shortest path used to cover and unit area is called
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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} {\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})\in V\times V\times \cdots \times V} such that {\displaystyle v_{i}} v_{i} is adjacent to {\displaystyle v_{i+1}} v_{i+1} for {\displaystyle 1\leq i<n} 1\leq i<n. Such a path {\displaystyle P} P is called a path of length {\displaystyle n-1} n-1 from {\displaystyle v_{1}} v_{1} to {\displaystyle v_{n}} v_{n}. (The {\displaystyle v_{i}} 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}} e_{i,j} be the edge incident to both {\displaystyle v_{i}} v_{i} and {\displaystyle v_{j}} v_{j}. Given a real-valued weight function {\displaystyle f:E\rightarrow \mathbb {R} } f:E\rightarrow \mathbb {R} , and an undirected (simple) graph {\displaystyle G} G, the shortest path from {\displaystyle v} v to {\displaystyle v'} v' is the path {\displaystyle P=(v_{1},v_{2},\ldots ,v_{n})} P=(v_{1},v_{2},\ldots ,v_{n}) (where {\displaystyle v_{1}=v} v_{1}=v and {\displaystyle v_{n}=v'} v_{n}=v') that over all possible {\displaystyle n} n minimizes the sum {\displaystyle \sum _{i=1}^{n-1}f(e_{i,i+1}).} \sum _{i=1}^{n-1}f(e_{i,i+1}). When each edge in the graph has unit weight or {\displaystyle f:E\rightarrow \{1\}} f:E\rightarrow \{1\}, 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.
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