0.7 Graphs  (Page 13/21)

• 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.

• Optimality

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 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.

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, Cheney's algorithm
• Finding the shortest path between two nodes u and v (in an unweighted graph)
• Testing a graph for bipartiteness
• (Reverse) Cuthill–McKee mesh numbering

Finding connected components

The set of nodes reached by a BFS are the largest connected component containing the start node.

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.

Usage in 2d grids for computer games

BFS has been applied to pathfinding problems in computer games, such as Real-Time Strategy games, where the graph is represented by a tilemap, and each tile in the map represents a node. Each of that node is then connected to each of its neighbour (neighbour in north, north-east, east, south-east, south, south-west, west, and north-west).

It is worth mentioning that when BFS is used in that manner, the neighbour list should be created such that north, east, south and west get priority over north-east, south-east, south-west and north-west. The reason for this is that BFS tends to start searching in a diagonal manner rather than adjacent, and the path found will not be the correct one. BFS should first search adjacent nodes, then diagonal nodes.

7.3.4. bellman-ford algorithms

(From Wikipedia, the free encyclopedia)

The Bellman–Ford algorithm computes single-source shortest paths in a weighted digraph (where some of the edge weights may be negative). Dijkstra's algorithm accomplishes the same problem with a lower running time, but requires edge weights to be non-negative. Thus, Bellman–Ford is usually used only when there are negative edge weights.