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

2. 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

3. 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

4. …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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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