1 Lower-Stretch Spanning Trees Daniel A. Spielman Joint work with Michael Elkin (Ben-Gurion) Yuval Emek (Weizmann) Shang-Hua Teng (BU)
2 Spanning tree T G
3 Stretch of a spanning tree v v u u T G stretchT(uv) length of uv path in T 4 Average stretch of a spanning tree v v 2 3 1 u 2 u 1 2 1 1 1 2 T G stretchT(uv) length of uv path in T ave-stretch(T) ave(uv) in G stretchT(uv) 5 Example n-cycle 6 Example n-cycle n-1 1 1 1 1 1 1 1 Every spanning tree has some edge of stretch n-1. ave-stretch 7 How low average stretch Alon-Karp-Peleg-West 92 This paper Conjecture 8 Why AKPW Analysis of k-server algorithm Bartal 95 Approximate metrics by tree metrics Polynomial approximation algorithms Boman-Hendrickson 01 Solve diagonally dominant linear systems in time S-Teng 03 Solve in time 9 Why AKPW Analysis of k-server algorithm Bartal 95 Approximate metrics by tree metrics Polynomial approximation algorithms Boman-Hendrickson 01 Solve diagonally dominant linear systems in time S-Teng 03 Solve in time 10 Once more v v 2 3 1 u 2 u 1 2 1 1 1 2 T G stretchT(uv) length of uv path in T ave-stretch(T) ave(uv) in G stretchT(uv) 11 Star decomposition at x0 V1 V7
Partition V into
V2 x0 V6 V0 V3
x0 in V0
each Vi touches V0
12 d-star decomposition at x0 For Require 13 Cost of star decomposition cost number of edges between components V1 x1 V7 x7 V2 x2 x0 x6 V6 V0 x3 V3 x4 x5 V5 V4 14 Star decomposition theorem Build a d-star decomposition at x0 of cost where in time 15 Star decomposition theorem Build a d-star decomposition at x0 of cost where in time large r decreases cost small d increases cost 16 Trees from star decompositions 17 Trees from star decompositions 18 Radius of the tree If we choose Also true in each component 19 Total stretch of the tree Stretch of edges cut at first level Sum over levels 20 Total stretch of the tree Stretch of edges cut at first level Sum over levels improve to 21 How to build a low-cost star decomposition 22 Classic technique ball growing (Awerbuch 84) For r Such that for bdry decreases as r increases 23 Classic technique ball growing (Awerbuch) Grow level-by-level until small Boundary at one level is inside interior at next All boundaries big implies ball grows quickly 24 Classic technique ball growing (Awerbuch) Theorem Such that for we have Proof. If for then contradiction 25 Star-decomposition first attempt Grow ball remove grow ball in remaining graph etc. Wont be star-connected! 26 Star-decomposition idea Cone from x1 everything reach from x1 moving away from x0 v some shortest path to x0 goes though x1 x1 v x0 Removing cone 1. Sets touch 2. Do not increase dist of any node to x0 27 Star-decomposition second attempt Grow ball remove cut remaining graph into cones from boundary Could cut too many edges! x1 x0 Cone from x1 everything reach from x1 moving away from x0 28 Star decomposition technique Grow cones! Cone(S) all can reach from S moving away from x0 v some shortest path to x0 goes though S x1 29 Cone growing theorem For r r0 x1 r-r0 dr Such that for And can go up to dr steps back so at most drr-r0 forward 30 Remove grown cones until nothing left x3 x2 31 Star decomposition theorem Build a d-star decomposition at x0 of cost in time 32 Low-stretch tree from d-star decompositions ave-stretch 33 Edges with lengths Issue shrinking radius does not bound recursion depth Solution contract all edges of length lt radius/n2 each edge active in at most rounds 34 How to improve such that for If V1 very small lowers recursion depth If V1 big grow another dr levels for smaller boundary ave-stretch 35 what if planar 36 Implementation Heuristic (Jan 04) Star-decomposition linsolve 2-4x faster huh
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