An f-edge fault-tolerant distance sensitive oracle (f-DSO) with stretch
σ≥1 is a data structure that preprocesses a given undirected,
unweighted graph G with n vertices and m edges, and a positive integer
f. When queried with a pair of vertices s,t and a set F of at most f
edges, it returns a σ-approximation of the s-t-distance in G−F. We
study f-DSOs that take subquadratic space. Thorup and Zwick [JACM 2015]
showed that this is only possible for σ≥3. We present, for any
constant f≥1 and α∈(0,21), and any ε>0, an f-DSO with stretch 3+ε that takes
O(n2−f+1α/ε)⋅O(logn/ε)f+1 space and has an O(nα/ε2) query time.
We also give an improved construction for graphs with diameter at most D. For
any constant k, we devise an f-DSO with stretch 2k−1 that takes
O(Df+o(1)n1+1/k) space and has O(Do(1)) query time,
with a preprocessing time of O(Df+o(1)mn1/k). Chechik, Cohen, Fiat,
and Kaplan [SODA 2017] presented an f-DSO with stretch 1+ε and
preprocessing time Oε(n5+o(1)), albeit with a super-quadratic
space requirement. We show how to reduce their preprocessing time to
Oε(mn2+o(1)).Comment: accepted at STOC 202