Compact Routing for Graphs Excluding a Fixed Minor

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Compact Routing for Graphs Excluding a Fixed Minor
Succinct Routing Tables for Planar Graphs

• Ittai Abraham
• (Hebrew Univ. of Jerusalem)
• Cyril Gavoille
• (LaBRI, University of Bordeaux)
• Dahlia Malkhi
• (Hebrew Univ. of Jerusalem, Microsoft Research)

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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
(3,2)
(8,5)
Routes are constructed in a distributed manner
according to some local routing tables (or
routing algorithms)
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Quality Complexity Measures

• Time vs. Space
• Near-shortest paths
• route(x,y) stretch . dG(x,y)
• Size of the local routing tables
• Goal constant stretch polylog size tables

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Labeled vs. Name-independentRouting Schemes

• Name-independent Node identifiers are chosen by
  an adversary (the input is a graph with the IDs)
• Labeled Node IDs can be chosen by the designer
  of the scheme (as a routing label whose length is
  a parameter)

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in a Path

• Name-independent Fixed IDs in 1,,n

Routing from 5 to any target t?
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Labeled routing is trivial! stretch 1 with O(1)
space

• Stretch 9 with O(1) space BYCR93
• Stretch 1? with polylog(n) space AM05
• Stretch 1 implies ?(n) bit space

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Main Contribution

• Theorem 1

Every unweighted graph G with n nodes excluding a
fixed Kr,r minor has a name-independent routing
scheme with constant stretch and polylog(n) space
local routing tables.
Rem the scheme is polynomially constructible,
even if r is not known
Rem unknown for trees (r2). Best result
O(n1/k) space for stretch 2O(k) Laing04
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Graph Minor Theory
H is a minor of G if H is a subgraph of a graph
obtained by edge constractions of G
Edge conctraction
Edge conctraction
K4 is a minor of K3,3 Kr1 is a minor of Kr,r
A graph G without Kr,r minor excludes any H minor
with r1 nodes (or less)
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Well known H-free minor graphs

• Trees ? K3-free minor graphs
• Series-parallel graphs ? K4-free minor graphs
• Planar graphs ? excludes K5 (and without K3,3)
• Genus-g graphs ? excludes KO(?g)
• Treewidth-r graphs ? excludes Kr2
• Not only!
• There are K5-free minor graphs with unbounded
  treewidth and unbounded genus
• The Minor Graph Theorem R S
• ? Every family of graphs F closed under minor
  taking excludes some fixed minor HH(F)

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Try Fail Technique

• Design a (name-independent) routing scheme for
  distance at most r nodes such that
• For any source s and target t ? G
• If t is at distance ? r from s, then t is
  discovered after a route of length O(r)
• If t is at distance gt r from s, a negative answer
  is reported back to s after a walk of length O(r)
• ? Trying with r 1,2,4,,2i , any t will be
  found with a constant stretch factor and with an
  increasing factor of logn on the space.

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The Weak Diameter Cover

• Theorem 2 For G excluding a Kr,r minor and rgt0,
  one can construct a collection of clusters H
  (connected subgraphs) and a collection of trees T
  of G such that
• cover the ball of u of radius r/4 is contained
  in some cluster H in H
• sparse ? u ? to at most 2r clusters and 2rlogr
  trees
• weak diameter ?u,v?H ? H are r-tail-connected
  with trees of T

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Tail-Connections with Trees in T
T2
T5
G
T3
T6
T4
T1
w1
w2
w3
? r r
? r r
x5
x3
x2
x4
vx6
ux1
? u,v ? H
At most r nodes wis xis may be adjacent
? dG(u,v) O(r2r)
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Routing in a Cluster H
If diamH(H) lt r2r, then the source routes to the
root of a BFS tree T0 for H, then looks for the
target with a single-source routing in
T0 (doable using the single-source
name-independent routing scheme in trees AGM04
with constant stretch and polylog space per node
of H) However, if diamH(H) ? r2r, then still
doable via tail-connections, and with some
efforts DeVos-Ding-Sanders-Reed-Robertson-Seymo
ur 04 H-free minor graphs edge-partition in 2
bounded treewidth graphs
Unfortunately, open problem even for planar
graphs (r3) to find strong diameter cluster
decomposition KPR93
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Weak Diameter Covering
Based on a Partitioning Algorithm Input a
graph G without Kr,r minor Output a partition in
r-tail-connected clusters Inspired by
Klein-Plotkin-Rao decomposition S(T,j,i) v ?
T (j-i)r ? dT(v,x0) lt (ji1)r where x0 is
the root of a tree T
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For i1r, construct T, A, and B
T

• ABG
• For i1 to r do
• Ta BFS tree of B rooted in A
• Aa CC of B?S(T,j,0)
• Ba CC of B?S(T,j,i) ? A
• HA

A
B
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Weak Diameter Covering (end)
Lemma Either G contains a Kr,r minor, or every
two nodes in H are r-tail-connected with trees
TT1,T2,,Tr.
QED
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Conclusion

• A new intrusion of Minor Theory in Computer
  Science, here in Distributed Computing.
• Surprising for routing and related problems
  because edge-contraction and near-shortest
  path are a priori two opposite concepts.
• Open problems understand the shortest path
  metric of Planar graphs.

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Labeled Routing Planar Graphs
Thorup JACM 04
Planar graphs have 1? stretch labeled routing
schemes with polylog labels.
Theorem 3
There are bounded degree planar triangulations
with n nodes for which every shortest-path
labeled routing scheme requires labels of ?(n1/6)
bits.