Abstract
An $(\alpha,\beta)$-spanner of a weighted graph $G=(V,E)$, is a subgraph $H$ such that for every $u,v\in V$, $d_G(u,v) \le d_H(u,v)\le\alpha\cdot d_G(u,v)+\beta$. The main parameters of interest for spanners are their size (number of edges) and their lightness (the ratio between the total weight of $H$ to the weight of a minimum spanning tree).
In this paper we focus on near-additive spanners, where $\alpha=1+\eps$ for arbitrarily small $\eps>0$.
We show the first construction of {\em light} spanners in this setting. Specifically, for any integer parameter $k\ge 1$, we obtain an $(1+\eps,O(k/\eps)^k\cdot W(\cdot,\cdot))$-spanner with lightness $\Oish(n^{1/k})$ (where $W(\cdot,\cdot)$ indicates for every pair $u, v \in V$ the heaviest edge in some shortest path between $u,v$). In addition, we can also bound the number of edges in our spanner by $O(kn^{1+3/k})$.
In this paper we focus on near-additive spanners, where $\alpha=1+\eps$ for arbitrarily small $\eps>0$.
We show the first construction of {\em light} spanners in this setting. Specifically, for any integer parameter $k\ge 1$, we obtain an $(1+\eps,O(k/\eps)^k\cdot W(\cdot,\cdot))$-spanner with lightness $\Oish(n^{1/k})$ (where $W(\cdot,\cdot)$ indicates for every pair $u, v \in V$ the heaviest edge in some shortest path between $u,v$). In addition, we can also bound the number of edges in our spanner by $O(kn^{1+3/k})$.
Original language | American English |
---|---|
Title of host publication | WG 2024 is the 50th International Workshop on Graph-Theoretic Concepts in Computer Science |
Number of pages | 19 |
State | Published - 19 Jun 2024 |
Keywords
- spanners lightness shortest path weighted graph