Shortest path problem

In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.

The problem of finding the shortest path between two intersections on a road map (the graph's vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of its road segment) may be modeled by a special case of the shortest path problem in graphs.

The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge.

Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices ${\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})\in V\times V\times \cdots \times V}$ such that ${\displaystyle v_{i}}$ is adjacent to ${\displaystyle v_{i+1}}$ for ${\displaystyle 1\leq i<n}$. Such a path ${\displaystyle P}$ is called a path of length ${\displaystyle n-1}$ from ${\displaystyle v_{1}}$ to ${\displaystyle v_{n}}$. (The ${\displaystyle v_{i}}$ are variables; their numbering here relates to their position in the sequence and needs not to relate to any canonical labeling of the vertices.)

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