Download presentation

Presentation is loading. Please wait.

Published byJena Capell Modified over 2 years ago

1
All-or-Nothing Multicommodity Flow Chandra Chekuri Sanjeev Khanna Bruce Shepherd Bell Labs U. Penn Bell Labs

2
Routing connections in networks 25 NY SE DE 10 20 5 6 NY – SF 10 Gb/sec NY – SF 20 SE – DE 5 SF – DE 6 Core Optical Network

3
Multicommodity Routing Problem Network – graph with edge capacities Requests: k pairs, (s i, t i ) with demand d i Objective: find a feasible routing for all pairs Optimization: maximize number of pairs routed

4
All-or-Nothing Flow Problems Pair is routed only if all of d i satisfied Single path for routing: unsplittable flow (connection oriented networks) Fractional flow paths: all-or-nothing flow (packet routing networks) Integer flow paths: all-or-nothing integer flow (wavelength paths)

5
Complexity of AN-Flow d i = 1 for all i Single path: edge disjoint paths problem (EDP) classical problem, NP-hard only polynomial approx ratios AN-MCF: APX-hard on trees approximation ?

6
Approximating EDP/AN-MCF O(min(n 2/3,m 1/2 )) approx in dir/undir graphs (EDP/UFP) [Kleinberg 95, Srinivasan 97, Kolliopoulos-Stein 98, C-Khanna 03, Varadarajan-Venkataraman 04] EDP is (n 1/2 - )-hard to approx in directed graphs [Guruswami-Khanna-Rajaraman-Shepherd-Yannakakis 99] LP integrality gap for EDP is (n 1/2 ) [GVY 93] AN-MCF: APX-hard on trees [Garg-Vazirani-Yannakakis 93]

7
Results In undirected graphs AN-MCF has an O(log 3 n log log n) approximation Polynomial factor to poly-logarithmic factor Approx via LP, integrality gap not large For planar graphs O(log 2 n log log n) approx Same ratios for arbitrary demands: d max · u min Online algorithm with same ratio

8
LP Relaxation x i : amount of flow routed for pair (s i, t i ) max i x i s.t x i flow is routed for (s i,t i )1 · i · k 0 · x i · 11 · i · k

9
A Simple Fact Given AN-MCF instance: all d i = 1 Can find (OPT) pairs such that each pair routes 1/log n flow each How? rand rounding of LP and scaling down Problem: we need pairs that send 1 unit each

10
Nice Flow Paths Suppose all flow paths use a single vertex v v s1s1 s2s2 s3s3 s4s4 t1t1 t2t2 t3t3 t4t4

11
Routing via Clustering v cluster has log n terminals cluster induces a connected component clusters are edge disjoint

12
Clustering Finding connected edge-disjoint clusters? G is connected: use a spanning tree for a rough grouping of terminals New copy of G for clustering: congestion 2 1 for clustering, 1 for routing Congestion 1 using complicated clustering

13
How to find nice flow paths? Algorithmic tool: Racke’s hierarchical graph decomposition for oblivious routing [Räcke02]

14
Räcke’s Graph Decomposition Represent G as a capacitated tree T leaves of T are vertices of G internal node v: G(v) is induced graph on leaves of T(v) 10 3 2 4 7 4 v

15
Räcke’s Result T is a proxy for G For all D c*(D,G) · c(D,T) · (G) c*(D,G) Routing in T is unique (G) = O(log 3 n) [Räcke 02] (G) = O(log 2 n log log n) [Harrelson-Hildrum-Rao 03]

16
Routing details With each v there is distribution v on G(v) s.t i 2 G(v) v (i) = 1 s distributes 1 unit of flow to G(v) according to v t distributes 1 unit of flow to G(v) according to v v s t

17
Back to Nice Flow Paths v s1s1 s2s2 s3s3 s4s4 t1t1 t2t2 t3t3 t4t4 G(v), v s1s1 s2s2 s3s3 s4s4 t1t1 t2t2 t3t3 t4t4 X(v): pairs with v as their least common ancestor (lca) Routing in TRouting in G

18
Algorithm Find set of pairs X that can be routed in T (use tree algorithm [GVY93,CMS03] ) Each pair (s i,t i ) in X has a level L(i) Choose level L* at which most pairs turn Route pairs independently in subgraphs at L* L* v

19
Algorithm cont’d v at L*, X(v) pairs in X that turn at v Can route 1/ (G) flow for each pair in X(v) using nice flow paths Use clustering to route X(v)/ (G) pairs Approx ratio is (G) depth(T) = O(log 3 n log log n)

20
Open Problems Improve approximation ratio What is integrality gap of LP ? No super-constant gap known Extend ideas to EDP Recent result: Poly-log approximation for EDP/UFP in planar graphs with congestion 3

Similar presentations

OK

Online Packet Admission and Oblivious Routing in Sensor Networks Mohamed Aly Department of Computer Science University of Pittsburgh And John Augustine.

Online Packet Admission and Oblivious Routing in Sensor Networks Mohamed Aly Department of Computer Science University of Pittsburgh And John Augustine.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on metro cash and carry Ppt on sources of energy class 10 Ppt on intelligent manufacturing ppt Ppt on movable bridge Ppt on semi solid casting video View ppt on ipad Ppt on data handling in maths Ppt on chapter 3 atoms and molecules youtube Ppt on line drawing algorithm in computer Ppt on rotating magnetic field