Generic Algorithms for SSSP

Let G be a directed graph, s a node of G, and cost a cost function on the edges of G. Edge costs may be positive or negative. A single source shortest path algorithm computes the shortest paths from s to all other nodes of G with respect to cost.

Remark: Finding the shortest path from one node to one other node is not significantly faster than computing the shortest paths from the source to all other nodes.

SHORTEST_PATH_T() is the generic LEDA function for computing single source shortest paths in a directed graph. SHORTEST_PATH_T() is a template function. The template parameter NT can be instantiated with any number type.

Example of How to Use SHORTEST_PATH_T()

SHORTEST_PATH() is the name of the preinstantiated versions of SHORTEST_PATH_T(). There is one function SHORTEST_PATH() for edge costs of type int and one for double.

Example of How to Use SHORTEST_PATH()

Important Notice: SHORTEST_PATH() only performs correctly if all arithmetic is performed without rounding errors and without overflows. If you are not sure about this, you should use SHORTEST_PATH_T() with one of the LEDA number types.

Running time

The running time of SHORTEST_PATH_T() and SHORTEST_PATH() for a graph G=(V,E) is

• linear in the size of the graph (O(|V|+|E|)) for acyclic graphs
• O(|E|+|V|log|V|) if all edge costs are non-negative
• O(|V||E|) otherwise

Tips

• SHORTEST_PATH_T() provides a convenient interface to the algorithms for solving the SSSP. You do not have to think about cycles and negative edge costs beforehand.
• If you know that there can be no overflows and no rounding errors you can use SHORTEST_PATH().
• If your graph is acyclic, you can also call the SSSP Algorithm for Acyclic Graphs directly.
• If your edge costs are non-negative, you can also call Dijkstra's Algorithm for SSSP directly.
• If your graph contains cycles and there are edges with negative costs, you can also call the Bellman-Ford Algorithm for SSSP directly.
• LEDA also provides functions for checking the results of an SSSP Algorithm.

See also:

Shortest Path Algorithms

Checking the Results of an SSSP Algorithm

SSSP Algorithm for Acyclic Graphs

Dijkstra's Algorithm for SSSP

Bellman-Ford Algorithm for SSSP

Number Types

Graphs and Related Data Types

Graph Algorithms

Manual Entries:

Manual Page Shortest Path Algorithms