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A Subquadratic-Time Algorithm for Decremental Single-Source Shortest Paths


We study dynamic (1+ε)-approximation algorithms for the single-source shortest paths problem in an unweighted undirected n-node m-edge graph under edge deletions. The fastest algorithm for this problem is an algorithm with O(n^{2+o(1)}) total update time and constant query time by Bernstein and Roditty (SODA 2011). In this paper, we improve the total update time to O(n^{1.8+o(1)} + m^{1+o(1)}) while keeping the query time constant. This running time is essentially tight when m=\Omega(n^{1.8}) since we need \Omega(m) time even in the static setting. For smaller values of m, the running time of our algorithm is subquadratic, and is the first that breaks through the quadratic time barrier. In obtaining this result, we develop a fast algorithm for what we call center cover data structure. We also make non-trivial extensions to our previous techniques called lazy-update and monotone Even-Shiloach trees (ICALP 2013 and FOCS 2013). As by-products of our new techniques, we obtain two new results for the decremental all-pairs shortest-paths problem. Our first result is the first approximation algorithm whose total update time is faster than O(mn) for all values of m. Our second result is a new trade-off between the total update time and the additive approximation guarantee.

Paper in Conference Proceedings or in Workshop Proceedings (Full Paper in Proceedings)
Event Title
SODA 2014
Theory and Applications of Algorithms
Theoretische Informatik
Event Location
Portland, USA
Event Type
Event Dates
January 5-7, 2014
January 2014
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Faculty of Computer Science
University of Vienna

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