We study dynamic (1+ϵ)-approximation algorithms for the all-pairs
shortest paths problem in unweighted undirected n-node m-edge graphs under
edge deletions. The fastest algorithm for this problem is a randomized
algorithm with a total update time of O~(mn/ϵ) and constant
query time by Roditty and Zwick [FOCS 2004]. The fastest deterministic
algorithm is from a 1981 paper by Even and Shiloach [JACM 1981]; it has a total
update time of O(mn2) and constant query time. We improve these results as
follows: (1) We present an algorithm with a total update time of O~(n5/2/ϵ) and constant query time that has an additive error of 2
in addition to the 1+ϵ multiplicative error. This beats the previous
O~(mn/ϵ) time when m=Ω(n3/2). Note that the additive
error is unavoidable since, even in the static case, an O(n3−δ)-time
(a so-called truly subcubic) combinatorial algorithm with 1+ϵ
multiplicative error cannot have an additive error less than 2−ϵ,
unless we make a major breakthrough for Boolean matrix multiplication [Dor et
al. FOCS 1996] and many other long-standing problems [Vassilevska Williams and
Williams FOCS 2010]. The algorithm can also be turned into a
(2+ϵ)-approximation algorithm (without an additive error) with the
same time guarantees, improving the recent (3+ϵ)-approximation
algorithm with O~(n5/2+O(log(1/ϵ)/logn)) running
time of Bernstein and Roditty [SODA 2011] in terms of both approximation and
time guarantees. (2) We present a deterministic algorithm with a total update
time of O~(mn/ϵ) and a query time of O(loglogn). The
algorithm has a multiplicative error of 1+ϵ and gives the first
improved deterministic algorithm since 1981. It also answers an open question
raised by Bernstein [STOC 2013].Comment: A preliminary version was presented at the 2013 IEEE 54th Annual
Symposium on Foundations of Computer Science (FOCS 2013