Succinct Routing Tables for Planar Graphs Compact Routing for Graphs Excluding a Fixed Minor Ittai Abraham (Hebrew Univ. of Jerusalem) Cyril Gavoille (LaBRI, 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 Routes are constructed in a distributed manner … according to some local routing tables (or routing algorithms) (3,2) (8,5)

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)

… in a Path ► Name-independent: Fixed IDs in {1,…,n} Routing from 5 to any target t? ► Stretch 9 with O(1) space [BYCR93] ► Stretch 1+  with polylog(n) space [AM05] ► Stretch 1 implies  (n) bit space Labeled routing is trivial! stretch 1 with O(1) space

Rem: the scheme is polynomially constructible, even if r is not known [Theorem 1] Every unweighted graph G with n nodes excluding a fixed K r,r minor has a name- independent routing scheme with constant stretch and polylog(n) space local routing tables. Main Contribution Rem: unknown for trees (r=2). Best result: O(n 1/k ) space for stretch 2 O(k) [Laing04]

Graph Minor Theory K 4 is a minor of K 3,3 K r+1 is a minor of K r,r H is a minor of G if H is a subgraph of a graph obtained by edge constractions of G Edge conctraction Edge conctraction A graph G excluding any H minor of r+1 nodes (or less) excludes K r,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 K O(  g) ► Treewidth-r graphs  excludes K r+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  nodes such that: For any source s and target t ∈ G 1. If t is at distance   from s, then t is discovered after a route of length O(  ) 2. If t is at distance >  from s, a negative answer is reported back to s after a walk of length O(  )  Trying with  = 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 K r,r minor and  >0, one can construct a collection of “clusters” H (connected subgraphs) and a collection of trees T of G such that: 1. [cover] the ball of u of radius  /4 is contained in some cluster H in H 2. [sparse]  u  to at most 2 r clusters and 2 rlogr trees 3. [weak diameter]  u,v  H  H are r-tail-connected with trees of T

Tail-Connections with Trees in T G   u,v  H v=x 6 u=x 1 T1T1 T2T2 T3T3 T4T4 T5T5 T6T6  r  x 5 x 4 x 3 x 2 w 3 w 1 w 2 At most r nodes w i ’s x i ’s may be adjacent  r 2   d G (u,v) = O(r 2  )

r 2  If diam H (H) < r 2 , 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 [AGM04] with constant stretch and polylog space per node of H) r 2  However, if diam H (H)  r 2 , 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 Routing in a Cluster H Unfortunately, open problem even for planar graphs (r=3) to find “strong” diameter cluster decomposition [KPR93]

Based on a Partitioning Algorithm: Input: a graph G without K r,r minor Output: a partition in r-tail-connected clusters Inspired by Klein-Plotkin-Rao decomposition S(T,j,i) := {v ∈ T | (j-i)   d T (v,x 0 ) < (j+i+1)  } where x 0 is the root of a tree T Weak Diameter Covering

 2222 2222 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 H:=A For i=1…r, construct T, A, and B … A B T   

Weak Diameter Covering (end) [Lemma] Either G contains a K r,r minor, or every two nodes in H are r-tail-connected with trees T ={T 1,T 2,…,T r }.

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.

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