Small distortion and volume preserving embedding for Planar and Euclidian metrics Satish Rao presented by Fjóla Rún Björnsdóttir CSE 254 - Metric Embeddings Winter 2007 CSE 254 - Metric Embeddings - Winter 2007 p. 1/2
Overview Definitions Main Results Proof Further Results Open Problems CSE 254 - Metric Embeddings - Winter 2007 p. 2/2
Main Result Definition. Let G be a class of graphs and let G G. A graph metric is the shortest distance metric d on the vertices V (G) of G. Definition. A planar metric is a graph metric on the class of all planar graphs. Theorem. (Rao s Theorem) Any finite planar metric of cardinality n can be embedded into l 2 with distortion O( log n). This improves on the general O(log n) distortion bound obtained by Bourgain for all metrics. CSE 254 - Metric Embeddings - Winter 2007 p. 3/2
Proof - Outline We will outline a decomposition method which has some nice properties (for planar graphs in particular) a repeated number of decompositions provide coordinates for embedding distant vertices will have independent coordinates Each decomposition satisfies our purpose with a constant probability We then estimate the distortion of the composed embedding CSE 254 - Metric Embeddings - Winter 2007 p. 4/2
Decomposition Pick {1,2,4,...n} Pick v 0 V (G) arbitrarily Pick r {0,1, 2,... 1} uniformly at random Let S 1 = {v V (G) : d(v 0,v) r (mod )} Partition G \ S 1 into connected components For each component, repeat this procedure twice (with the same ) Let finally S = S 1 S 2 S 3 Decomposition is connected components of G\S CSE 254 - Metric Embeddings - Winter 2007 p. 5/2
Properties Each connected component in the decomposition has diameter at most O( ) This results from a theorem by Klein, Plotkin and Rao We will sketch the proof shortly For each x V (G) we have P[d(x,S) c 1 ] c 2 Given a, d(v 1,x) (mod ) will depend upon the choice of BFS-tree root v 1 CSE 254 - Metric Embeddings - Winter 2007 p. 6/2
Property 1 - Outline of proof We do a proof by contradiction We assume the existence of a component of diameter greater than k We will use the BFS-trees on which we constructed the decomposition to expose a K 3,3 minor in G This implies that the graph can not be planar The diameter of each component therefore has to be bounded CSE 254 - Metric Embeddings - Winter 2007 p. 7/2
Property 1 - Proofsketch Suppose there is a component C containing u, v such that d(u, v) 34 Let w be the midpoint of the path between them (within C) d(u, w), d(w, v) 17 Let v 3 be the root of the last BFS-tree used to obtain the component disjoint paths ut 1, wt 2, vt 3 of length 4 in the tree Let h 1,h 2,h 3 be their midpoints, i.e. d(u, h 1 ) = 2, etc We then have that d(h i, h j ) > 12 for all i j CSE 254 - Metric Embeddings - Winter 2007 p. 8/2
Property 1 - Diagram 1 0 1 0 1 v 00 11 0 1 00 11 C 17 0 1 0 1 0 1 00 11 0 1 00 11 w 0 1 00 11 0 1 0 1 2 00 11 17 00 11 0 1 00 11 00 11 00 11 u00 11 0 1 2 h1 2 00 11 2 h3 2 t1 h2 2 t3 t2 T3 v3 CSE 254 - Metric Embeddings - Winter 2007 p. 9/2
Property 1 - Proofsketch Now, let v 2 be the root of the BFS-tree of the previous level It has disjoint paths h 1 t 1, h 2 t 2, h 3 t 3 of length 4 and if we let h 1,h 2,h 3 be their midpoints d(h i,h j) > 8 for all i j Similarly, for the first BF S-tree of the decomposition Look at disjoint paths in the tree Define h 1,h 2,h 3 as before h 1,h 2,h 3 are pairwise more than 4 apart CSE 254 - Metric Embeddings - Winter 2007 p. 10/2
Property 1 - Diagram 2 v2 T2 t 1 t 3 t 2 h 3 h 2 h 1 2 00 11 2 00 11 v 0 12 00 11 00 11 C >12 2 w 00 11 2 h3 >12 00 11 0 1 >12 00 11 u 0 12 h1 h2 v3 CSE 254 - Metric Embeddings - Winter 2007 p. 11/2
Property 1 - Diagram 3 v2 v1 T1 t 3 t 2 t 1 h 3 h 2 h 1 2 2 2 >8 h 1 h 3 >8 00 11 00 11 v >8 0 1 00 11 00 11 C w 00 11 h3 00 11 0 1 00 11 u 0 1 h1 h 2 h2 v3 CSE 254 - Metric Embeddings - Winter 2007 p. 12/2
Red super nodes Definition. A super node of a graph G is a connected subgraph Let us define the following red super nodes: A(v 1 ), A(v 2 ), A(v 3 ) A(v 3 ) is the union of three paths of the tree T 3 from the root v 3 to vertices t 1, t 2, t 3 (but not including t 1, t 2, t 3 ) each of t 1, t 2, t 3 is at distance 4 from one of u, v, w Similarly, define A(v 2 ), A(v 1 ) with respect to the t i s and t i s CSE 254 - Metric Embeddings - Winter 2007 p. 13/2
Blue super nodes Let us define the following blue super nodes: A(u) is the union of A(u), A(v), A(w) the path on tree T 3 joining u and t 1 the path on tree T 2 joining h 1 and t 1 the path on tree T 1 joining h 1 and t 1 A(v) and A(w) are defined similarly CSE 254 - Metric Embeddings - Winter 2007 p. 14/2
Super nodes v2 v1 h 1 h 3 h 2 00 11 00 11 v 0 1 00 11 00 11 C w 00 11 h3 00 11 0 1 00 11 u 0 1 h1 h2 v3 CSE 254 - Metric Embeddings - Winter 2007 p. 15/2
Super nodes are disjoint Claim. A(u), A(v), A(w) (Blue super nodes) are pairwise disjoint Proof. Each blue node is only 8 in diameter (see diagram) Yet, they each contain one of u, v, w, any two of which are > 16 apart Claim. A(v 1 ), A(v 2 ), A(v 3 ) (Red super nodes) are pairwise disjoint Proof. A(v 1 ), A(v 2 ), A(v 3 ) are separated by the decomposition Each of h 1, h 2, h 3 is > 4 away from A(v 2 ) Thus A(v 2 ) T 3 = and a fortiori A(v 2 ) A(v 3 ) = Same argument applies to A(v 1 ) with respect to either of A(v 2 ) and A(v 3 ) Finally, similar arguments will show that any red super node is disjoint from any blue super node CSE 254 - Metric Embeddings - Winter 2007 p. 16/2
Super nodes 00 00 0 00 00 00 00 00 00 00 0 11 11 1 11 11 11 11 11 11 11 1 0 0 0 0 0 0 00 0 0 00 1 1 1 1 1 1 11 1 1 11 00 0 0 00 0 00 00 0 0 0 0 0 0 11 1 1 11 1 11 11 1 1 1 1 1 1 00 0 00 0 0 00 0 0 0 00 11 1 11 1 1 11 1 1 1 11 00 00 0 0 0 00 00 11 11 1 1 1 11 11 0 00 00 0 0 0 00 00 0 0 1 11 11 1 1 1 11 11 1 1 v3 v1 v2 u w v h1 h2 h3 h 3 h 1 h 2 C t 3 t 2 t 1 t 2 t 3 t 1 t2 t3 t1 CSE 254 - Metric Embeddings - Winter 2007 p. 17/2
Red nodes, blue nodes Claim. A(v 1 ) is disjoint from blue nodes A(u), A(v), A(w) Proof. Visibly, the parts that belong to trees T 2 and T 3 can not intersect with A(v 1 ) (because they are all within the same 1 levels of vertices 4 apart from A(v 1 )) Question is: could a vertex x of A(v 1 ) belong to one of t 1h 1, t 2h 2, t Then, d(x, h 1) 4, thus d(x, u) 8 2h 2, say t 1h 1? Without loss of generality, let s say x is on path v 1 h 2 Now d(x, h 2) 5 1 (because h 1 and h 2 are within consecutive 1 levels) So, d(x, v) 9 1 and thus d(u, v) 17 1, a contradiction CSE 254 - Metric Embeddings - Winter 2007 p. 18/2
Red nodes, blue nodes Claim. A(v 2 ) is disjoint from blue nodes A(u), A(v), A(w) Proof. Repeating the arguments for A(v 1 ), we only need to worry about paths t 1h 1, t 2h 2, t 2h 2 intersecting A(v 2) Pick an x in T 2 on t 1h 1 and y in A(v 2) Without loss of generality, let s say y is on path v 2 h 2 Then, restricting our distance metric to T 2 we get: d(v 2, x) d(v 2, h 1) ( 1) (consecutive 1 levels) So, d(v 2, x) d(v 2, h 1 ) 2 ( 1) = d(v 2, h 1 ) 3 + 1 But then d(v 2, x) d(v 2, h 2 ) ( 1) 3 + 1 = d(v 2, h 2 ) 4 + 2 But, y being in A(v 2 ), d(y, h 2 ) 4, so d(v 2, h 2 ) 4 + 2 d(v 2, y) + 2 Thus, d(v 2, x) d(v 2, y) + 2, x and y are therefore distinct CSE 254 - Metric Embeddings - Winter 2007 p. 19/2
Red nodes, blue nodes Claim. A(v 3 ) is disjoint from blue nodes A(u), A(v), A(w) Proof. The exact same technique as in the previous proof covers all the cases we need to consider... CSE 254 - Metric Embeddings - Winter 2007 p. 20/2
Property 1 - End of Proofsketch By contracting the super nodes, we observe a K 3,3 This violates the assumption of the graph being planar (Kuratowski) For each component C, we have Diam(C) < 34 By induction, Klein, Plotkin and Rao in their paper actually proof the following stronger statement: Theorem. If G excludes K r,r as a minor, any connected component obtained through r iterations of the described decomposition method has diameter O(r 3 ) CSE 254 - Metric Embeddings - Winter 2007 p. 21/2
Properties Given a random decomposition, with parameter Each component in the decomposition has diameter at most O( ) For each x V (G) we have P[d(x, S) c 1 ] c 2 We now furthermore have: For any x, y V (G), with d(x, y) 34 x, y / S with constant probability x, y are in different connected components C i, C j d(x, S), d(y, S) c 1 with constant probability Now, for r i, r j random numbers chosen uniformly from [1, 2] r i d(x, S) r j d(y, S) c 1 with constant probability CSE 254 - Metric Embeddings - Winter 2007 p. 22/2
Embedding We will now define the embedding: For each {2 j 1 2 j Diam(G)} perform 4 log n random decompositions For each component C k in a decomposition uniformly pick a random r k from [1, 2] For x C k define its coordinate as r k d(x,s) This defines a mapping f,i : x r k d(x,s) for all and all i {2 j 1 2 j log n} Finally, let f : x «1 2log n f,i(x) :, i CSE 254 - Metric Embeddings - Winter 2007 p. 23/2
Embedding The embedding is a contraction Let x, y in V (G), then f(x) f(y) 2 =,i 1 (2 log n) 2 (f,i(x) f,i (y)) 2 1 4 log 2 n (2d(x, y)) 2,i 1 4 log 2 n 4 log2 n(4d(x, y) 2 ) = d(x, y) 2 CSE 254 - Metric Embeddings - Winter 2007 p. 24/2
Embedding The embedding has distortion O( log n) Let x, y in V (G), and pick a such that 34 < d(x, y) < 68 then f(x) f(y) 2 i 1 (2 log n) 2 (f,i(x) f,i (y)) 2 i 1 (Ω(1)d(x, y))2 (2 log n) 2 1 (d(x, y))2 Ω(1) log n CSE 254 - Metric Embeddings - Winter 2007 p. 25/2
Applications Using this result, we can obtain a O( log n)-approximative max flow min cut theorem for multicommodity flow problems in planar graphs CSE 254 - Metric Embeddings - Winter 2007 p. 26/2
Further results Definition. For a set of k points S in R L the volume Evol(S) is the k 1 dimensional l 2 - volume of the convex hull of S Definition. The volume of a k-point metric space (S,d) is V ol(s) = sup f:s l 2 Evol(f(S)) (the maximum being taken over all contractions f ) CSE 254 - Metric Embeddings - Winter 2007 p. 27/2
Further results Definition. A (k,c)-volume preserving embedding of a metric space (S,d) is a contraction f : X l 2 where for all P S with P = k, (1) ( ) 1/(k 1) V ol(s) c Evol(f(S) The k-distortion of f is (2) ( ) 1/(k 1) V ol(s) sup P S, P =k Evol(f(S) With the help of some results from Feige, we can prove the following: Theorem. Rao s Theorem For every finite planar metric of cardinality n there exists a (k,c)-volume preserving embedding of k-distortion O( log n) CSE 254 - Metric Embeddings - Winter 2007 p. 28/2
References S. Rao. Small distortion and volume preserving embeddings for planar and euclidean metrics. In Proceedings of the 15th Annual Symposium on Computational Geometry, ACM Press, 1999 P. Klein, S. Rao and S. Plotkin. Excluded minors, network decompositions, and multicommodity flow. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993 U. Feige. Approximating the bandwidth via volume respecting embeddings. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998 J. Matousek. Lectures on Discrete Geometry. Springer-Verlag, New York, 2002 CSE 254 - Metric Embeddings - Winter 2007 p. 29/2