Fully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds

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 (O) over tilde (n(2)) amortized update time, and Thorup showed in [STOC '05] that worst-case update time (O) over tilde (n(2+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 (O) over tilde (n(2+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 (O) over tilde (n(2+5/7)) = (O) over tilde (n(2.71..)) and to (O) over tilde (n(2+3/5)) = (O) over tilde (n(2.6)) for unweighted graphs. We present a simple deterministic algorithm with (O) over tilde (n(2+3/4)) worst-case update time ((O) over tilde (n(2+2/3)) for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time (O) over tilde (n(2+2/3)) ((O) over tilde (n(2+1/2)) for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space (O) over tilde (n(2)). 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].