Deterministic improved round-trip spanners

Chun Jiang Zhu, Kam-yiu Lam

Abstract: Highlights • We study the first deterministic algorithm for constructing round-trip spanners. • Our deterministic algorithm out-performs the state-of-the-art randomized algorithm. • We re-raise the question that how best we can hope for the stretch-size trade-off. Abstract In this paper, we study the deterministic construction of round-trip spanners for weighted directed graphs. We propose a deterministic algorithm which constructs, for any n-vertex graph G ( V , E ) , a round-trip spanner H ( V , E ′ ⊆ E ) of stretch 2 k + ϵ and size O ( ( k / ϵ ) ⋅ n 1 + 1 / k log ⁡ ( n w ) ) , where w is the maximum edge weight of G. Notably, this is the first deterministic construction of round-trip spanners and its stretch-size trade-off even improves the previous state-of-the-art randomized algorithm by Roditty et al. More specifically, the size is asymptotically reduced by a factor of k while the stretch factor remains the same. The result is the first clear improvement on round-trip spanners after about ten years and re-raises the open question that how best we can hope for the stretch-size trade-off of round-trip spanners in digraphs.