Cutting corners cheaply: How to remove Steiner points

1. Cutting corners cheaply: How to remove Steiner points LiorKamma, Weizmann Institute of Science Joint work with Robert Krauthgamer and Huy L. Nguyễn TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

2. Compressing Graphs • Vast literature on “compression” (succinct representation) of graphs • We focus on preserving specific features – distances, cuts, flows, etc. • Edge sparsification: • Cut and spectral sparsifiers [Benczur-Karger, …, Batson-Spielman-Srivastava] • Spanners and distance oracles [Peleg-Schaffer, …, Thorup-Zwick,…] • Vertex sparsification (keep only the terminals) • Cut/multicommodity-flow sparsifier [Moitra,…,Chuzhoy] • Distances [Gupta, Coppersmith-Elkin] approximately this talk Fast query time Graphical representation or mostly Cutting corners cheaply: How to remove Steiner points

3. Terminal Distances • Graph with edge weights w:+. (“huge” network) • terminals (“important”vertices) • We care about terminal distances: • G,w(,) is the shortest-path distance according to w in . “important” Cutting corners cheaply: How to remove Steiner points

4. Steiner Point Removal (SPR) only terminals • Objective: Find a minor with edge-lengths w’ such that 8s,t2T, ,w≤ w’≤ α∙,w. • Why require a minor? To maintain structure, e.g. planarity. • Otherwise – problem is trivial. A 5 B 6 C 3 “Distortion” G’,w’ dominates G,w 4 G’: (Here α = 1.6) G: 3 2 8 8 A B C (edges of weight 2) Cutting corners cheaply: How to remove Steiner points

5. Main Result Theorem 1.Every -terminal graph with edge-weight w admits a minor w that contains only the terminals and has distortion . Moreover, this minor can be found in polynomial time. Previously: No upper bound was known for general graphs, although a simple argument achieves distortion of . Cutting corners cheaply: How to remove Steiner points

6. Prior Work • Theorem [Gupta’01]:Every treeG has a tree G’ only on T½V with approximation ®=8. • [Chan-Xia-Konjevod-Richa’06]: • The algorithm of [Gupta’01] produces a minor of G. • The factor of 8 is optimal for trees. • Defined the problem for general graphs. • [Basu-Gupta’08]: Algorithm for outerplanar graphs achieving constant distortion. • [Englert-Gupta-Krauthgamer-Räcke-TalgamCohen-Talwar’10]: • For general graphs: a randomized minor with expected stretch O(log |T|). • Improved to O(1) for planar graphs. • Application to sparsifiers preserving multicommodity flow. Cutting corners cheaply: How to remove Steiner points

7. Terminal-Centered Minor • a partial partition such that or all • . • is connected. • Contracting to a single “super-node” produces a minor of . Cutting corners cheaply: How to remove Steiner points

8. Terminal-Centered Minor • Associating “super-node” with we get . • For every edge set w’=,w. Claim: dG’,w’dominatesdG,won. 3 1 1 2 1 3 Cutting corners cheaply: How to remove Steiner points

9. Naïve Approach • for • set to contain all unassigned vertices that are closest to . Claim: Algorithm induces a terminal centered minor. 2 3 Ties are broken by the ordering of Cutting corners cheaply: How to remove Steiner points

10. Naïve Approach • Consider two distinct terminals . • Let be a shortest -path in . • Induce an -path in : • When leaves a cluster and enters another, add an edge in . 2 3 Cutting corners cheaply: How to remove Steiner points

11. Naïve Approach • Eliminating cycles (if there are) leaves with at most edges. 2 3 Cutting corners cheaply: How to remove Steiner points

12. Length of an edge in • is an edge in and By the contraction scheme To get a better bound, we allow more flexibility when constructing The length of in is at most Cutting corners cheaply: How to remove Steiner points

13. Algorithm for Terminal-Centered Minor Approach: Gradually grow a connected cluster “around” each terminal • Proceed in iterations. • Start with for all . • In every iteration, each cluster grows by a random radius • but not into previously-clustered vertices • expected radius increases with iterations. • Stop when . Radius in iteration is distributed exponentially with parameter Cutting corners cheaply: How to remove Steiner points

14. Analysis • Fix two terminals , and a shortest -path P* in G. • Show that with probability the final G’ has an -path of length . • Apply union bound over all terminal-pairs. Denoted . Previous results only analyzed distortion of a single edge (which suffices to bound the expected distortion). Cutting corners cheaply: How to remove Steiner points

15. Analysis of a Single Path P* For sake of analysis, we maintain a path P in G. Start with Updating P with each iteration. • Must “control” how d(P) increases. Why? Detours (paths in G) Cutting corners cheaply: How to remove Steiner points

16. Why bound d(P)? Assume unit length edges in G. Next, contract . Detours (paths in G) Cutting corners cheaply: How to remove Steiner points

17. Why bound d(P)? G,wis always an upper bound on G’,w’. Edges in G’ Cutting corners cheaply: How to remove Steiner points

18. Updating P Throughout Execution G,wis always an upper bound on G’,w’. Cutting corners cheaply: How to remove Steiner points

19. Updating P Throughout Execution Cutting corners cheaply: How to remove Steiner points

20. Updating P Throughout Execution Cutting corners cheaply: How to remove Steiner points

21. Updating P Throughout Execution Number of clusters Number of Intersecting P detours Goal: Reduce the number of clusters intersecting P Joining detours – implies at most concurrent detours. Cutting corners cheaply: How to remove Steiner points

22. Known Technique: Metric Decomposition • A stochastic decomposition of a metric space is a distribution over partitions of . • Theorem [Bartal’96]: For every n-point metric space and ¢>0, there is a distribution μ over partitions of such that: • Diameter bound: For every partition , every cluster has diameter at most . • Separation probability: For every , x y … Cutting corners cheaply: How to remove Steiner points

23. Degree of Separation • , namely • Given a partition of X, define the degree of separation of w.r.t as number of clusters meeting P: P Cutting corners cheaply: How to remove Steiner points

24. Decomposition with Concentration • Theorem 2: For every n-point metric space and >0, there is a distribution μ over partitions of satisfying: • Diameter bound … • Separation probability … • Degree of separation: For every path of length , • Compare: “separation probability” gives a weaker bound (which implies a bound of ). as we discussed Cutting corners cheaply: How to remove Steiner points

25. Sampling a Partition ¦ exponential distribution with parameter ¢/log n truncated at ¢. • Iteratively, for every point do: • Sample a random radius . • Create new cluster from respective ball. is bounded and “almost” memoryless. • only not-yet clustered points a “center” can be outside “its” cluster Cutting corners cheaply: How to remove Steiner points

26. Proving Concentration • Fix a shortest path and . • Divide into two sets: • = points that are “far” from P. • . • For all , Pr meets ≤ . • Using Chernoff, we get: P Cutting corners cheaply: How to remove Steiner points

27. Proving Concentration – cont’d • For all , use “almost” memoryless Prcovers | it reached ≥ . • By independence of the radii, P Cutting corners cheaply: How to remove Steiner points

28. Back to SPR: Bounding d(P) Lemma. With probability , at the end of the final iteration d(P) is at most . Cutting corners cheaply: How to remove Steiner points

29. Future Questions • Close the gap for SPR between 8 and O(log6 k) • For general graphs. • For excluded minors graph families. • Achieve 1+ distortion (“bypass” the LB)? • Allow “few” non-terminals. • Minor based sparsifiers for other graph features (cuts, flows, Steiner trees?) Thank You! Cutting corners cheaply: How to remove Steiner points