Nonnegative Integer Weights: Difference between revisions
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== Time Complexity | == Time Complexity Graph == | ||
[[File:Shortest Path (Directed Graphs) - Nonnegative Integer Weights - Time.png|1000px]] | [[File:Shortest Path (Directed Graphs) - Nonnegative Integer Weights - Time.png|1000px]] | ||
== Space Complexity | == Space Complexity Graph == | ||
[[File:Shortest Path (Directed Graphs) - Nonnegative Integer Weights - Space.png|1000px]] | [[File:Shortest Path (Directed Graphs) - Nonnegative Integer Weights - Space.png|1000px]] | ||
== Pareto | == Pareto Frontier Improvements Graph == | ||
[[File:Shortest Path (Directed Graphs) - Nonnegative Integer Weights - Pareto Frontier.png|1000px]] | [[File:Shortest Path (Directed Graphs) - Nonnegative Integer Weights - Pareto Frontier.png|1000px]] | ||
Revision as of 13:04, 15 February 2023
Description
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. Here, the weights are restricted to be nonnegative integers.
Related Problems
Generalizations: nonnegative weights
Related: General Weights, Nonnegative Weights, Second Shortest Simple Path, st-Shortest Path, 1-sensitive (3/2)-approximate ss-shortest paths, 2-sensitive (7/5)-approximate st-shortest paths, 1-sensitive decremental st-shortest paths, 2-sensitive decremental st-shortest paths, Replacement Paths Problem
Parameters
V: number of vertices
E: number of edges
L: maximum absolute value of edge cost
Table of Algorithms
| Name | Year | Time | Space | Approximation Factor | Model | Reference |
|---|---|---|---|---|---|---|
| Dijkstra's algorithm with Fibonacci heap (Johnson ; Karlsson & Poblete 1983) | 1981 | $O(E log log L)$ | $O(V+L)$ | Exact | Deterministic | Time & Space |
| Gabow Ahuja Algorithm | 1990 | $O(E + V*((log(L))^{0.5}) )$ | $O(m + log C)$ | Exact | Deterministic | Time & Space |
| Thorup's algorithm | 2004 | $O(E + V min(log log V, log log L))$ | $O(V)$? ("linear-space queue") | Exact | Deterministic | Time & Space |
Time Complexity Graph
_-_Nonnegative_Integer_Weights_-_Time.png)
Space Complexity Graph
_-_Nonnegative_Integer_Weights_-_Space.png)
Pareto Frontier Improvements Graph
_-_Nonnegative_Integer_Weights_-_Pareto_Frontier.png)