TY - GEN

T1 - On Additive Spanners in Weighted Graphs with Local Error

AU - Ahmed, Reyan

AU - Bodwin, Greg

AU - Hamm, Keaton

AU - Kobourov, Stephen

AU - Spence, Richard

N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - An additive + β spanner of a graph G is a subgraph which preserves distances up to an additive + β error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error β= cW, where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error + cW(s, t) for each vertex pair (s, t), where W(s, t) is the maximum edge weight along the shortest s–t path in G. These include pairwise + (2 + ε) W(·, · ) and + (6 + ε) W(·, · ) spanners over vertex pairs P⊆ V× V on Oε(n| P|1 / 3) and Oε(n| P|1 / 4) edges for all ε> 0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs + 4 W(·, · ) spanner on O~ (n7 / 5) edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of an MST of G. We provide a + εW(·, · ) spanner with Oε(n) lightness, and a + (4 + ε) W(·, · ) spanner with Oε(n2 / 3) lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

AB - An additive + β spanner of a graph G is a subgraph which preserves distances up to an additive + β error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error β= cW, where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error + cW(s, t) for each vertex pair (s, t), where W(s, t) is the maximum edge weight along the shortest s–t path in G. These include pairwise + (2 + ε) W(·, · ) and + (6 + ε) W(·, · ) spanners over vertex pairs P⊆ V× V on Oε(n| P|1 / 3) and Oε(n| P|1 / 4) edges for all ε> 0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs + 4 W(·, · ) spanner on O~ (n7 / 5) edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of an MST of G. We provide a + εW(·, · ) spanner with Oε(n) lightness, and a + (4 + ε) W(·, · ) spanner with Oε(n2 / 3) lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

UR - http://www.scopus.com/inward/record.url?scp=85115856911&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85115856911&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-86838-3_28

DO - 10.1007/978-3-030-86838-3_28

M3 - Conference contribution

AN - SCOPUS:85115856911

SN - 9783030868376

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 361

EP - 373

BT - Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Revised Selected Papers

A2 - Kowalik, Lukasz

A2 - Pilipczuk, Michal

A2 - Rzazewski, Pawel

PB - Springer Science and Business Media Deutschland GmbH

T2 - 47th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2021

Y2 - 23 June 2021 through 25 June 2021

ER -