Compact Routing for Graphs Excluding a Fixed Minor

1. 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)

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.

3. 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)

4. 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

5. 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)

6. … in a Path • Name-independent: Fixed IDs in {1,…,n} Routing from 5 to any target t? 17 19 14 9 2 8 4 7 5 15 6 10 11 12 1 13 18 16 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

7. 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 (r=2). Best result: O(n1/k) space for stretch 2O(k)[Laing04]

8. 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 Kr+1 is a minor of Kr,r A graph G without Kr,r minor excludes any H minor with r+1 nodes (or less)

9. 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 Kr+2 • 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 H=H(F)

10. Try & Fail Technique Design a (name-independent) routing scheme for distance at mostr 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 > 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.

11. The Weak Diameter Cover [Theorem 2] For G excluding a Kr,r minor and r>0, 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

13. Tail-Connections with Trees in T T2 T5 T3 G T6 T4 T1 w1 w2 w3 r r r r x2 x3 x5 x4 v=x6 u=x1 u,vH At most r nodes wi’s xi’s may be adjacent dG(u,v) = O(r2r)

14. Routing in a Cluster H If diamH(H) < 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-Seymour ’04]: H-free minor graphs edge-partition in 2 bounded treewidth graphs Unfortunately, open problem even for planar graphs (r=3) to find “strong” diameter cluster decomposition [KPR93]

15. 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) < (j+i+1)r} where x0 is the root of a tree T

16. r 2r 2r r r r For i=1…r, construct T, A, and B … T A:=B:=G For i:=1 to r do T:=a BFS tree of B rooted in A A:=a CC of BS(T,j,0) B:=a CC of BS(T,j,i)  A H:=A A B

17. Weak Diameter Covering (end) [Lemma] Either G contains a Kr,r minor, or every two nodes in H are r-tail-connected with trees T={T1,T2,…,Tr}. QED

18. 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.

19. 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-pathlabeled routing scheme requires labels of (n1/6) bits.