Download presentation

Presentation is loading. Please wait.

Published byDeanna Mealer Modified over 3 years ago

1
Vertex sparsifiers: New results from old techniques (and some open questions) Robert Krauthgamer (Weizmann Institute) Joint work with Matthias Englert, Anupam Gupta, Harald Räcke, Inbal Talgam-Cohen and Kunal Talwar. Presented: Newton Institute, Jan. 2011 (w/minor corrections) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A AAAAA A A A A

2
Vertex sparsifiers: New results from old techniques (and some open questions) 2 Graph Bisection Input: Graph G=(V,E) Goal: partition the vertex set into V 1,V 2 with |V 1 |=|V 2 |, so as to minimize e(V 1,V 2 ). (may allow edge-capacities) Central problem, well-studied, NP-hard … Polynomial-time algorithm [Räcke08]: O(log n) approximation

3
Vertex sparsifiers: New results from old techniques (and some open questions) 3 Terminal ( or Steiner ) Bisection Input: Graph G=(V,E) and terminals K µ V Goal: partition the vertex set into V 1,V 2 with |V 1 Å K|=| V 2 Å K |, so as to minimize e(V 1,V 2 ). Same O(log n) approximation [Räcke08]. But can we do f(k) where k=|K|? Similarly, Steiner versions of Linear Arrangement, Oblivious Routing, etc.

4
Vertex sparsifiers: New results from old techniques (and some open questions) 4 Vertex Sparsifiers (w.r.t. Cuts) Input: Graph G=(V,E) and terminals K µ V Goal: A graph H on vertex set K, such that for every partition K=S [ T, MinCut G (S,T) ¼ MinCut H (S,T). (we allow edge-capacities) Why compress graph G onto terminal set K? Information-theory: Efficiently represent 2 k values Computation: Reduce problem size/approximation Stronger version: preserve all multi-commodity flows among terminals K. 5 9 4 8 3 5 2 1 9 8 G 4 3 1 H

5
Vertex sparsifiers: New results from old techniques (and some open questions) 5 Vertex Sparsifiers – Previous work [Moitra09, Leighton-Moitra10] There are (flow) sparsifiers with quality. Can efficiently find one with quality. Yields approximation for Terminal Bisection Similarly, approximation for other problems But H is not simple Even if G is

6
Vertex sparsifiers: New results from old techniques (and some open questions) 6 Our Results Can efficiently find a sparsifier with quality. Can efficiently find a tree-based sparsifier with quality. Yields approximation for Terminal Bisection. Similar improvements for other problems If G is planar, then quality is O(1) and H is planar-based In fact, only use minors of G Holds for every minor-closed family Convex combination of dominating trees Similar results (and lower bounds) were proved independently by Makarychev-Makarychev and by Charikar-Leighton-Li-Moitra.

7
Vertex sparsifiers: New results from old techniques (and some open questions) 7 Best Previous Result Our ResultBest result for k=n Efficient Flow sparsifier- Tree-based flow sparsifier Minor-based flow sparsifier - - Steiner Oblivious Routing Steiner Min Lin. Arr. Steiner MLA in planar graphs Steiner Min cut Lin. Arr. Steiner Bisection

8
Vertex sparsifiers: New results from old techniques (and some open questions) 8 Flow–Distance Duality Connection between: Sparsifier: faithful representation of flows Embedding: faithful representation of distances Transfer Theorem [Räcke08, Andersen-Feige09]. Fix a graph G and a collection M of mappings M:E P(E). Then: For all edge-lengths l :E R + there is a probabilistic mapping with stretch (distortion) ½ ¸ 1 m For all edge-capacities c :E R + there is a probabilistic mapping with quality (congestion) ½ ¸ 1 Moreover, there is efficient algorithm for one iff for the other. convex combination of mappings

9
Vertex sparsifiers: New results from old techniques (and some open questions) 9 Edge Mappings Fix G=(V,E), and let P(E) be all multisets of E (typically paths). A mapping M:E P(E) can be represented as a matrix M in Z E £ E where M e,f = number of occurrences of f in M(e). Illustration: Embed V to a dominating tree T=(V,E T ) For xy 2 E T fix x-y path in G (e.g. shortest) Let M(uv 2 E) = {map u-v path in T into G}. But how to choose M ? G 1 n 2 3 s 13 e M(e) s 1n 2 3 n 1 T …

10
Vertex sparsifiers: New results from old techniques (and some open questions) 10 0-extensions Defn: A 0-extension of (G=(V,E), l G ) with terminals K µ V to be a retraction f:V K; along with a graph (H=(K,E H ), l H ) where l H (x,y)=d G (x,y) for all (x,y) 2 E H. ) d H dominates d G [on pairs in K] G 1 n 2 3 2 3 n 1 H Defn: Stretch of a probabilistic 0-extension is the minimum ® ¸ 1 s.t. E H [d H (f(x),f(y))] · ® d G (x,y)for all x,y 2 V 30 10 20

11
Vertex sparsifiers: New results from old techniques (and some open questions) 11 Tree 0-extensions Example 1: Graph H is a tree call it a tree 0-extension Corollary of [Gupta-Nagarajan-Ravi10]: There is an algorithm that produces tree 0-extensions with stretch ® =O(log k) Idea: Use variant of [Fakcharoenphol-Rao-Talwar04] but Allow distance between non-terminals to contract Remap non-terminals leaves to terminals Purge internal (Steiner) nodes [Gupta01] Now use the Transfer Theorem: M = all tree 0-extensions Distance mappings exist with stretch O(log k) Thus get a tree-based sparsifier with quality O(log k) G 1 n 2 3 2 3 n 1 H 30 10 20

12
Vertex sparsifiers: New results from old techniques (and some open questions) 12 Induced 0-extensions Example 2: Graph H is induced by G via E H ={ (f(u),f(v)) : (u,v) 2 E } call this H=H f an induced 0-extension. Theorem [Fakcharoenphol-Harrelson-Rao-Talwar04]: There is an algorithm producing induced 0-extensions with ® =O(log k / loglog k) Now use the Transfer Theorem: M = all induced 0-extensions Distance mappings exist with stretch O(log k / loglog k) Thus get a sparsifier with quality O(log k / loglog k) G 1 n 2 3 10 20 2 3 n 1 HfHf

13
Vertex sparsifiers: New results from old techniques (and some open questions) 13 Planar Graphs Theorem [Calinescu-Karloff-Rabani04]: There is an algorithm producing induced 0-extensions with ® =O(1) Now use the Transfer Theorem: M = all induced 0-extensions Distance mappings exist with stretch O(1) Thus get a sparsifier with quality O(1) Idea: Make sure H f is a minor of G. Hence planarity is guaranteed. We would like the sparsifier to be planar!!

14
Vertex sparsifiers: New results from old techniques (and some open questions) 14 Connected 0-extension Defn. A 0-extension f:V K is called connected if each f -1 (x) induces a connected subgraph of G. Observe: f is connected ) H f is a minor of G ) H f is planar We give first algorithms for connected 0-extension: For planar graphs: we achieve stretch O(1) For ¯ -decomposable metrics: stretch O( ¯ log ¯ ) For general metrics: stretch O(log k) Via Transfer Theorem: planar-based sparsifier with quality O(1) etc. Not connected: 1 n 2 3 10 20 Connected: 1 n 2 3 10 20

15
Vertex sparsifiers: New results from old techniques (and some open questions) 15 Implications to Metric Embedding Theorem [Gupta01]: For every tree T and terminals K, there is a tree on K that represents all distances faithfully (factor 8) 2 3 n 1 T … This work: For every planar graph G and terminals K, there is a (probabilistic) planar graph on K that represents all distances faithfully (expected O(1) stretch) Simplifies embedding results 3 4 n 2 T … 2 2 2

16
Vertex sparsifiers: New results from old techniques (and some open questions) 16 Connected 0-extension in Planar Metrics Algorithm (Input: Graph G with edge-lengths l and terminals K) 1. Init: f(v)=v for v 2 K and f(v)= ? for v 2 V n K. 2. For each r=1,2,…,2 i,…,diam(V) 3. sample ¯ -decomposition P of d G with diameter r 4. for each C 2 P containing both mapped and unmapped vertices 5. delete from C mapped vertices 6. for each connected component C in C 7. choose vertex w C 2 C that was deleted and has edge to C 8. reset f(u)=f(w C ) for all u 2 C G Connectivity: by construction Diameter: at time r, vertices are mapped to terminals within O(r) Stretch: Prob. to settle (u,v) at late time r is 1/r 2 (must be separate twiced) Pr[P(x) P(y)] · ¯ d G (x,y) / r

17
Vertex sparsifiers: New results from old techniques (and some open questions) 17 Open Problems Steiner Points Removal: Given planar graph G and terminals K, build a single planar graph only on K that represents all distances faithfully Apparently possible for outerplanar graphs [Basu-Gupta08] More generally: same for general G, using minors s -sparse extension: Given a graph G and terminals K, choose S ¶ K of size s, and a 0-extension (retraction) into this S Is there a poly(k)-sparse extension of expected stretch O(1)? Is there a single (non-probabilistic) planar sparsifier graph? More generally: extend duality between Distances and Capacities, perhaps to level of a single graph, or to preserve minors Analogous questions for cuts (e.g. SPR, few pseudo-terminals) Analogous questions for Euclidean metrics (e.g. what is minor)

Similar presentations

Presentation is loading. Please wait....

OK

Theory of Computing Lecture 12 MAS 714 Hartmut Klauck.

Theory of Computing Lecture 12 MAS 714 Hartmut Klauck.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on water pollution in delhi Ppt on purchase order Ppt on bluetooth devices The heart anatomy and physiology ppt on cells Ppt on hanging gardens of babylon Ppt on food adulteration in india Ppt on robert frost poems Ppt on first conditional activities Ppt on image sensor By appt only business cards