TY - GEN
T1 - Weighted Additive Spanners
AU - Ahmed, Reyan
AU - Bodwin, Greg
AU - Darabi Sahneh, Faryad
AU - Kobourov, Stephen
AU - Spence, Richard
N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
PY - 2020/1/1
Y1 - 2020/1/1
N2 - A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic + 2 and + 4 unweighted spanners (both all-pairs and pairwise) to + 2 W and + 4 W weighted spanners, where W is the maximum edge weight. Specifically, we show that a weighted graph G contains all-pairs (pairwise) + 2 W and + 4 W weighted spanners of size O(n3 / 2) and O(n7 / 5) (O(np1 / 3) and O(np2 / 7) ) respectively. For a technical reason, the + 6 unweighted spanner becomes a + 8 W weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that G contains all-pairs (pairwise) + 8 W weighted spanners of size O(n4 / 3) (O(np1 / 4) ).
AB - A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic + 2 and + 4 unweighted spanners (both all-pairs and pairwise) to + 2 W and + 4 W weighted spanners, where W is the maximum edge weight. Specifically, we show that a weighted graph G contains all-pairs (pairwise) + 2 W and + 4 W weighted spanners of size O(n3 / 2) and O(n7 / 5) (O(np1 / 3) and O(np2 / 7) ) respectively. For a technical reason, the + 6 unweighted spanner becomes a + 8 W weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that G contains all-pairs (pairwise) + 8 W weighted spanners of size O(n4 / 3) (O(np1 / 4) ).
KW - Additive spanner
KW - Pairwise spanner
KW - Shortest-path neighborhood
UR - http://www.scopus.com/inward/record.url?scp=85093862218&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-60440-0_32
DO - 10.1007/978-3-030-60440-0_32
M3 - Conference contribution
AN - SCOPUS:85093862218
SN - 9783030604394
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 401
EP - 413
BT - Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Revised Selected Papers
A2 - Adler, Isolde
A2 - Müller, Haiko
PB - Springer Science and Business Media Deutschland GmbH
T2 - 46th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2020
Y2 - 24 June 2020 through 26 June 2020
ER -