TY - GEN
T1 - On the Size Overhead of Pairwise Spanners
AU - Neiman, Ofer
AU - Shabat, Idan
N1 - Publisher Copyright:
© 2024 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.
PY - 2024/1/1
Y1 - 2024/1/1
N2 - Given an undirected possibly weighted n-vertex graph G = (V,E) and a set P V 2 of pairs, a subgraph S = (V,E) is called a P-pairwise α-spanner of G, if for every pair (u, v) P we have dS(u, v) α dG(u, v). The parameter α is called the stretch of the spanner, and its size overhead is define as |E| |P| . A surprising connection was recently discussed between the additive stretch of (1+, β)-spanners, to the hopbound of (1 + , β)-hopsets. A long sequence of works showed that if the spanner/hopset has size n1+1/k for some parameter k 1, then β 1 log k. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if |P| n1+1/k, then a P-pairwise (1 + )-spanner must have size at least β |P| with β 1 log k (a near matching upper bound was recently shown in [18]). That is, the size overhead of pairwise spanners has similar bounds to the hopbound of hopsets, and to the additive stretch of spanners. We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for P-pairwise α-spanners. In particular, we show that if |P| n1+1/k, then the size overhead is β k α. A source-wise spanner is a special type of pairwise spanner, for which P = A × V for some A V . A prioritized spanner is given also a ranking of the vertices V = (v1, . . . , vn), and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions: from pairwise spanners to source-wise spanners to prioritized spanners, we improve on the state-of-The-Art results for source-wise and prioritized spanners. Since our spanners can be equipped with a path-reporting mechanism, we also substantially improve the known bounds for path-reporting prioritized distance oracles. Specifically, we provide a path-reporting distance oracle, with size O(n (log log n)2), that has a constant stretch for any query that contains a vertex ranked among the first n1 vertices (for any constant 0). Such a result was known before only for non-path-reporting distance oracles.
AB - Given an undirected possibly weighted n-vertex graph G = (V,E) and a set P V 2 of pairs, a subgraph S = (V,E) is called a P-pairwise α-spanner of G, if for every pair (u, v) P we have dS(u, v) α dG(u, v). The parameter α is called the stretch of the spanner, and its size overhead is define as |E| |P| . A surprising connection was recently discussed between the additive stretch of (1+, β)-spanners, to the hopbound of (1 + , β)-hopsets. A long sequence of works showed that if the spanner/hopset has size n1+1/k for some parameter k 1, then β 1 log k. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if |P| n1+1/k, then a P-pairwise (1 + )-spanner must have size at least β |P| with β 1 log k (a near matching upper bound was recently shown in [18]). That is, the size overhead of pairwise spanners has similar bounds to the hopbound of hopsets, and to the additive stretch of spanners. We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for P-pairwise α-spanners. In particular, we show that if |P| n1+1/k, then the size overhead is β k α. A source-wise spanner is a special type of pairwise spanner, for which P = A × V for some A V . A prioritized spanner is given also a ranking of the vertices V = (v1, . . . , vn), and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions: from pairwise spanners to source-wise spanners to prioritized spanners, we improve on the state-of-The-Art results for source-wise and prioritized spanners. Since our spanners can be equipped with a path-reporting mechanism, we also substantially improve the known bounds for path-reporting prioritized distance oracles. Specifically, we provide a path-reporting distance oracle, with size O(n (log log n)2), that has a constant stretch for any query that contains a vertex ranked among the first n1 vertices (for any constant 0). Such a result was known before only for non-path-reporting distance oracles.
KW - Graph Algorithms
KW - Shortest Paths
KW - Spanners
UR - http://www.scopus.com/inward/record.url?scp=85184137964&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ITCS.2024.83
DO - 10.4230/LIPIcs.ITCS.2024.83
M3 - Conference contribution
AN - SCOPUS:85184137964
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 15th Innovations in Theoretical Computer Science Conference, ITCS 2024
A2 - Guruswami, Venkatesan
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 15th Innovations in Theoretical Computer Science Conference, ITCS 2024
Y2 - 30 January 2024 through 2 February 2024
ER -