TY - GEN
T1 - Light Tree Covers, Routing, and Path-Reporting Oracles via Spanning Tree Covers in Doubling Graphs
AU - Chang, Hsien Chih
AU - Conroy, Jonathan
AU - Le, Hung
AU - Solomon, Shay
AU - Than, Cuong
N1 - Publisher Copyright:
© 2025 Copyright is held by the owner/author(s). Publication rights licensed to ACM.
PY - 2025/6/15
Y1 - 2025/6/15
N2 - A (1+ϵ)-stretch tree cover of an edge-weighted n-vertex graph G is a collection of trees, where every pair of vertices has a (1+ϵ)-stretch path in one of the trees. The celebrated Dumbbell Theorem by Arya et. al. [STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+ϵ)-stretch tree cover with a constant number of trees, where the constant depends on ϵ and the dimension d. This result was generalized for arbitrary doubling metrics by Bartal et. al. [ICALP'19]. While the total number of edges in the tree covers of Arya et. al. and Bartal et. al. is O(n), all known tree cover constructions incur a total lightness of ω(logn); whether one can get a tree cover of constant lightness has remained a longstanding open question, even for 2-dimensional point sets. In this work we resolve this fundamental question in the affirmative, as a direct corollary of a new construction of (1+ϵ)-stretch spanning tree cover for doubling graphs; in a spanning tree cover, every tree may only use edges of the input graph rather than the corresponding metric. To the best of our knowledge, this is the first constant-stretch spanning tree cover construction (let alone for (1+ϵ)-stretch) with a constant number of trees, for any nontrivial family of graphs. Concrete applications of our spanning tree cover include: - A (1+ϵ)-stretch tree cover construction, where both the number of trees and lightness are bounded by O(1), for doubling graphs. In doubling metrics, we can also bound the maximum degree of each vertex by O(1) (which is impossible in doubling graphs). - A compact (1+ϵ)-stretch routing scheme in the labeled model for doubling graphs, which uses the asymptotically optimal (up to the dependencies on ϵ and d) bound of O(logn) bits on all the involved measures (label, header, and routing tables sizes). This is a significant improvement over the works of Chan et. al. [SODA'05], Abraham et. al. [ICDCS'06], Konjevod et. al. [SODA'07], where the local memory usage either depends on the aspect ratio of the graph or is ω(log3 n). - The first path-reporting distance oracle for doubling graphs achieving optimal bounds for all important parameters: O(n) space, (1+ϵ)-stretch, and O(1) query time for constant d and ϵ.
AB - A (1+ϵ)-stretch tree cover of an edge-weighted n-vertex graph G is a collection of trees, where every pair of vertices has a (1+ϵ)-stretch path in one of the trees. The celebrated Dumbbell Theorem by Arya et. al. [STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+ϵ)-stretch tree cover with a constant number of trees, where the constant depends on ϵ and the dimension d. This result was generalized for arbitrary doubling metrics by Bartal et. al. [ICALP'19]. While the total number of edges in the tree covers of Arya et. al. and Bartal et. al. is O(n), all known tree cover constructions incur a total lightness of ω(logn); whether one can get a tree cover of constant lightness has remained a longstanding open question, even for 2-dimensional point sets. In this work we resolve this fundamental question in the affirmative, as a direct corollary of a new construction of (1+ϵ)-stretch spanning tree cover for doubling graphs; in a spanning tree cover, every tree may only use edges of the input graph rather than the corresponding metric. To the best of our knowledge, this is the first constant-stretch spanning tree cover construction (let alone for (1+ϵ)-stretch) with a constant number of trees, for any nontrivial family of graphs. Concrete applications of our spanning tree cover include: - A (1+ϵ)-stretch tree cover construction, where both the number of trees and lightness are bounded by O(1), for doubling graphs. In doubling metrics, we can also bound the maximum degree of each vertex by O(1) (which is impossible in doubling graphs). - A compact (1+ϵ)-stretch routing scheme in the labeled model for doubling graphs, which uses the asymptotically optimal (up to the dependencies on ϵ and d) bound of O(logn) bits on all the involved measures (label, header, and routing tables sizes). This is a significant improvement over the works of Chan et. al. [SODA'05], Abraham et. al. [ICDCS'06], Konjevod et. al. [SODA'07], where the local memory usage either depends on the aspect ratio of the graph or is ω(log3 n). - The first path-reporting distance oracle for doubling graphs achieving optimal bounds for all important parameters: O(n) space, (1+ϵ)-stretch, and O(1) query time for constant d and ϵ.
KW - Distance oracle
KW - Metric sketching
KW - Routing
KW - Tree cover
UR - https://www.scopus.com/pages/publications/105009821256
U2 - 10.1145/3717823.3718312
DO - 10.1145/3717823.3718312
M3 - Conference contribution
AN - SCOPUS:105009821256
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 2257
EP - 2268
BT - STOC 2025 - Proceedings of the 57th Annual ACM Symposium on Theory of Computing
A2 - Koucky, Michal
A2 - Bansal, Nikhil
PB - Association for Computing Machinery
T2 - 57th Annual ACM Symposium on Theory of Computing, STOC 2025
Y2 - 23 June 2025 through 27 June 2025
ER -