Succinct Routing Tables Compact Routing forfor Graphs Planar

Succinct Routing Tables Compact Routing forfor Graphs Planar Graphs a Fixed Minor Excluding Ittai Abraham (Hebrew Univ. of Jerusalem) Cyril Gavoille (La. BRI, University of Bordeaux) Dahlia Malkhi (Hebrew Univ. of Jerusalem, Microsoft Research)

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.

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)

Quality & Complexity Measures Time vs. Space ► Near-shortest paths: |route(x, y)| ≤ stretch. d. G(x, y) ► Size of the local routing tables ► Goal: constant stretch & polylog size tables

Labeled vs. Name-independent Routing 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)

7 … in a Path ► Name-independent: Fixed IDs in {1, …, n} Routing from 5 to any target t? 19 14 9 2 8 4 7 5 15 6 10 11 12 1 13 18 Labeled routing is trivial! stretch 1 with O(1) space [BYCR 93] ► Stretch 9 with O(1) space ► Stretch 1+ with polylog(n) space [AM 05] ► Stretch 1 implies (n) bit space 16

Main Contribution [Theorem 1] Every unweighted graph G with n nodes excluding a fixed Kr, r minor has a nameindependent routing scheme with constant stretch and polylog(n) space local routing tables. Rem: unknown the scheme foristrees polynomially (r=2). Best result: 1/k) space for O(k) known constructible, O(n even stretch if r is 2 not [Laing 04]

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 A graph K 4 is a. Gminor excluding of K 3, 3 any H of r+1 nodes (or Kminor is a minor of K r+1 r, r less) excludes Kr, r

Well known H-free minor graphs Trees K 3 -free minor graphs ► Series-parallel graphs K 4 -free minor graphs ► Planar graphs excludes K 5 (and without K 3, 3) ► Genus-g graphs excludes KO( g) ► Treewidth-r graphs excludes Kr+2 ► ► Not only! There are K 5 -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)

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 1. If t is at distance r from s, then t is discovered after a route of length O(r) 2. 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, …, 2 i …, any t will be found with a constant stretch factor and with an increasing factor of logn on the space.

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: 1. 2. 3. [cover] the ball of u of radius r/4 is contained in some cluster H in H [sparse] u to at most 2 r clusters and 2 rlogr trees [weak diameter] u, v H H are r-tail-connected with trees of T

Tail-Connections with Trees in T T 2 T 3 T 1 T 5 T 4 w 1 G T 6 w 2 w 3 rr rr u=x 1 At most r nodes wi’s xi’s may be adjacent x 2 x 3 x 4 x 5 v=x 6 u, v H d. G(u, v) = O(r 2 r)

Routing in a Cluster H If diam. H(H) < r 2 r, then the source routes to the root of a BFS tree T 0 for H, then looks for the target with a single-source routing in T 0 (doable using the single-source name-independent routing scheme in trees [AGM 04] with constant stretch and polylog space per node of H) 2 r, then However, if diam (H) r Unfortunately, open problem evenstill fordoable planar H via tail-connections, with diameter some efforts … graphs (r=3) to find and “strong” cluster [De. Vos-Ding-Sanders-Reed-Robertson-Seymour ’ 04]: H-free decomposition [KPR 93] minor graphs edge-partition in 2 bounded treewidth graphs

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 d. T(v, x 0) < (j+i+1)r}

For i=1…r, construct T, A, and B … r r T 2 r r r 2 r A: =B: =G For i: =1 to r do 1. T: =a BFS tree of B rooted in A 2. A: =a CC of B S(T, j, 0) 3. B: =a CC of B S(T, j, i) A 4. H: =A A B

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={T 1, T 2, …, Tr}.

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.

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 (n 1/6) bits.