Forbidden-Set Distance Labels for Graphs of Bounded Doubling Dimension Presented by: Shiri Chechik Joint with: Ittai Abraham, Cyril Gavoille, David Peleg

Distance Labeling Graph Ls Lt … Labels s Distance Query 8 t 2 2 2 8 4 3 Distance Query 8 3 3 3 2 t

Distance Labeling Preprocessing Phase: preprocess G and assign to each vertex v of G a label L(v). Query Phase: given the labels L(s) and L(t), answer a query concerning the distance between s and t in G.

Distance Labeling Computing exact distances may be costly in term of memory requirements. Typically, polynomial space is needed for exact distances in sparse graphs like planar graphs [Gavoille, Peleg, Pérennés, and Raz, 04], whereas poly-logarithmic labels are possible if a small error is tolerated ([Peleg 99, Talwar 04, Thorup 04]).

Approximate Distance Labeling Graph Labels 2 s 2 2 8 … Ls … Lt … 4 3 3 3 3 2 t Approximate Distance Query ≈ d’ We require that d’ satisfy: dG(x,y) ≤ d’ ≤ (1+ε)dG(x,y)

Compact Routing Graph s 2 2 2 8 4 3 3 3 3 2 t

Compact Routing The situation is similar for compact routing: Ω(n1/2)-bit routing tables are required for shortest path routing in bounded growth networks ([Abraham et al. 2006]) or planar graphs. Routing tables of size (ε-1logn)O(1) suffice for (1+ ε)-stretch routing scheme in bounded doubling dimension graphs ([Abraham et al. 06, , Konjevod et al. 07, , Konjevod et al. 08, Slivkins 05, Slivkins 07] or in minor-free graphs ([Abraham and Gavoille 06]).

Forbidden-Set Approx. Distance Labeling Graph Ls Lt … Labels Lz s 2 2 2 8 4 3 Distance Query ≈ 11 3 z 3 3 2 t

Forbidden-Set Distance Labeling Preprocessing Phase: preprocess G and assign to each vertex v of G a label L(v). Query Phase: given the set of labels {L(f) : f  F}, and the two labels L(u) and L(v), answer a query concerning the distance between u and v in G\F.

Related Work In the failure-free setting, there is a vast literature on labeling schemes and information oracles, and it covers many aspects: distance, routing, exact queries, approximation, global or localized data- structures.

Related Work [Demetrescu and Thorup, 02] General directed graphs. A scheme for answering exact distance queries. Single edge failure. Oracle of size O(n2logn) Query time O(log n)

Related Work Exact case, cont: This has been extended to a single edge and/or vertex failure [Bernstein and Karger, 09], and only recently to dual- failure [Duan and Pettie, 09] Approximated case: multiple edge failures [Chechik, Langberg, Peleg, and Roditty, 10]. The routing problem has been explored in [Khanna and Baswana, 10] for single vertex failure and in [Chechik, Langberg, Peleg, and Roditty, 10] for two edge failures.

Related Work A scheme for the forbidden-set distance labeling problem that allows the failure of arbitrary sub-graphs is given in [Courcelle and Twigg 07, Twigg 06]. The label size depends on the tree-width or the clique-width of the graph.

Results - Forbidden-Set Distance Labeling Unweighted graphs of doubling dimension α. Stretch of (1+ ε). Label size O(1+ ε-1)2α log2n. All the labels can be computed in polynomial time. Each query can be answered in time polynomial in the length of the labels occurring in the query.

Results - Forbidden-Set Routing The scheme extends to a forbidden- set routing labeling scheme with: stretch 1+ε and O(1+ ε-1)2α log2n-bit routing tables.

Results – Lower Bound The exponential term in α appearing in the label length bound in our schemes is in fact necessary. We show that any forbidden-set connectivity labeling scheme on the family of unweighted graphs of doubling dimension α requires labels of length Ω(2α +logn).

Doubling Dimension The dimension of a metric space is the smallest α > 0 such that every ball of radius 2r can be covered by 2α balls of radius r.

Doubling Dimension Fact For a graph with doubling dimension α, there is an efficiently constructible r-dominating set W(r), such that for every vertex v  V(G) and radius R≥r, the set B(v,R,G)W(r) has size at most (4R/r)α.

Hierarchy of Nets N0=V(G) Ni  Ni-1, for all i>0. Define a hierarchy of nets, namely, vertex sets in G, denoted by Ni, one for each level 0≤i≤logn, with the following properties: N0=V(G) Ni  Ni-1, for all i>0. Ni is a 2i-dominating set for G. N0 N1 N2 N3

Hierarchy of Nets For every vertex v, let Mi(v) be the net-point in Ni closest to v. Note that dG(v,Mi(v)) ≤ 2i. N0 N1 M2(v) M1(v) v N2 N3 M3(v)

Failure-free case: Preprocessing The label L(v) of each vertex v consists of the distances from v to all vertices in B(v,2i+2,G) Ni-c for every c≤i≤logn and Mj(v) for every 0≤j≤logn. N0 N1 v N2 N3

Failure-free case: Query phase Find the smallest index i such that Mi-c(t) is in B(s,2i+2,G) We then return d'=dG(s,Mi-c(t)) + dG(t, Mi-c(t)). w N0 N1 s N2 t N3

Failure-free case Setting c=max{log(2/ε),0} yields the desired stretch of 1+ε

Forbidden-Set Distance Labeling w N0 x N1 s N2 t N3

Forbidden-Set Distance Labeling w N0 x N1 s N2 t N3

Forbidden-Set Distance Labeling w N0 N1 s x N2 t N3

Forbidden-Set Distance Labeling w N0 x N1 s N2 t N3

Forbidden-Set Distance Labeling w N0 x1 N1 s N2 x2 t N3

Forbidden-Set Distance Labeling Unweighted graphs of doubling dimension α have a forbidden-set (1+ ε)-approximate distance labeling scheme of label O(1+ ε-1)2αlog2n. All the labels can be computed in polynomial time, and each query can be answered in time polynomial in the length of the labels occurring in the query.

Results The scheme extends to a forbidden-set routing labeling scheme with stretch 1+ε and O(1+ ε-1)2αlog2n-bit routing tables. The exponential term in α appearing in the label length bound in our schemes is in fact necessary, even for a connectivity labeling scheme. We show that any forbidden-set connectivity labeling scheme on the family of unweighted graphs of doubling dimension α requires labels of length Ω(2α +logn).

Thank You!