Shiri Chechik

f-sensitivity distance Oracles and routing schemes

An f-sensitivity distance oracle for a weighted undirected graph G(V, E) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s, t, F),where s and t are vertices and F is a set of forbidden edges such that |F| ≤ f, returns an estimate of the distance between s and t in G(V, E\ F). For an integer parameter k ≥ 1, the size of the data structure is O(fkn1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k-2)(f +1), and the query time is O(|F|ċlog2nċlog log nċlog log d), where d is the distance between s and t in G(V, E\F). The paper also considers f-sensitive compact routing schemes, namely,routing schemes that avoid a given set of forbidden (or failed) edges. It presents a scheme capable of withstanding up to two edge failures. Given a message M destined to t at a source vertex s, in the presence of a forbidden edge set F of size |F| ≤ 2 (unknown to s), our scheme routes M from s to t in a distributed manner, over a path of length at most O(k) times the length of the optimal path (avoiding F). The total amount of information stored in vertices of G is O(kn1+1/k log (nW) log n).

Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels

This paper considers fully dynamic (1+ε) distance oracles and (1+ε) forbidden-set labeling schemes for planar graphs. For a given n-vertex planar graph G with edge weights drawn from [1,M] and parameter ε>0, our forbidden-set labeling scheme uses labels of length λ = O(ε-1 log2n log(nM) • maxlogn). Given the labels of two vertices s and t and of a set F of faulty vertices/edges, our scheme approximates the distance between s and t in G \ F with stretch (1+ε), in O(|F|2 λ) time. We then present a general method to transform (1+ε) forbidden-set labeling schemas into a fully dynamic (1+ε) distance oracle. Our fully dynamic (1+ε) distance oracle is of size O(n log{n} • maxlogn) and has ~O(n1/2) query and update time, both the query and the update time are worst case. This improves on the best previously known (1+ε) dynamic distance oracle for planar graphs, which has worst case query time ~O(n2/3) and amortized update time of ~O(n2/3). Our (1+ε) forbidden-set labeling scheme can also be extended into a forbidden-set labeled routing scheme with stretch (1+ε).

Fault Tolerant Spanners for General Graphs

This paper concerns graph spanners that are resistant to vertex or edge failures. In the failure-free setting, it is known how to efficiently construct a

Approximate Distance Oracles with Improved Bounds

(2k-1)

Fault tolerant additive spanners

-spanner of size

f-Sensitivity Distance Oracles and Routing Schemes

O(n^{1+1/k})

Deterministic decremental single source shortest paths: beyond the o(mn) bound

, and this size-stretch trade-off is conjectured to be tight. The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [C. Levcopoulos, G. Narasimhan, and M. Smid, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 186-195]. A subgraph

ROBUST FAULT TOLERANT UNCAPACITATED FACILITY LOCATION

H

Fully dynamic all-pairs shortest paths with worst-case update-time revisited

is an

Fault tolerant additive and ( µ , α ) -spanners

f