Internet Algorithms. Lecture 10 06/24/11

Transcription

1 Internet Algorithms Lecture 10 06/24/11

2 (Oblivious) Routing Given a network, withedgelengthsl and demands(requirements), forall vertex pairs,, afeasibleroutingisa multicommodityflow, satisfying the requirements, i.e.,,,,, Wereferto,,,,, and forall. cong max, l,,,, asthecongestionof. 2

3 (Oblivious) Routing An oblivious routing scheme is a multicommodity flow, with, 1forall,. Eachflow, definesa weightedsystemof - -paths,,,,,, where 1. Givenrequirements,, weroute flow, along eachpath,. Alternatively, wecanthinkof astheprobability, thata packet from to isroutedalongpath,. Let, denotethecongestionwhenrouting demands accordingtoobliviousroutingscheme. 3

4 (Oblivious) Routing For demands, letopt bethecongestionofan optimal adaptive (i.e., non-oblivious) routing. The obliviousroutingscheme issaidtobe -competitive, if forall requirements. cong, opt 4

5 DecompositionTrees A decompositiontreeofgraph, withedgelengths l isa rootedtree,, whoseleafnodes correspondto. isembeddedin : mapsverticesof totheirrepresentatives in, mapsedgesof topathsbetweenthe representativesoftheirendpointsin. In the opposite direction, mapsverticesin tothecorresponding leafnodesin, mapsedgesof tothe(unique) path betweenthecorrespondingleafnodesin. 5

6 DecompositionTrees Note, that decomposition trees are unweighted. Nevertheless, wewill associatea lengthl,, definedas l, l,, i.e., thelengthoftheassociatedpathin, witheachedge, ofthetree. Wedenoteby theshortest-pathsmetricon inducedby edgelengthsl. 6

7 DecompositionTrees bijection between leaf nodesofthetreeand graph nodes eachinternalnodeofthe treeismappedtosome graph node treeedgesaremapped topathsin thegraph between the corresponding endpoints d a e c b f g h j i a b c d e f g h i j 7

8 Communication Trees In theminimum CostCommunication Tree(MCCT) Problem, wearegivena graph, withedgelengthsl and requirements, forall,. Wewanttofind a decompositiontree, minimizing cost,,., Theorem 13 Given an instance,l, ofthemcct problem, a solutionof cost log,,, can be computed in polynomial time. 8

9 Communication Trees Routing demandbetweena pair of nodes according to the communication tree. In the example, tree edge, ismappedtopath,, ismappedto,, ismapped to,, is mappedto. d b a h c g e f i c e h j i a b c d e f g h i j 9

10 Communication Trees Theorem 13 follows from our result on approximating arbitrary metrics by tree metrics. Our decomposition procedurereturnsa randomtree, such that holdsforall,. E, log, Toobtaina deterministictreewitha guaranteeon the weighted average stretch, we need to derandomize the algorithm. This requires two standard techniques: 10

11 Communication Trees (1) Enumeration: Thereareatmost valuesof that can lead to different outcomes. Why? Fora fixed, orderverticesin increasing distancefrom, say,,. On level, forany,,,, canbesettledbythe same set of vertices. (2) ConditionalExpectation: Fix someprefix ofthe permutation. Forall,, a., isdetermined, ifon somelevel ofthe decompositionexactlyoneof, issettledbythe prefix, and b., dependsonlyon theprobabilitythat, areseparatedon somelevel. 11

12 Communication Trees A treeedge, partitionstheleafnodesof (thus, ) intotwodisjointsets,. Let,, denotethetotal requirementthathastocrossthiscutand load : thetotal loadon edge inducedby. Then cost load l. 12

13 ApproximatingBottlenecks So far, wehavelookedatroutingwiththetotal costof communication(i.e., sumofcongestions) asobjective. A more standard objective is the maximum congestion on any edge. Can we use decomposition trees to approximate the bottlenecks of a network, too? Let a graph, withedgecapacities,. We let,, if, and, 0, else. Givena multicommodityflow, wetothemaximumrelative load cong max, asthecongestionof.,,, /, 13

14 ApproximatingBottlenecks To obtain a low congestion oblivious routing scheme based on decomposition trees, we need to do two things: 1. Constructa decompositiontree thathasbetter communicationperformancethan. Any multicommodity flow instance that can be routed with congestion in canberoutedwithcongestion in. 2. Show that cansimulate (constructively).givena multicommodityflowwithcongestion in, mapping thisflowto (via themappingof to ) resultsin congestionatmost. 14

15 DecompositionforCongestionMinimization Givena decompositiontree, of, we define thecapacity, ofa treeedge, as,,,. This takes care of condition(1). Lemma 9 Let bea multicommodityflowin withcongestion cong. Let bea decompositiontreeof and theflowobtainedbymapping to. Then has congestioncong cong. 15

16 Decomposition for Congestion Minimization c e b a b c d e f g h i j h Capacities of tree edges are chosen large enoughtoroute anyflowin without increased congestion. d a e c b f g h j i This is immediate as all flowcrossingan edgeof hasto cross the corresponding cutin. 16

17 Decomposition for Congestion Minimization Let us define load : astheloadinducedon edge by. Note, thatthisis thesame loadasin thesolutionofthemcct problemwith requirements,, forall,. Definetherelative loadofan edge inducedby as rload load. 17

18 Decomposition for Congestion Minimization What about condition(2)? Impossible! High capacity tree edges aremappedtoa single path. c e b a b c d e f g h i j h a b h i d c g e f j 18

19 Decomposition for Congestion Minimization Similar to the approximation of general metrics by trees (where a probability distribution on trees yields small expectedstretchforeachedge), thesolutionhereustouse convex combinations of decomposition trees. Let,, bedecompositiontreesof, 1 and consider the convex combination We would like to find such a convex combination minimizing. max rload. 19

20 Decomposition for Congestion Minimization Lemma 10 Let a convexcombination ofdecomposition treesof withmaximumexpectedrelative load begiven. Furthermore, letforeachtree a multicommodityflow withcongestioncong 1in begiven. Thenthe multicommodity flow hascongestionatmost in. Lemmas 9 and 10 yield a straightforward(oblivious) routing scheme: 20

21 Decomposition for Congestion Minimization Given requirements thatcanberoutedwithcongestion opt in, computethecorrespondingoptimumflow in each (whichistrivial, because isa tree, so thereisa unique path between each pair of leaf nodes). Thenmaptheseflowsto. Theirconvexcombination has congestion at most cong max opt cong in. 21