Cyril Gavoille

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Forbidden-set labelling in graphs

• Cyril Gavoille
• Bruno Courcelle
• Mamadou Kanté
• (LaBRI, Bordeaux U)
• Andy Twigg
• (Cambridge U, Thomson Research Paris)

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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
  destinations network identifier.

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Ex Grid with X,Y-coordinates
(2,3)
(5,8)
according to some local routing tables (or
routing algorithms)
Routes are constructed in a distributed manner
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and subgraphs of the grid?
(x,y)-coordinates no longer sufficient routing
in planar graphs
(2,3)
(5,8)
according to some local routing tables (or
routing algorithms)
Routes are constructed in a distributed manner
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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

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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.

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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,
kn1/2.
Stretch gt 1 Thorup 04
Planar graphs have (1e)-stretch labeled routing
schemes with O(e-1 log2n) bit labels
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Forbidden-set routing
Shortest path avoiding forbidden blue nodes
(2,3)
(5,8)
according to some local routing tables (or
routing algorithms)
Routes are constructed in a distributed manner
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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?
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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

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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
XØ) i.e. O(n) for general graphs, O(n1/2) for
planar, O(k) for twd-k
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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 T(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?
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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

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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?

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Future work

• Open problems
• O(1)-stretch planar fs-routing with Õ(1) bit
  labels?
• Simplifications? Restrict choices of X, eg
• X lt k (bounded size)
• d(u,X) lt k (bounded distance)
• dG\X(u,v) lt k dG(u,v) (max path inflation k)
• Other simplifications, eg e-slack