweighted graph)
- Testing a graph for bipartiteness
[edit] Finding connected Components
The set of nodes reached by a BFS are the largest connected component containing the start node.
[edit] Testing bipartiteness
BFS can be used to test bipartiteness, by starting the search at any vertex and giving alternating labels to the vertices visited during the search. That is, give label 0 to the starting vertex, 1 to all its neighbours, 0 to those neighbours' neighbours, and so on. If at any step a vertex has (visited) neighbours with the same label as itself, then the graph is not bipartite. If the search ends without such a situation occurring, then the graph is bipartite.
[edit] See also
[edit] References
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 22.2: Breadth-first search, pp.531–539.
