APSP on Dense Undirected Graphs with Arbitrary Weights: Difference between revisions
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== Parameters == | == Parameters == | ||
n: number of vertices | $n$: number of vertices | ||
m: number of edges | $m$: number of edges | ||
== Table of Algorithms == | == Table of Algorithms == | ||
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| [[Pettie & Ramachandran (APSP on Dense Undirected Graphs with Arbitrary Weights; APSP on Sparse Undirected Graphs with Arbitrary Weights All-Pairs Shortest Paths (APSP))|Pettie & Ramachandran]] || 2002 || $O( | | [[Pettie & Ramachandran (APSP on Dense Undirected Graphs with Arbitrary Weights; APSP on Sparse Undirected Graphs with Arbitrary Weights All-Pairs Shortest Paths (APSP))|Pettie & Ramachandran]] || 2002 || $O(mn \log \alpha(m,n))$ || || Exact || Deterministic || [https://dl.acm.org/citation.cfm?id=545417 Time] | ||
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| [[Chan (APSP on Dense Directed Graphs with Arbitrary Weights; APSP on Dense Undirected Graphs with Arbitrary Weights All-Pairs Shortest Paths (APSP))|Chan]] || 2009 || $O( | | [[Chan (APSP on Dense Directed Graphs with Arbitrary Weights; APSP on Dense Undirected Graphs with Arbitrary Weights All-Pairs Shortest Paths (APSP))|Chan]] || 2009 || $O(n^{3} \log^{3} \log n / \log^{2} n)$ || $O(n^{2})$ || Exact || Deterministic || [http://tmc.web.engr.illinois.edu/moreapsp.pdf Time] | ||
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[[File:All-Pairs Shortest Paths (APSP) - APSP on Dense Undirected Graphs with Arbitrary Weights - Time.png|1000px]] | [[File:All-Pairs Shortest Paths (APSP) - APSP on Dense Undirected Graphs with Arbitrary Weights - Time.png|1000px]] | ||
Latest revision as of 10:06, 28 April 2023
Description
In this case, the graph $G=(V,E)$ that we consider is dense ($m = O(n^2)$), is undirected, and has arbitrary weights.
Related Problems
Generalizations: APSP
Related: APSP on Dense Directed Graphs with Arbitrary Weights, APSP on Geometrically Weighted Graphs, APSP on Dense Undirected Graphs with Positive Integer Weights, APSP on Sparse Directed Graphs with Arbitrary Weights, APSP on Sparse Undirected Graphs with Positive Integer Weights, APSP on Sparse Undirected Graphs with Arbitrary Weights, APSP on Dense Directed Unweighted Graphs, APSP on Dense Undirected Unweighted Graphs, APSP on Sparse Directed Unweighted Graphs, APSP on Sparse Undirected Unweighted Graphs, (5/3)-approximate ap-shortest paths
Parameters
$n$: number of vertices
$m$: number of edges
Table of Algorithms
| Name | Year | Time | Space | Approximation Factor | Model | Reference |
|---|---|---|---|---|---|---|
| Pettie & Ramachandran | 2002 | $O(mn \log \alpha(m,n))$ | Exact | Deterministic | Time | |
| Chan | 2009 | $O(n^{3} \log^{3} \log n / \log^{2} n)$ | $O(n^{2})$ | Exact | Deterministic | Time |
Time Complexity Graph
_-_APSP_on_Dense_Undirected_Graphs_with_Arbitrary_Weights_-_Time.png)