A fast algorithm for source-wise round-trip spanners
Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review
Author(s)
Related Research Unit(s)
Detail(s)
Original language | English |
---|---|
Pages (from-to) | 34-44 |
Journal / Publication | Theoretical Computer Science |
Volume | 876 |
Online published | 24 May 2021 |
Publication status | Published - 12 Jul 2021 |
Link(s)
Abstract
In this paper, we study the problem of fast constructions of source-wise round-trip spanners in weighted directed graphs. For a source vertex set S ⊆ V in a graph G(V, E), an S-sourcewise round-trip spanner of G of stretch k is a subgraph H of G such that for every pair of vertices u, v ∈ S × V, their round-trip distance in H is at most k times of their round-trip distance in G. We show that for a graph G(V, E) with n vertices and m edges, an s-sized source vertex set S ⊆ V and an integer k > 1, there exists an algorithm that in time O (ms1/k log5 n) constructs an S-sourcewise round-trip spanner of stretch O (k log n) and O (ns1/k log2 n) edges with high probability. Compared to the fast algorithms for constructing all-pairs round-trip spanners [26,12], our algorithm improves the running time and the number of edges in the spanner when k is super-constant. Compared with the existing algorithm for constructing source-wise round-trip spanners [36], our algorithm significantly improves their construction time Ω(min{ms, nω}) (where ω ∈ [2, 2.373) and 2.373 is the matrix multiplication exponent) to nearly linear O (ms1/k log5 n), at the expense of paying an extra O (log n) in the stretch. As an important building block of the algorithm, we develop a graph partitioning algorithm to partition G into clusters of bounded radius and prove that for every u, v ∈ S × V at small round-trip distance, the probability of separating them in different clusters is small. The algorithm takes the size of S as input and does not need the knowledge of S. With the algorithm and a reachability vertex size estimation algorithm, we show that the recursive algorithm for constructing standard round-trip spanners [26] can be adapted to the source-wise setting. We rigorously prove the correctness and computational complexity of the adapted algorithms. Finally, we show how to remove the dependence on the edge weight in the source-wise case.
Research Area(s)
- Graph algorithms, Graph partitioning, Graph spanners, Round-trip spanners
Citation Format(s)
A fast algorithm for source-wise round-trip spanners. / Zhu, Chun Jiang; Han, Song; Lam, Kam-Yiu.
In: Theoretical Computer Science, Vol. 876, 12.07.2021, p. 34-44.
In: Theoretical Computer Science, Vol. 876, 12.07.2021, p. 34-44.
Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review