Internet Algorithms. (Oblivious) Routing. Lecture 10 06/24/11. Wereferto. demands(requirements), forall vertex pairs,,
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1 Intrnt Algoritms Ltur 10 06/24/11 (Olivious) Routing Givn a ntwork, witdglngtsl and dmands(rquirmnts), forall vrtx pairs,, afasilroutingisa multiommodityflow, satisfying t rquirmnts, i..,,,,, Wrfrto,,,,, and forall. ong max, l,,,, astongstionof. 2
2 (Olivious) Routing An olivious routing sm is a multiommodity flow, wit, 1forall,. Eaflow, dfinsa wigtdsystmof - -pats,,,,,, wr 1. Givnrquirmnts,, wrout flow, along apat,. Altrnativly, wantinkof astproaility, tata pakt from to isroutdalongpat,. Lt, dnottongstionwnrouting dmands aordingtooliviousroutingsm. 3 (Olivious) Routing For dmands, ltopt tongstionofan optimal adaptiv (i.., non-olivious) routing. T oliviousroutingsm issaidto -omptitiv, if forall rquirmnts. ong, opt 4
3 Domposition Trs A dompositiontrofgrap, witdglngts l isa rootdtr,, woslafnods orrspondto. ismdddin : mapsvrtisof totirrprsntativs in, mapsdgsof topatstwnt rprsntativsoftirndpointsin. In t opposit dirtion, mapsvrtisin totorrsponding lafnodsin, mapsdgsof tot(uniqu) pat twntorrspondinglafnodsin. 5 Domposition Trs Not, tat domposition trs ar unwigtd. Nvrtlss, wwill assoiata lngtl,, dfindas l, l,, i.., tlngtoftassoiatdpatin, witadg, ofttr. Wdnoty tsortst-patsmtrion indudy dglngtsl. 6
4 Domposition Trs ijtion twn laf nodsofttrand grap nods aintrnalnodoft trismappdtosom grap nod trdgsarmappd topatsin tgrap twn t orrsponding ndpoints d a f g j i a d f g i j 7 Communiation Trs In tminimum CostCommuniation Tr(MCCT) Prolm, wargivna grap, witdglngtsl and rquirmnts, forall,. Wwanttofind a dompositiontr, minimizing ost,,., Torm 13 Givn an instan,l, oftmcct prolm, a solutionof ost log,,, an omputd in polynomial tim. 8
5 Communiation Trs Routing dmandtwna pair of nods aording to t ommuniation tr. In t xampl, tr dg, ismappdtopat,, ismappdto,, ismappd to,, is mappdto. d a g f i j i a d f g i j 9 Communiation Trs Torm 13 follows from our rsult on approximating aritrary mtris y tr mtris. Our domposition produrrturnsa randomtr, su tat oldsforall,. E, log, Tootaina dtrministitrwita guaranton t wigtd avrag strt, w nd to drandomiz t algoritm. Tis rquirs two standard tniqus: 10
6 Communiation Trs (1) Enumration: Traratmost valusof tat an lad to diffrnt outoms. Wy? Fora fixd, ordrvrtisin inrasing distanfrom, say,,. On lvl, forany,,,, ansttldyt sam st of vrtis. (2) ConditionalExptation: Fix somprfix oft prmutation. Forall,, a., isdtrmind, ifon somlvl oft dompositionxatlyonof, issttldyt prfix, and., dpndsonlyon tproailitytat, arsparatdon somlvl. 11 Communiation Trs A trdg, partitionstlafnodsof (tus, ) intotwodisjointsts,. Lt,, dnotttotal rquirmnttatastorosstisutand load : ttotal loadon dg indudy. Tn ost load l. 12
7 Approximating Bottlnks So far, wavlookdatroutingwitttotal ostof ommuniation(i.., sumofongstions) asojtiv. A mor standard ojtiv is t maximum ongstion on any dg. Can w us domposition trs to approximat t ottlnks of a ntwork, too? Lt a grap, witdgapaitis,. W lt,, if, and, 0, ls. Givna multiommodityflow, wtotmaximumrlativ load ong max, astongstionof.,,, /, 13 Approximating Bottlnks To otain a low ongstion olivious routing sm asd on domposition trs, w nd to do two tings: 1. Construta dompositiontr tatasttr ommuniationprformantan. Any multiommodity flow instan tat an routd wit ongstion in anroutdwitongstion in. 2. Sow tat ansimulat (onstrutivly).givna multiommodityflowwitongstion in, mapping tisflowto (via tmappingof to ) rsultsin ongstionatmost. 14
8 Domposition for Congstion Minimization Givna dompositiontr, of, w dfin tapaity, ofa trdg, as,,,. Tis taks ar of ondition(1). Lmma 9 Lt a multiommodityflowin witongstion ong. Lt a dompositiontrof and tflowotaindymapping to. Tn as ongstionong ong. 15 Domposition for Congstion Minimization a d f g i j Capaitis of tr dgs ar osn larg nougtorout anyflowin witout inrasd ongstion. d a f g j i Tis is immdiat as all flowrossingan dgof asto ross t orrsponding utin. 16
9 Domposition for Congstion Minimization Lt us dfin load : astloadindudon dg y. Not, tattisis tsam loadasin tsolutionoftmcct prolmwit rquirmnts,, forall,. Dfintrlativ loadofan dg indudy as rload load. 17 Domposition for Congstion Minimization Wat aout ondition(2)? Impossil! Hig apaity tr dgs armappdtoa singl pat. a d f g i j a i d g f j 18
10 Domposition for Congstion Minimization Similar to t approximation of gnral mtris y trs (wr a proaility distriution on trs yilds small xptdstrtforadg), tsolutionrustous onvx ominations of domposition trs. Lt,, dompositiontrsof, 1 and onsidr t onvx omination. W would lik to find su a onvx omination minimizing max rload. 19 Domposition for Congstion Minimization Lmma 10 Lt a onvxomination ofdomposition trsof witmaximumxptdrlativ load givn. Furtrmor, ltforatr a multiommodityflow witongstionong 1in givn. Tnt multiommodity flow asongstionatmost in. Lmmas 9 and 10 yild a straigtforward(olivious) routing sm: 20
11 Domposition for Congstion Minimization Givn rquirmnts tatanroutdwitongstion opt in, omputtorrspondingoptimumflow in a (wiistrivial, aus isa tr, so trisa uniqu pat twn a pair of laf nods). Tnmaptsflowsto. Tironvxomination as ongstion at most in. ong max opt ong 21
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