Forbidden-set labelling in graphs Cyril Gavoille Bruno Courcelle Mamadou Kanté (LaBRI, Bordeaux U) Andy Twigg (Cambridge U, Thomson Research Paris)

The Compact Routing Problem Input: a network G (a connected graph) Output: a routing scheme for G A routing scheme allows any source node to route messages to any destination node, given the destination’s network identifier. Forbidden-set labelling in graphs

Ex: Grid with X,Y-coordinates (2,3) (5,8) Routes are constructed in a distributed manner … according to some local routing tables (or routing algorithms) Forbidden-set labelling in graphs

…and subgraphs of the grid? (x,y)-coordinates no longer sufficient; routing in planar graphs… (2,3) (5,8) Routes are constructed in a distributed manner … according to some local routing tables (or routing algorithms) Forbidden-set labelling in graphs

Quality & Complexity Measures Near-shortest paths: |route(x,y)| ≤ stretch . dG(x,y) Size of the labels and routing tables Goal: constant stretch & compact (polylog) tables Trivial upper bound: Each node x stores the neighbour on the next-hop towards each destination y  O(n log n) bits Forbidden-set labelling in graphs

Labeled vs. Name-independent Labeled: Node IDs can be chosen by the designer of the scheme (as a routing label whose length is a parameter) Name-independent: Node identifiers are chosen by an adversary (the input is a graph with the IDs) Name-independent is harder than labeled variant. This talk: labeled schemes only. Forbidden-set labelling in graphs

Forbidden-set labelling in graphs Routing / distances on planar graphs Stretch-1 [Gavoille et al, J Alg ’04] Shortest-path labeled routing on weighted planar graphs requires labels of (n1/2) bits. Treewidth-k graphs have stretch-1 labeled routing schemes with O(k log2n) bit labels. For planar, k=n1/2. Stretch > 1 [Thorup ’04] Planar graphs have (1+ε)-stretch labeled routing schemes with O(ε-1 log2n) bit labels Forbidden-set labelling in graphs

Forbidden-set routing Shortest path avoiding forbidden blue nodes (2,3) (5,8) Routes are constructed in a distributed manner … according to some local routing tables (or routing algorithms) Forbidden-set labelling in graphs

Forbidden-set labelling in graphs Forbidden-set routing Input: a network G (a connected graph) Output: a forbidden-set routing scheme for G A forbidden-set routing scheme allows any source node to route messages to any destination node v, avoiding any set X of forbidden nodes, given the identifier of v and the identifiers of nodes in X. e.g. Are u,v connected in G\X? What is dG\X(u,v)? Next hop? Forbidden-set labelling in graphs

Forbidden-set labelling in graphs Motivation Routing around failures Routing schemes are generally static; recomputation of labels / routing tables is costly. The set X can be a set of failed nodes/edges Best known techniques only handle single failures e.g. “fast reroute”, Cisco not-via Internet routing ASes want control over where their packets travel; shortest-path routing not expressive enough BGP allows AS i to specify that its packets avoid AS j Forbidden-set labelling in graphs

Forbidden-set labelling in graphs Known results (forbidden-set) Upper bounds O(n log n) no longer trivial! The trivial upper bound is to store the entire graph at each node  O(n2) bits. Lower bounds Distance labeling lower bounds apply (take X=Ø) i.e. Ω(n) for general graphs, Ω(n1/2) for planar, Ω(k) for twd-k Forbidden-set labelling in graphs

Forbidden-set labelling in graphs Known results (forbidden-set) [Courcelle, T, STACS ’07] Treewidth-k & cliquewidth-k graphs: forbidden-set stretch-1 routing schemes with O(k2 log2n) bit labels. Compare to Θ(k) for vanilla routing [Gavoille, T, 2007] Planar graphs: forbidden-set stretch-1 labeled routing scheme with labels of Õ(n1/2) bits. Equals optimal bound for vanilla stretch-1 planar distances! [This paper] Planar graphs: forbidden-set connectivity labeling scheme with labels of O(log n) bits. Can u reach v in G\X? Forbidden-set labelling in graphs

Planar forbidden-set connectivity Fact: every planar graph G has a planar dual G*. A set of edges E is a cut in G iff the dual edges E* form a cycle in G*. Construct new planar graph M by subdividing edges of G* and taking union with G Associate with each edge e of G the coordinates of its dual edge M has a straight-line embedding in an n x n grid [Schneider], hence the labels are O(log n) bits Forbidden-set labelling in graphs

Planar forbidden-set connectivity Let X be a set of edges of G, and G 3-connected. u,v are reachable in G\X iff X* contains a cycle separating u,v in G* Can be extended to handle forbidden vertices Question: time to answer queries? Is O(|X|) possible? Forbidden-set labelling in graphs

Forbidden-set labelling in graphs Conclusions A new collection of problems in compact routing Open problems O(1)-stretch planar fs-routing with Õ(1) bit labels? … Simplifications? Restrict choices of X, eg |X| < k (bounded size) d(u,X) < k (bounded distance) dG\X(u,v) < k dG(u,v) (max path inflation k) Other simplifications, eg ε-slack… Forbidden-set labelling in graphs