Fully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds
Maximilian Probst Gutenberg, Christian Wulff-Nilsn · Society for Industrial and Applied Mathematics eBooks · 2019
Given a directed weighted graph G = (V, E) undergoing vertex insertions and deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of G. In two breakthrough results, Italiano and Demetrescu [STOC'03] presented an algorithm that requires Õ(n2) amortized update time, and Thorup showed in [STOC'05] that worst-case update time Õ(n2+3/4) can be achieved. In this article, we make substantial progress on the problem. We present the following new results: We present the first deterministic data structure that breaks the Õ(n2+3/4) worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time to Õ(n2+5/7) = Õ(n2.71) and to Õ(n2+3/5) = Õ(n2.6) for unweighted graphs. We present a simple deterministic algorithm with Õ(n2+3/4) worst-case update time (Õ(n2+2/3) for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time Õ(n2+2/3) (Õ(n2+1/2) for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space Õ(n2). These are the first exact dynamic algorithms with trulysubcubic update time and space usage. This makes significant progress on an open question posed in multiple articles [COCOON'01, STOC '03, ICALP '04, Encyclopedia of Algorithms '08] and is critical to algorithms in practice [TALG '06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA '17].