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• 10:55, 15 February 2023 ‎Reduction from Positive Betweenness Centrality to Diameter (hist | edit) ‎[766 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Positive Betweenness Centrality TO: Diameter == Description == == Implications == if: to-time: $\tilde{O}(T(n,m,M))$ for $n$-node $m$-edge directed (resp. undirected) graph with integer weights in $(-M,M)$<br/>then: from-time: $\tilde{O}(T(n,m,M))$ for $n$-node $m$-edge directed (resp. undirected) graph with integer weights in $(-M,M)$ == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic eq...")
• 10:55, 15 February 2023 ‎Reduction from Diameter to Positive Betweenness Centrality (hist | edit) ‎[696 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Diameter TO: Positive Betweenness Centrality == Description == == Implications == if: to-time: $\tilde{O}(T(n,m))$ for $n$-node $m$-edge directed (resp. undirected) graph<br/>then: from-time: $\tilde{O}(T(n,m))$ for $n$-node $m$-edge directed (resp. undirected) graph == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Pr...")
• 10:55, 15 February 2023 ‎Reduction from Diameter to Reach Centrality (hist | edit) ‎[681 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Diameter TO: Reach Centrality == Description == == Implications == if: to-time: $\tilde{O}(T(n,m))$ for $n$-node $m$-edge directed (resp. undirected) graph<br/>then: from-time: $\tilde{O}(T(n,m))$ for $n$-node $m$-edge directed (resp. undirected) graph == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of th...")
• 10:55, 15 February 2023 ‎Reduction from Undirected, Weighted APSP to All-Nodes Median Parity (hist | edit) ‎[587 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Undirected, Weighted APSP TO: All-Nodes Median Parity == Description == == Implications == if: to-time: Truly subcubic<br/>then: from-time: Truly subcubic == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, Ja...")
• 10:55, 15 February 2023 ‎Reduction from Directed, Weighted APSP to All-Nodes Median Parity (hist | edit) ‎[585 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Directed, Weighted APSP TO: All-Nodes Median Parity == Description == == Implications == if: to-time: Truly subcubic<br/>then: from-time: Truly subcubic == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, Janu...")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to Radius (hist | edit) ‎[643 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: Radius == Description == == Implications == if: to-time: $\tilde{O}(T(n,m,M))$ for $n$-node $m$-edge graph with integer weights in $(-M,M)$<br/>then: from-time: $\tilde{O}(T(n,m,M))$ == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposi...")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to Median (hist | edit) ‎[639 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: Median == Description == == Implications == if: to-time: $\tilde{O}(T(n,M))$ for $n$-node $m$-edge graph with integer weights in $(-M, M)$<br/>then: from-time: $\tilde{O}T(n,M))$ == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium o...")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to All-Nodes Median Parity (hist | edit) ‎[656 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: All-Nodes Median Parity == Description == == Implications == if: to-time: $\tilde{O}(T(n,M))$ for $n$-node $m$-edge graph with integer weights in $(-M, M)$<br/>then: from-time: $\tilde{O}T(n,M))$ == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM...")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to Betweenness Centrality (BC) (hist | edit) ‎[626 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: Betweenness Centrality (BC) == Description == == Implications == if: to-time: $\tilde{O}(T(n,m))$ for $n$-node $m$-edge graph<br/>then: from-time: $\tilde{O}T(n,m))$ == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Al...")
• 10:55, 15 February 2023 ‎Reduction from CNF-SAT to Approximate Diameter (hist | edit) ‎[493 bytes] ‎Admin (talk | contribs) (Created page with "FROM: CNF-SAT TO: Approximate Diameter == Description == == Implications == if: to-time: $O(m^{2-\epsilon})$ for some $\epsilon > {0}$ for a $({3}/{2} - \epsilon)$-approximation<br/>then: from-time: $O*(({2}-\delta)^n)$ for constant $\delta > {0}$ == Year == 2013 == Reference == L. Roditty and V. Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In STOC, pages 515–524, 2013. https://people.csail.mit....")
• 10:55, 15 February 2023 ‎Reduction from Reach Centrality to Diameter (hist | edit) ‎[737 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Reach Centrality TO: Diameter == Description == == Implications == == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681–1697, 2015. https://epubs.siam.org/doi/10.1137/1.9781611...")
• 10:55, 15 February 2023 ‎Reduction from Betweenness Centrality (BC) to Undirected, Weighted APSP (hist | edit) ‎[514 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Betweenness Centrality (BC) TO: Undirected, Weighted APSP == Description == == Implications == == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681–1697, 2015. https://epubs.si...")
• 10:55, 15 February 2023 ‎Reduction from Betweenness Centrality (BC) to Directed, Weighted APSP (hist | edit) ‎[512 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Betweenness Centrality (BC) TO: Directed, Weighted APSP == Description == == Implications == == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681–1697, 2015. https://epubs.siam...")
• 10:55, 15 February 2023 ‎Reduction from Directed Median to Directed, Weighted APSP (hist | edit) ‎[500 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Directed Median TO: Directed, Weighted APSP == Description == == Implications == == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681–1697, 2015. https://epubs.siam.org/doi/10....")
• 10:55, 15 February 2023 ‎Reduction from Undirected Median to Undirected, Weighted APSP (hist | edit) ‎[504 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Undirected Median TO: Undirected, Weighted APSP == Description == == Implications == == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681–1697, 2015. https://epubs.siam.org/doi...")
• 10:55, 15 February 2023 ‎Reduction from Undirected Radius to Undirected, Weighted APSP (hist | edit) ‎[504 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Undirected Radius TO: Undirected, Weighted APSP == Description == == Implications == == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681–1697, 2015. https://epubs.siam.org/doi...")
• 10:55, 15 February 2023 ‎Reduction from Directed Radius to Directed, Weighted APSP (hist | edit) ‎[500 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Directed Radius TO: Directed, Weighted APSP == Description == == Implications == == Year == 2015 == Reference == Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681–1697, 2015. https://epubs.siam.org/doi/10....")
• 10:55, 15 February 2023 ‎Reduction from All-Integers 3SUM to 3SUM (hist | edit) ‎[398 bytes] ‎Admin (talk | contribs) (Created page with "FROM: All-Integers 3SUM TO: 3SUM == Description == == Implications == if: to-time: $O(n^{2-\epsilon})$ for some $\epsilon > {0}$<br/>then: from-time: $O(n^{1.5} + n^{2-\epsilon / 2})$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 8.1")
• 10:55, 15 February 2023 ‎Reduction from 3SUM to All-Integers 3SUM (hist | edit) ‎[177 bytes] ‎Admin (talk | contribs) (Created page with "FROM: 3SUM TO: All-Integers 3SUM == Description == == Implications == if: to-time: $T(n)$<br/>then: from-time: $O(T(n))$ == Year == == Reference == Trivial")
• 10:55, 15 February 2023 ‎Reduction from Nondecreasing Triangle to Triangle in Unweighted Graph (hist | edit) ‎[403 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Nondecreasing Triangle TO: Triangle in Unweighted Graph == Description == == Implications == if: to-time: $T(n)$ for unweighted graph<br/>then: from-time: $O(n^{3/2} \sqrt{T(O(n))})$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 7.1")
• 10:55, 15 February 2023 ‎Reduction from $(\min, \leq)$ Product to Triangle in Unweighted Graph (hist | edit) ‎[410 bytes] ‎Admin (talk | contribs) (Created page with "FROM: $(\min, \leq)$ Product TO: Triangle in Unweighted Graph == Description == == Implications == if: to-time: $T(n)$ for unweighted graph<br/>then: from-time: $O(n^{3/2} \sqrt{T(O(n))} \log n)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 7.1")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle to Price Query (hist | edit) ‎[503 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle TO: Price Query == Description == == Implications == if: to-time: $O(n^{2} / f(n))$ to answer any subsequent price query after $n$-node edge-weighted graph is preprocessed in $O(^k)$ time for some constant $k > {0}$<br/>then: from-time: $O(n^{3} / f(n^{1/({2}k)})$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/...")
• 10:55, 15 February 2023 ‎Reduction from BMM to Independent Set Queries (hist | edit) ‎[513 bytes] ‎Admin (talk | contribs) (Created page with "FROM: BMM TO: Independent Set Queries == Description == == Implications == if: to-time: $O(n^{2} / \log^c n)$ to answer all subsequent batches of $\log n$ independent set queries from a graph that takes $O(n^k)$ time to preprocess for some $c,k > {0}$<br/>then: from-time: $O(n^{3} / \log^{c+1} n)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.or...")
• 10:55, 15 February 2023 ‎Reduction from Triangle Detection to Independent Set Queries (hist | edit) ‎[528 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Triangle Detection TO: Independent Set Queries == Description == == Implications == if: to-time: $O(n^{2} / \log^c n)$ to answer all subsequent batches of $\log n$ independent set queries from a graph that takes $O(n^k)$ time to preprocess for some $c,k > {0}$<br/>then: from-time: $O(n^{3} / \log^{c+1} n)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. ht...")
• 10:55, 15 February 2023 ‎Reduction from Directed, Weighted APSP to Replacement Paths Problem (RPP) (hist | edit) ‎[494 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Directed, Weighted APSP TO: Replacement Paths Problem (RPP) == Description == == Implications == if: to-time: $T(n) polylog M$ in graphs with integer weights in $({0}, M)$<br/>then: from-time: $O(n^{2} T(O(n^{1/3})) polylog M)$ in graphs with integer weights in $(-M, M)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/318...")
• 10:55, 15 February 2023 ‎Reduction from Distance Product to Second Shortest Simple Path (hist | edit) ‎[503 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Distance Product TO: Second Shortest Simple Path == Description == == Implications == if: to-time: $T(n,W)$ where there are $n$ nodes and integer weights in $({0}, W)$<br/>then: from-time: $O(n^{2} T(O(n^{1/3}), O(nW)) \log W)$ for two $n\times n$ matrices with weights in $(-W, W)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/...")
• 10:55, 15 February 2023 ‎Reduction from Directed, Weighted APSP to Second Shortest Simple Path (hist | edit) ‎[497 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Directed, Weighted APSP TO: Second Shortest Simple Path == Description == == Implications == if: to-time: $T(n,W)$ where there are $n$ nodes and integer weights in $({0}, W)$<br/>then: from-time: $O(n^{2} T(O(n^{1/3}), O(n^{2}W)) \log Wn)$ for graphs with weights in $(-W, W)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.114...")
• 10:55, 15 February 2023 ‎Reduction from Minimum Triangle to Second Shortest Simple Path (hist | edit) ‎[479 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Minimum Triangle TO: Second Shortest Simple Path == Description == == Implications == if: to-time: $T(n,W)$ where there are $n$ nodes and integer weights in $({0}, W)$<br/>then: from-time: $T(O(n), O(nW))$ for $n$ node graph with integer weights in $(-W, W)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem...")
• 10:55, 15 February 2023 ‎Reduction from Maximum Subarray to Negative Triangle Detection (hist | edit) ‎[411 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Maximum Subarray TO: Negative Triangle Detection == Description == == Implications == == Year == 1998 == Reference == Hisao Tamaki and Takeshi Tokuyama. 1998. Algorithms for the maximum subarray problem based on matrix multiplication. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’98). 446–452. https://dl.acm.org/doi/abs/10.5555/314613.314823")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to Maximum Subarray (hist | edit) ‎[307 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: Maximum Subarray == Description == == Implications == == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 5.4")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to Shortest Cycle (hist | edit) ‎[463 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: Shortest Cycle == Description == == Implications == if: to-time: $T(n,M)$ where there are $n$ nodes and weights in $({1}, M)$<br/>then: from-time: $T(n, O(M))$ where there are $n$ nodes and weights in $(-M, M)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 5.3")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to Metricity (hist | edit) ‎[494 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: Metricity == Description == == Implications == if: to-time: $O(n^{2}) + T(O(n), O(M))$ where the metricity problem is on $(n)$ s.t. all distances are in $(-M, M)$, and $T(n,M)$ is nondecreasing<br/>then: from-time: $O(n^{2}) + T(O(n), O(M))$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3...")
• 10:55, 15 February 2023 ‎Reduction from Metricity to Negative Triangle Detection (hist | edit) ‎[495 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Metricity TO: Negative Triangle Detection == Description == == Implications == if: to-time: $O(n^{2}) + T(O(n), O(M))$ where $T(n,M)$ is nondecreasing<br/>then: from-time: $O(n^{2}) + T(O(n), O(M))$ where the metricity problem is on $(n)$ s.t. all distances are in $(-M, M)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/...")
• 10:55, 15 February 2023 ‎Reduction from Directed, Weighted APSP to Undirected, Weighted APSP (hist | edit) ‎[450 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Directed, Weighted APSP TO: Undirected, Weighted APSP == Description == == Implications == if: to-time: $T(n, \Theta(M))$ where digraph has weights in $(-M, M)$<br/>then: from-time: $T(n, M)$ where graph has weights in $({0}, M)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 5.1")
• 10:55, 15 February 2023 ‎Reduction from Undirected, Weighted APSP to Directed, Weighted APSP (hist | edit) ‎[450 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Undirected, Weighted APSP TO: Directed, Weighted APSP == Description == == Implications == if: to-time: $T(n, M)$ where graph has weights in $({0}, M)$<br/>then: from-time: $T(n, \Theta(M))$ where digraph has weights in $(-M, M)$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 5.1")
• 10:55, 15 February 2023 ‎Reduction from All Pairs Minimum Witness (APMW) to Negative Triangle Detection (hist | edit) ‎[382 bytes] ‎Admin (talk | contribs) (Created page with "FROM: All Pairs Minimum Witness (APMW) TO: Negative Triangle Detection == Description == == Implications == if: to-time: $T(n)$<br/>then: from-time: $O(T(n^{1/3})n^{2})$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Lemma 4.5")
• 10:55, 15 February 2023 ‎Reduction from Minimum Witness Finding to Negative Triangle Detection (hist | edit) ‎[408 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Minimum Witness Finding TO: Negative Triangle Detection == Description == == Implications == if: to-time: $T(n)$ where $n$ is the number of nodes in the graph<br/>then: from-time: $O(T(n))$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Lemma 4.4")
• 10:55, 15 February 2023 ‎Reduction from Maximum Subarray to Distance Product (hist | edit) ‎[499 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Maximum Subarray TO: Distance Product == Description == == Implications == if: to-time: $O(n^{3-\epsilon})$ for some $\epsilon > {0}$<br/>then: from-time: $O(n^{3-\epsilon})$ == Year == 1998 == Reference == Hisao Tamaki and Takeshi Tokuyama. 1998. Algorithms for the maximum subarray problem based on matrix multiplication. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’98). 446–452. https://dl.acm.org/doi/a...")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Listing to Negative Triangle Detection (hist | edit) ‎[578 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Listing TO: Negative Triangle Detection == Description == == Implications == if: to-time: $O(n^{3-\epsilon}\log^c M)$ for some $\epsilon > {0}$ and where $M$ is the maxint of $R$<br/>then: from-time: $O(n^{3-\epsilon'}\log^c M)$ for some $\epsilon' > {0}$ for listing $\Delta = O(n^{3-\delta})$ negative triangles with fixed $\delta > {0}$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Betw...")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Search to Negative Triangle Detection (hist | edit) ‎[392 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Search TO: Negative Triangle Detection == Description == == Implications == if: to-time: $T(n)$ where $T(n)/n$ is decreasing<br/>then: from-time: $O(T(n))$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Lemma 4.1")
• 10:55, 15 February 2023 ‎Reduction from Matrix Product to Negative Triangle Detection (hist | edit) ‎[524 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Matrix Product TO: Negative Triangle Detection == Description == Generic runtimes for two $n \times n$ matrices == Implications == if: to-time: $T(n)$ where $T(n)/n$ is decreasing<br/>then: from-time: $O(n^{2} \cdot T(n^{1/3})\log W)$ for two $n\times n$ matrices where $W$ is the maxint of $R$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.o...")
• 10:55, 15 February 2023 ‎Reduction from Negative Triangle Detection to Matrix Product Verification (hist | edit) ‎[371 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Negative Triangle Detection TO: Matrix Product Verification == Description == == Implications == if: to-time: $T(n)$<br/>then: from-time: $O(T({2}n))$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893, Theorem 4.1")
• 10:55, 15 February 2023 ‎Reduction from BMM to CFG Parsing (hist | edit) ‎[467 bytes] ‎Admin (talk | contribs) (Created page with "FROM: BMM TO: CFG Parsing == Description == == Implications == if: to-time: $O(gn^{3-\epsilon})$ for some $\epsilon > {0}$ where $g$ is the size of the CFG and $n$ is the size of the string<br/>then: from-time: $O(n^{3-\epsilon/3})$ where $n \times n$ matrix == Year == 2002 == Reference == L. Lee. 2002. Fast context-free grammar parsing requires fast boolean matrix multiplication. J. ACM 49, 1 (2002), 1–15. https://arxiv.org/abs/cs/0112018")
• 10:55, 15 February 2023 ‎Reduction from CFG Parsing to BMM (hist | edit) ‎[467 bytes] ‎Admin (talk | contribs) (Created page with "FROM: CFG Parsing TO: BMM == Description == == Implications == if: to-time: $O(n^{3-\epsilon})$ for some $\epsilon > {0}$ where $n \times n$ matrix<br/>then: from-time: $O(gn^{3-\epsilon})$ where $g$ is the size of the CFG == Year == 1975 == Reference == L. G. Valiant. 1975. General context-free recognition in less than cubic time. J. Comput. Syst. Sci. 10 (1975), 308–315. https://www.sciencedirect.com/science/article/pii/S0022000075800468")
• 10:55, 15 February 2023 ‎Reduction from CNF-SAT to Multiple Local Alignment (hist | edit) ‎[607 bytes] ‎Admin (talk | contribs) (Created page with "FROM: CNF-SAT TO: Multiple Local Alignment == Description == == Implications == if: to-time: $N^{k-\epsilon}$ for some $\epsilon > {0}$ on $k$ binary strings of length $n$ with $k$-wise scoring<br/>then: from-time: ${2}^{(n-\epsilon')}$ for some $\epsilon' > {0}$ == Year == 2014 == Reference == Abboud, Amir, Virginia Vassilevska Williams, and Oren Weimann. "Consequences of faster alignment of sequences." In International Colloquium on Automata, Langu...")
• 10:55, 15 February 2023 ‎Reduction from Directed, Weighted APSP to Negative Triangle Detection (hist | edit) ‎[421 bytes] ‎Admin (talk | contribs) (Created page with "FROM: Directed, Weighted APSP TO: Negative Triangle Detection == Description == == Implications == if: to-time: $n^{3-\epsilon}$ for some $\epsilon >{0}$<br/>then: from-time: $n^{3-\epsilon'}$ for some $\epsilon' > {0}$ == Year == 2018 == Reference == V. V. Williams, R. R. Williams, Subcubic Equivalences Between Path, Matrix, and Triangle Problems. 2018. https://dl.acm.org/doi/pdf/10.1145/3186893")
• 10:55, 15 February 2023 ‎Reduction from CNF-SAT to Longest Common Substring with don't cares (hist | edit) ‎[566 bytes] ‎Admin (talk | contribs) (Created page with "FROM: CNF-SAT TO: Longest Common Substring with don't cares == Description == == Implications == if: to-time: $N^{2-\epsilon}$ for some $\epsilon > {0}$<br/>then: from-time: ${2}^{(n-\epsilon')}$ for some $\epsilon' > {0}$ == Year == 2014 == Reference == Abboud, Amir, Virginia Vassilevska Williams, and Oren Weimann. "Consequences of faster alignment of sequences." In International Colloquium on Automata, Languages, and Programming, pp. 39-51. Springe...")
• 10:55, 15 February 2023 ‎Reduction from CNF-SAT to Local Alignment (hist | edit) ‎[576 bytes] ‎Admin (talk | contribs) (Created page with "FROM: CNF-SAT TO: Local Alignment == Description == == Implications == if: to-time: $N^{2-\epsilon}$ for some $\epsilon > {0}$ on two binary strings of length $N$<br/>then: from-time: ${2}^{(n-\epsilon')}$ for some $\epsilon' > {0}$ == Year == 2014 == Reference == Abboud, Amir, Virginia Vassilevska Williams, and Oren Weimann. "Consequences of faster alignment of sequences." In International Colloquium on Automata, Languages, and Programming, pp. 39-5...")
• 10:55, 15 February 2023 ‎Reduction from 3SUM to Local Alignment (hist | edit) ‎[625 bytes] ‎Admin (talk | contribs) (Created page with "FROM: 3SUM TO: Local Alignment == Description == == Implications == if: to-time $N^{2-\delta-\epsilon} for two strings of size $n$ and alphabet of size $n^{1-\delta}$ for some $\espilon > {0}$,$\delta \in ({0},{1})$<br/>then: from-time: $n^{2-\epsilon'}$ for some $\epsilon' > {0}$ == Year == 2014 == Reference == Abboud, Amir, Virginia Vassilevska Williams, and Oren Weimann. "Consequences of faster alignment of sequences." In International Colloquium...")
• 10:55, 15 February 2023 ‎Reduction from 3SUM' to Static Dihedral Rotation Queries (hist | edit) ‎[491 bytes] ‎Admin (talk | contribs) (Created page with "FROM: 3SUM' TO: Static Dihedral Rotation Queries == Description == == Implications == if: to-time $N^{2-\epsilon}$ for some $\epsilon > {0}$<br/>then: from-time: $N^{2-\epsilon'}$ for some $\epsilon' > {0}$ == Year == 2003 == Reference == Michael Soss, Jeff Erickson, Mark Overmars, Preprocessing chains for fast dihedral rotations is hard or even impossible, Computational Geometry, Volume 26, Issue 3, 2003, Pages 235-246 https://doi.org/10.1016/S0925...")

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