Nonnegative Integer Weights: Difference between revisions
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(Created page with "{{DISPLAYTITLE:Nonnegative Integer Weights (Shortest Path (Directed Graphs))}} == 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...") |
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== Parameters == | == Parameters == | ||
$V$: number of vertices | |||
E: number of edges | |||
L: maximum absolute value of edge cost | $E$: number of edges | ||
$L$: maximum absolute value of edge cost | |||
== Table of Algorithms == | == Table of Algorithms == | ||
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| [[Dijkstra's algorithm with Fibonacci heap (Johnson 1981; Karlsson & Poblete 1983) (Nonnegative Integer Weights Shortest Path (Directed Graphs))|Dijkstra's algorithm with Fibonacci heap (Johnson ; Karlsson & Poblete 1983)]] || 1981 || $O(E log log L)$ || $O(V+L)$ || Exact || Deterministic || [https://www-sciencedirect-com.ezproxy.canberra.edu.au/science/article/pii/0166218X8390104X?via%3Dihub Time & Space] | | [[Dijkstra's algorithm with Fibonacci heap (Johnson 1981; Karlsson & Poblete 1983) (Nonnegative Integer Weights Shortest Path (Directed Graphs))|Dijkstra's algorithm with Fibonacci heap (Johnson ; Karlsson & Poblete 1983)]] || 1981 || $O(E \log \log L)$ || $O(V+L)$ || Exact || Deterministic || [https://www-sciencedirect-com.ezproxy.canberra.edu.au/science/article/pii/0166218X8390104X?via%3Dihub Time & Space] | ||
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| [[Gabow Ahuja Algorithm (Nonnegative Integer Weights Shortest Path (Directed Graphs))|Gabow Ahuja Algorithm]] || 1990 || $O(E + V*((log(L))^{0.5}) )$ || $O( | | [[Gabow Ahuja Algorithm (Nonnegative Integer Weights Shortest Path (Directed Graphs))|Gabow Ahuja Algorithm]] || 1990 || $O(E + V*((\log(L))^{0.5}) )$ || $O(E + \log C)$ || Exact || Deterministic || [https://dl-acm-org.ezproxy.canberra.edu.au/doi/10.1145/77600.77615 Time & Space] | ||
|- | |- | ||
| [[Thorup's algorithm (Nonnegative Integer Weights Shortest Path (Directed Graphs))|Thorup's algorithm]] || 2004 || $O(E + V min(log log V, log log L))$ || $O(V)$? ("linear-space queue") || Exact || Deterministic || [https://dl-acm-org.ezproxy.canberra.edu.au/doi/10.1016/j.jcss.2004.04.003 Time & Space] | | [[Thorup's algorithm (Nonnegative Integer Weights Shortest Path (Directed Graphs))|Thorup's algorithm]] || 2004 || $O(E + V min(log log V, log log L))$ || $O(V)$? ("linear-space queue") || Exact || Deterministic || [https://dl-acm-org.ezproxy.canberra.edu.au/doi/10.1016/j.jcss.2004.04.003 Time & Space] | ||
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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]] | ||
== References/Citation == | == References/Citation == | ||
https://dl-acm-org.ezproxy.canberra.edu.au/doi/10.1016/j.jcss.2004.04.003 | https://dl-acm-org.ezproxy.canberra.edu.au/doi/10.1016/j.jcss.2004.04.003 | ||
Latest revision as of 09:06, 28 April 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 | $O(E + V*((\log(L))^{0.5}) )$ | $O(E + \log C)$ | 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
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References/Citation
https://dl-acm-org.ezproxy.canberra.edu.au/doi/10.1016/j.jcss.2004.04.003