A Shortest-Path Search Task is a optimal-path search task that requires shortest paths (in a graph).

## References

### 2013

• http://en.wikipedia.org/wiki/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.

This is analogous to 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.

• http://en.wikipedia.org/wiki/Shortest_path_problem#Definition
• 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 $\displaystyle{ P = (v_1, v_2, \ldots, v_n ) \in V \times V \times \ldots \times V }$ such that $\displaystyle{ v_i }$ is adjacent to $\displaystyle{ v_{i+1} }$ for $\displaystyle{ 1 \leq i \lt n }$.

Such a path $\displaystyle{ P }$ is called a path of length $\displaystyle{ n }$ 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.)

Let $\displaystyle{ e_{i, j} }$ be the edge incident to both $\displaystyle{ v_i }$ and $\displaystyle{ v_j }$.

Given a real-valued weight function $\displaystyle{ f: E \rightarrow \mathbb{R} }$, and an undirected (simple) graph $\displaystyle{ G }$, the shortest path from $\displaystyle{ v }$ to $\displaystyle{ v' }$ is the path $\displaystyle{ P = (v_1, v_2, \ldots, v_n ) }$ (where $\displaystyle{ v_1 = v }$ and $\displaystyle{ v_n = v' }$) that over all possible $\displaystyle{ n }$ minimizes the sum $\displaystyle{ \sum_{i =1}^{n-1} f(e_{i, i+1}). }$

When the graph is unweighted or $\displaystyle{ f: E \rightarrow \{c\},\ c \in \mathbb{R}^+ }$, this is equivalent to finding the path with fewest edges.

The problem is also sometimes called the single-pair shortest path problem, to distinguish it from the following variations:

• The single-source shortest path problem, in which we have to find shortest paths from a source vertex v to all other vertices in the graph.
• The single-destination shortest path problem, in which we have to find shortest paths from all vertices in the directed graph to a single destination vertex v. This can be reduced to the single-source shortest path problem by reversing the arcs in the directed graph.
• The all-pairs shortest path problem, in which we have to find shortest paths between every pair of vertices v, v' in the graph.
• These generalizations have significantly more efficient algorithms than the simplistic approach of running a single-pair shortest path algorithm on all relevant pairs of vertices.