- Breadth-first search
Order in which the nodes are expanded
optimal=yes (for unweighted graphs)
graph theory, breadth-first search (BFS) is a graph search algorithmthat 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.
How it works
BFS is an
uninformed searchmethod 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 FIFOqueue. 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".
# Put the root node on 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 (the direct child nodes) 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.
Note: Using a stack instead of a queue to store the nodes yet to be visited would turn this algorithm into a
Assume we have a graph made up of Node objects, each containing a value and a list of child Node objects:
Then this function performs a breadth-first search on that graph when passed the graph's root node (a Node instance) and any function to be applied to each node in the graph:
For example, this script finds the sum of the integer values of each node in the graph:
Algorithm of Breadth-first search：
Main Algorithm of apply Breadth-first search to graph G=(V,E)：
This is the implementation of the above informal algorithm, where the "so-far-unexamined" is handled by the parent array. For actual C++ applications, see the
Boost Graph Library.:Suppose we have a struct::and an array of vertices: (the algorithm will use the indices of this array, to handle the vertices):the algorithm starts from start and returns true if there is a directed path from start to end::it also stores the parents of each node, from which you can get the path.
Since all of the nodes of a level must be saved until their child nodes in the next level have been generated, the space complexity is proportional to the number of nodes at the deepest level. Given a
branching factorand graph depth the asymptotic space complexity is the number of nodes at the deepest level, . When the number of vertices and edges in the graph are known ahead of time, the space complexity can also be expressed as where is the cardinalityof the set of edges (the number of edges), and is the cardinality of the set of vertices. In the worst case the graph has a depth of 1 and all vertices must be stored. Since it is exponential in the depth of the graph, breadth-first search is often impractical for large problems on systems with bounded space.
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 which asymptotically approaches . The time complexity can also be expressed as since every vertex and every edge will be explored in the worst case.
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.
For unit-step cost, breadth-first search is optimal. 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 searchwhich 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.
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
* Copying Collection,
* Finding the
shortest pathbetween two nodes "u" and "v" (in an unweighted graph)
* Finding the
shortest pathbetween two nodes "u" and "v" (in a weighted graph: see talk page)
* Testing a graph for bipartiteness
* (Reverse) Cuthill–McKee mesh numbering
Finding connected Components
The set of nodes reached by a BFS (breadth-first search) are the largest connected component containing the start node.
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.
title=The Art Of Computer Programming Vol 1. 3rd ed.
* [http://www.martienus.com/code/t-sql-breadth-first-shortest-route-search.html Breadth-First shortest-route search implemented in T-SQL by Martin Withaar]
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