TY - GEN
T1 - A Unified Framework for Hopsets
AU - Neiman, Ofer
AU - Shabat, Idan
N1 - Publisher Copyright:
© 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.
PY - 2022/9/1
Y1 - 2022/9/1
N2 - Given an undirected graph G = (V,E), an (α, β)-hopset is a graph H = (V,E'), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, dG(u, v) ≤ d (β) G∪H(u, v) ≤ α dG(u, v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter α. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [3], given a parameter k, a (O(kϵ),O(k1-ϵ))-hopset of size O(n1+1/k) was shown for any n-vertex graph and parameter 0 < ϵ < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α, β)-hopset of size O(n1+1/k) must have α β ≥ Ω(k).
AB - Given an undirected graph G = (V,E), an (α, β)-hopset is a graph H = (V,E'), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, dG(u, v) ≤ d (β) G∪H(u, v) ≤ α dG(u, v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter α. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [3], given a parameter k, a (O(kϵ),O(k1-ϵ))-hopset of size O(n1+1/k) was shown for any n-vertex graph and parameter 0 < ϵ < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α, β)-hopset of size O(n1+1/k) must have α β ≥ Ω(k).
KW - Graph Algorithms
KW - Hopsets
KW - Shortest Paths
UR - http://www.scopus.com/inward/record.url?scp=85137547084&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ESA.2022.81
DO - 10.4230/LIPIcs.ESA.2022.81
M3 - Conference contribution
AN - SCOPUS:85137547084
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 30th Annual European Symposium on Algorithms, ESA 2022
A2 - Chechik, Shiri
A2 - Navarro, Gonzalo
A2 - Rotenberg, Eva
A2 - Herman, Grzegorz
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 30th Annual European Symposium on Algorithms, ESA 2022
Y2 - 5 September 2022 through 9 September 2022
ER -