Space-Efficient Fault-Tolerant Diameter Oracles

Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances

“…In case of a single failure, f = 1, we shorten this to FDO and FEO, respectively. The problem of designing FDOs was originally raised by Henzinger et al [27] and recently received some renewed interest by Bilò et al [7]. Although the major focus of the latter work was on undirected graphs, the authors also showed that, for directed graphs, one can compute, in O(mn + n 2 /ε) time, 1 an oracle of size 2 O(m) and constant query time that guarantees a stretch of 1 + ε, that is, it reports an upper bound on the value of the diameter within a factor of 1+ε, for any ε > 0.…”

“…Bilò et al [7] also gave a complementary space lower bound showing that any fault-tolerant diameter oracle with a sufficiently small stretch must take Ω(m) bits of space. However, this is not the full picture in that their construction only holds for diameter 2.…”

“…We show here that in reality there is a transition happening: the larger the diameter, the more space we can save, up to a point where even o(m)-space oracles become possible. We aim at pinpointing this transition, starting with a generalization of the bound in [7] to diameter up to n/ √ m .…”

“…We now move our attention to the case multiple edge failures and give bounds in terms of f and n on the minimum space requirement of f -FDOs. Bilò et al [7] designed an f -FDO for undirected graphs of stretch f + 2 that takes space O(f n). The size of this oracle is optimal up to polylogarithmic factors.…”

“…As already mentioned above, the Ω(2 f /2 n)-bits lower bound in Theorem 4 also holds for FEOs. On DAGs, however, we can improve upon this and obtain a space requirement that is reminiscent of the one Bilò et al [7] gave for undirected graphs. Note that in a DAG at most one vertex can have bounded eccentricity.…”