Space Information Flow: Multiple Unicast

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1 Spac Informaton Flow: Multpl Uncast Zongpng L Dpt. of Computr Scnc, Unvrsty of Calgary and Insttut of Ntwork Codng, CUHK zongpng@ucalgary.ca Chuan Wu Dpartmnt of Computr Scnc Th Unvrsty of Hong Kong cwu@cs.hku.hk Abstract Th multpl uncast ntwork codng conjctur stats that for multpl uncast n an undrctd ntwork, ntwork codng s quvalnt to routng. Smpl and ntutv as t s, th conjctur has rmand opn snc ts proposal n 004 [1], [], and s now a wll-known unsolvd problm n th fld of ntwork codng. In ths work, w provd a proof to th conjctur n ts spac/gomtrc vrson. Spac nformaton flow s a nw paradgm bng proposd [], [4]. It studs th transmsson of nformaton n a gomtrc spac, whr nformaton flows ar fr to propagat along any trajctors, and may b ncodd whrvr thy mt. Th goal s to mnmz a natural bandwdth-dstanc sum-product (ntwork volum), whl sustanng nd-to-nd uncast and multcast communcaton dmands among trmnals at known coordnats. Th conjctur s tru n ntworks only f t s tru n spac. Our man rsult s that ntwork codng s ndd quvalnt to routng n th spac modl. Bsds ts own mrt, ths partally vrfs th orgnal conjctur, and furthr lads to a gomtrc framwork [5] for a hopful proof to th conjctur. I. BACKGROUND AND INTRODUCTION A. Th Multpl Uncast Ntwork Codng Conjctur Dpartng from th classc stor-and-forward prncpl of data ntworkng, ntwork codng ncourags nformaton flows to b mxd n th mddl of a ntwork, va mans of codng [6], [7]. Whl ntwork codng for a sngl communcaton ssson (uncast, broadcast or multcast) s wll undrstood by now, th cas of multpl ndpndnt sssons (mult-sourc, mult-snk) s much hardr, wth lss rsults known [8]. A basc scnaro n th lattr s th multpl uncast sttng, whr multpl ndpndnt on-to-on communcaton dmands coxst n a ntwork. Wth routng, th optmal soluton can b computd by solvng a multcommodty flow (MCF) lnar program; wth ntwork codng, th structur and th computatonal complxty of th optmal soluton ar largly unknown. In drctd ntworks, ntwork codng can augmnt th capacty rgon of multpl uncast. For xampl, Fg. 1(A) shows a ntwork codng soluton for two uncast sssons n a drctd ntwork, whr ach ssson has a throughput of 1. Wthout ntwork codng, t s not hard to vrfy that achvng a throughput of 1 and 1 for both sssons concurrntly s nfasbl, gvn th pr-dfnd lnk drctons. In gnral, th codng advantag, th rato of th maxmum throughput achvabl wth ntwork codng ovr that wth routng, may grow lnarly wth th ntwork sz [9]. Howvr, no >1 codng advantag has bn obsrvd for multpl uncast n th undrctd sttng. For xampl, Fg. 1(B) shows a MCF wth nd-to-nd flow rat of 1 for s 1 t + x x + x + x s t 1 x s 1 t x x x Fg. 1. (Exampl from [1].) Two uncast sssons, from s 1 to t 1 and from s to t, ach wth targt rat 1. All lnk capacts ar 1. (A). Soluton wth ntwork codng. (B) Soluton wthout ntwork codng. ach of th two uncast sssons. L and L [1] and Harvy t al. [] conjcturd that ntwork codng s quvalnt to routng for multpl uncast n undrctd ntworks. Dspt a srs of rsarch ffort dvotd [10] [1], rathr lmtd progrsss hav bn mad towards sttlng ths fundamntal problm n ntwork codng. Bsds asy cass whr th cut st bounds can b achvd wthout ntwork codng [1], [], th conjctur has bn vrfd only n small, fxd ntworks and thr varatons, such as th Okamura- Symour ntwork [10], [11]. It s worth notng that such vrfcaton alrady nvolvs nw tools n ntwork nformaton thory such as nformaton domnanc [10], nput-output qualty and crypto qualty [11]. A growng agrmnt s that nw tools byond a smpl blnd of graph thory and nformaton thory ar rqurd for vntually sttlng th conjctur. In ths work, w prov th gomtrc vrson of th multpl uncast conjctur, by furthr ncorporatng matur tchnqus n gomtry nto th pctur. In 007, Mtznmachr t al. compld a lst of svn opn problms n ntwork codng [1], whr th multpl uncast conjctur appars as problm numbr 1. Chkur commntd that clamng an quvalnc btwn ntwork codng and routng for all undrctd ntworks s a bold conjctur, and that th problm of fully undrstandng ntwork codng for multpl uncast sssons s stll wld opn ( [14], p51-55). B. Spac Informaton Flow Spac nformaton flow s a nw subjct of study bng proposd [], [4]. It consdrs trmnals at known locatons n a gomtrc spac, wth uncast, broadcast or multcast communcaton dmands among thm. Informaton flows can b transmttd along any trajctors n th spac, and may b s t 1 x

2 rplcatd whrvr dsrd, or ncodd whrvr thy mt. Th goal s to mnmz th total bandwdth-dstanc sumproduct, whl sustanng gvn nd-to-nd communcaton rats. Bsds bng a concvabl thortcal problm of ntwork codng n spac, spac nformaton flow modls th mn-cost dsgn of a bluprnt of a communcaton ntwork, whch dsrvs rnwd rsarch attnton gvn ntwork codng []. As w wll dscuss latr, spac nformaton flow also opns th door to gomtrc approachs for studyng ntwork nformaton flow problms, ncludng n partcular th multpl uncast ntwork codng conjctur n graphs. x 1 t 0 1 s 1 s s t 1 t x 1 t 0 1 s 1 s s t 1 Fg.. A -D xampl of spac nformaton flow: mtng communcaton dmands among nods n spac. A mn-cost soluton s to b computd, for thr unt-dmand uncast sssons from s 1 to t 1, from s to t and from s to t, rspctvly (lft). Gvn ntwork codng, s thr a soluton bttr than MCF (rght)? For a quck fl of spac nformaton flow, consdr thr uncast sssons ach wth unt dmand, from s 1 to t 1, from s to t and from s to t, rspctvly, n a -D Eucldan spac as shown n Fg.1. W can rout an nformaton flow along any path n spac, nsrt rlay nods whrvr dsrd, and rplcat or ncod nformaton flows whrvr dsrd. W am to mnmz th volum of th soluton ntwork nducd, f(). Hr s a lnk mployd for flow transmsson, s th lngth of n spac, and f() s th rat of nformaton flow routd across. What s th optmal soluton for satsfyng th thr uncast dmands? Can ntwork codng lad to bttr solutons than routng (MCF)? Rcnt xampls show that ntwork codng can outprform routng whn th dmand n spac s multcast [], [4]. What about multpl uncast? C. Summary of Rsults Our man rsult n ths work s that ntwork codng s quvalnt to routng, for multpl uncast n th nw spac nformaton flow modl. W rstrct our attnton to Eucldan spacs; th cas of non-eucldan spacs, such as n- dmnsonal spacs undr Chbyshv dstanc, s stll bng nvstgatd [5]. W frst analyz th smpl cas of a 1-D spac, whr a sngl pont consttuts a cut. A natural rqurmnt on a vald multpl uncast soluton hr s that, at any gvn pont A, th total amount of flows at A, aggrgatd from both drctons, should b at last th total dmand of uncast sssons whos trmnals rsd on dffrnt sds of A. W tak ntgraton on both sds of ths nqualty along all ponts n th 1-D t spac, and prov that ntwork codng can not mprov upon an optmal soluton basd on routng (MCF). For th gnral cas of a h-d spac, h, our approach s to rduc th problm nto 1-D, by applyng th matur tool of projcton n gomtry. W prov that, f ntwork codng can outprform MCF n h-d, thn t can do so n 1-D, thrby ladng to a contradcton. Mor spcfcally, w show that n a gvn cas whr a ntwork codng basd soluton has a smallr cost than that of MCF, thr must xst a 1-D subspac, onto whch th projcton of th ntwork codng soluton s stll chapr than th projcton of th MCF soluton. Th challng hr s that such a good canddat subspac for projcton s hard to fnd. It s problm dpndnt and no fxd subspac always works. W prov th xstnc of such an lusv subspac wthout xplctly dntfyng t, through an argumnt of ntgratng th projctd ntwork codng and MCF solutons ovr all possbl 1-D rays from orgn. D. Rlvanc and Dscussons In Sc. II-C, w prov that th cost advantag, th potntal advantag of ntwork codng ovr routng n trms of rducng data transmsson cost, s always at last as hgh n ntworks than n spac. Thrfor, our rsult n ths papr partally vrfs th orgnal multpl uncast ntwork codng conjctur n ntworks. Prhaps mor ntrstng s that th nw spac nformaton flow prspctv provds a promsng drcton for attackng th orgnal conjctur tslf. In a sblng work [5], w dscrb a gomtrc framwork that s hopful for vntually rsolvng th orgnal conjctur. W brfly prvw ths gomtrc framwork, as wll ts conncton to ths work, n Sc. III-D. Gvn that ntwork codng s quvalnt to routng for multpl uncast n a Eucldan spac, t s ntrstng to ask whthr th sam holds for multcast. In two sblng work [], [4], w study th multcast problm n spac, wth ntwork codng xplctly consdrd. Thr w prsnt xampls that show ntwork codng and routng ar ndd dffrnt n spac, prov uppr-bounds on th cost advantag, analyz th achvablty of optmalty wth fnt solutons, and dscuss th complxty of optmal multcast n a gomtrc spac. A. Ntwork Informaton Flow II. PROBLEM MODELS W rprsnt an xstng ntwork, drctd or undrctd, usng a graph G =(V,E). Th vctor c Z E + stors capacts of lnks n E. Hr Z + s th st of postv ntgrs. Anothr vctor w Q E + rprsnts th dstanc or cost of lnks n E, and w can b ntrprtd as th cost of routng a unt flow through that lnk. Hr Q + rprsnts th st of postv ratonal numbrs. For th mn-cost multpl uncast problm, w consdr k uncast sssons co-xstng n ntwork G, and lt s and t b th sndr and rcvr of ssson {1,...,k}. W us r to dnot th targt throughput vctor of th k sssons, and r s th rqurd throughput of ssson. Wthout ntwork codng, a soluton to th multpl uncast problm s a multcommodty

3 flow (MCF), whch can b rprsntd usng a lnk flow vctor f Q E +. Th mn-cost MCF can b computd by solvng a lnar program [1], []. A ntwork codng soluton to th multpl uncast problm has two componnts: (A) a flow componnt, for how much flow to transmt ovr ach lnk, and (B) a codng componnt, for whr and how to ncod and dcod th nformaton flows. W dnot th undrlyng lnk flow vctor n (A) usng f Q E + too. In undrctd ntworks, a ntwork codng schm may b dynamc n that th transmsson schm s a tmslottd on (a convolutonal cod), and a dffrnt flow routng and codng schm s adoptd n ach dffrnt tm slot [6], [10]. In ths cas, w smply lt f b th tm-avrag flow rat at lnk. Thr s no known lnar program of polynomal sz that computs th mn-cost ntwork codng soluton. B. Spac Informaton Flow In th spac nformaton flow problm, w ar gvn a st of trmnal nods, wth (multpl) uncast or multcast communcaton dmand. Th spac w consdr n ths work s a h-d Eucldan spac, h 1. A nod u has coordnat (,u,x,u,...,x h,u ). Th Eucldan dstanc btwn two nods u and v s ( h uv h = =1 (x,u x,v ) ) 1/ Gvn a spac nformaton flow vctor f, a ntwork can b nducd, ovr th sam nods and lnks as n f, by vwng f as th capacty of. Th dstanc of s dnotd as h. Th cost of f s thn hf. Ths rflcts th gnral rul that th longr and th wdr a communcaton cabl, th mor xpnsv t s. For th sak of cost mnmzaton, apparntly, only straght ln sgmnts nd to b consdrd n f. Gvn two vctors p and q, p q = p q cos θ s th absolut valu of th nnr product of p and q, whr θ s th angl btwn p and q. C. Paradgm Comparson W can stablsh a conncton btwn th cost advantag n spac and that n graphs. Gvn a problm nstanc, n th form of thr multpl uncast or multcast, lt β d, β u and β s b th max cost advantag possbl n drctd ntworks, undrctd ntworks, and spac, rspctvly. Thn w hav th followng rlaton among th thr: Thorm.1. β d β u β s. Proof: W frst show that β d β u. Gvn th max cost advantag β u n undrctd ntworks, lt u b a problm nstanc whr ths cost advantag s achvd, and lt f b th undrlyng flow of th optmal ntwork codng soluton. W can crat a corrspondng problm nstanc d for th drctd sttng, by vwng f as th drctd ntwork, whl kpng th trmnal nods, lnk costs and targt throughput ntact. Wth ntwork codng, th cost of th optmal soluton s th sam n d and n u. Wthout ntwork codng, th cost of th optmal soluton can only ncras from u to d, snc th lattr s mor rstrctv. Thrfor β d β u. Th proof to β u β s s smlar, by vwng th undrlyng flow f of th optmal ntwork codng soluton for s as an undrctd ntwork. Th drctons n f ar gnord. Th cost of a lnk s takn as h. Gvn Thorm.1, w know that all uppr-bounds on th cost advantag provn for th undrctd modl ar stll vald n th spac modl. Convrsly, all lowr-bounds that w can prov for th spac modl wll also b vald for th undrctd modl. For xampl, an uppr-bound of s known for cost advantag n undrctd multcast ntworks [15] [17]. Ths bound automatcally holds for multcast n a spac of any dmnson. In Sc. III, w prov that th cost advantag for multpl uncast s always 1 n spac. Unfortunatly, ths dos not drctly mply th multpl uncast conjctur n undrctd ntworks. W dscuss how ths bound s connctd to th conjctur n Sc. III-D. III. SPACE INFORMATION FLOW: MULTIPLE UNICAST A. Th Multpl Uncast Conjctur for Ntwork Informaton Flow In thr orgnal work whr th multpl uncast conjctur was proposd [1], L and L frst formulatd th conjctur n th throughput doman, and thn appld lnar programmng dualty to translat t nto an quvalnt vrson n th cost doman. Th Multpl Uncast Conjctur [1], [] Throughput doman: For k ndpndnt uncast sssons n a capactatd undrctd ntwork (G, c), a throughput vctor r s fasbl wth ntwork codng f and only f t s fasbl wth routng. Cost doman: Lt f b th undrlyng flow vctor of a ntwork codng soluton for k ndpndnt uncast sssons wth throughput vctor r, n a cost-wghtd undrctd ntwork (G, w). Thn w f d r, whr d s th shortst path dstanc btwn th sndr and rcvr of ssson undr mtrc w. Intutvly, th throughput vrson of th conjctur clams that ntwork codng cannot hlp mprov throughput, whl th cost vrson clams that ntwork codng cannot hlp rduc transmsson cost. In th rst of ths scton, w prov th cost vrson of th multpl uncast conjctur for spac nformaton flow, whr th cost w bcoms, naturally, th Eucldan lngth of lnk. B. Multpl Uncast n 1-D Spac In a 1-D spac, ach ln sgmnt (or dg) btwn two nghborng vrtcs forms a cut of th ntwork. Th amount of flow f 1 ovr has to b at last th total throughput rqurmnt of trmnal pars sparatd by th rmoval of. W nxt prov that ths mpls th multpl uncast conjctur n 1-D spac.

4 s1 t x0 s Fg.. Thr uncast sssons n 1-D. Total flow crossng pont x 0, f 1 x 0, s lowr-boundd by Dmand((,x 0 ); (x 0, )) = r 1 + r. Thorm.1. Gvn k ndpndnt uncast sssons n 1-D spac, lt f 1 b th undrlyng flow vctor of a ntwork codng soluton achvng a rat vctor r. Thn ( 1f 1 ) ( s t 1 r ). Proof: For a gvn pont x n th 1-D spac, lt f 1 x b th total amount of flow crossng x, n both drctons. Not that th pont x consttuts a cut of th 1-D spac, and thrfor f 1 x s lowr-boundd by th flow dmand btwn th lft sub-spac (, x) and th rght sub-spac (x, ), dnotd as Dmand((,x); (x, )). W ntgrat both szs ovr th ntr 1-D spac, and obtan: x= f 1 xdx x= = t t1 s Dmand((,x); (x, ))dx s t 1 r Furthrmor, not that ( 1f 1 ) = x= f 1 xdx. W conclud that ( 1f 1 ) ( s t 1 r ). C. Multpl Uncast n h-d Spac W now consdr multpl uncast dmands n a h-d spac, for h. Whl only th cass of h =and h =allow ntutv ntrprtatons, th problm s as wll-dfnd for hghr dmnsons, whch s hlpful n connctng to th orgnal multpl uncast conjctur n graphs, bcaus mbddng a graph mtrc nto a gomtrc spac oftn rqurs a hgh dmnson spac [5]. W prov th multpl uncast conjctur by projctng th problm from h-d to 1-D, and thn apply Thorm.1. Th rqurmnt on th projcton s: a codng soluton has total cost lss than th spcfd bound n th conjctur, only f t dos so aftr th projcton. Th man dffculty of th proof s that an optmal or good drcton for projcton s actually hard to fnd. In partcular, t s not suffcnt to always projct onto on of th axs. W show ndrct vdnc nstad, for th xstnc of such a good drcton, by takng an ntgraton ovr all possbl rays at orgn for projcton. Thorm.. For k ndpndnt uncast sssons n a h-d spac, h, assum thr s a ntwork codng soluton wth undrlyng flow vctor f h, w hav (f h h ) ( s t h r ). Proof: Assum, by way of contradcton, that (f h h ) < ( s t h r ). W construct th k pars uncast nstanc and ts ntwork codng soluton n 1-D by projctng thr countr parts from h-d. Our goal s to show that thr xsts a 1-D subspac/drcton n th h-d spac, onto whch th projcton x satsfs (f 1 1 ) < ( s t 1 r ), and thrfor obtan a contradcton to Thorm.1. As shown n Fg. 4, lt b th surfac of th h-d unt hypr-sphr at th orgn. W can numrat all possbl drctons n h-d by travrsng all ponts on, and connctng to thr from th orgn. Lt p b th vctor from orgn to th corrspondng pont on, lt 1 b th unt vctor (1, 0, 0,...,0). Fg. 4. Projct and Intgrat ovr all possbl drtons p. Th ntgraton ovr th closd surfac for all th projctons of f h s: (f h ( p ))d = 1 f h ( p )d = f h h ( 1 p )d = (f h h ) ( 1 p )d Th nc proprty of ths ntgraton s that t s sparabl, n th sns that w can prform ntgraton for ach lnk flow sgmnt frst, and thn tak th summaton (= 1 ). Furthrmor, w obsrv that whn w ntgrat for ach ln sgmnt, th orntaton of that ln sgmnt dos not mattr, snc w vary th projcton drcton to tak all possbl valus (= ). Th ntgraton ovr th closd surfac for all th projctons of { s t =1,...,k} s: ( s )d = ( s )d = ( s t h ( 1 p ))d = s t h ( 1 p ))d Snc (f h h ) < s t h by assumpton, w clam that: (f h ( p ))d < ( s )d Snc th trms bng ntgratd on both sds ar nonngatv, w clam that, thr must xst a partcular drcton p, for whch (f h ( p )) < ( s t p ) p

5 D. Conncton to th Multpl Uncast Conjctur Comparng Thorm. wth th orgnal multpl uncast conjctur n undrctd ntworks, w not that th two statmnts ar rathr smlar. Th only dffrnc ls n th fact that Thorm. s basd on Eucldan dstancs, whras th conjctur s basd on a cost mtrc nducd from a graph. A natural drcton for sttlng th conjctur s thn to mbd th graph mtrc nto a gomtrc spac, and thn utlz Thorm.. An somtrc mbddng of a graph G nto a spac s on that prsrvs parws nod dstancs n G. Th dstanc btwn two nods u and v n G s th shortst path lngth btwn u and v n G. By Thorm., w can s that, f thr xsts a crtan spac to whch an somtrc mbddng of th ntwork s fasbl, and our projcton basd proof tchnqu for dmnson rducton can b adaptd to carry through, thn w can prov th multpl uncast conjctur. Whl no Eucldan spac always prmts somtrc mbddng of graphs, thr do xst non-eucldan spacs that satsfy ths proprty [5]. For mbddng nto Eucldan spacs, f w rlax th somtrc rqurmnt and allow a dstanc dstorton rato up to α n th mbddng, w can prov th cost advantag s uppr-boundd by α n th orgnal graph. Followng ths drcton, w prsnt a gomtrc framwork for studyng th multpl uncast ntwork codng conjctur n a sblng work [5]. Th framwork conssts of four major stps: () translatng th conjctur from throughput doman to cost doman, () mbd th ntwork nto an Eucldan or non-eucldan spac, n an somtrc or low-dstorton mannr, () rduc th problm from hgh-dmnson spac to low dmnson spac, and (v) prov that thr cannot b a countr xampl to th conjctur n low dmnson. Basd on ths framwork, w hav bn abl to formulat a unfd proof to a numbr of rsults, ncludng (1) th conjctur holds for two uncast sssons, () th gap btwn a ntwork codng soluton and a routng solutons s at most O(log k), () th conjctur holds for unform complt graphs, (4) th gap btwn a ntwork codng soluton and a routng solutons s at most n unform grd ntworks, (5) th conjctur s tru n star ntworks [18], and (6) th conjctur s tru n a class of nfntly many layrd or bpartt ntworks. Among ths, (1) and () wr known bfor, but wr provd usng dffrnt tchnqus. Rsults ()- (6) ar nw and not prvously known. Th proofs to rsults (), () and (4) rsort to mbddng nto a Eucldan spac, and drctly buld upon th man rsult n ths work. Th proofs to rsults (1), (5) and (6) rsort to mbddng nto a non-eucldan spac nstad, whr somtrc mbddng of a graph mtrc s fasbl. n ntwork codng. Ths rsult, togthr wth th nw spac nformaton flow prspctv, lads to a nw gomtrc framwork for studyng th orgnal multpl uncast conjctur. 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Mtznmachr, Ntwork Codng: Opn Problms, Tch. Rp., August 007. [14] C. Chkur, Routng vs. Ntwork Codng, Tch. Rp., Unvrsty of Illnos, Urbana-Champagn, tmpl.brs.ca/ 09w510/chkur 09w510 talk.pptx, 009. [15] Z. L and B. L, Ntwork Codng n Undrctd Ntworks, n Proc. of th 8th Annual Confrnc on Informaton Scncs and Systms (CISS), 004. [16] A. Agarwal and M. Charkar, On th Advantag of Ntwork Codng for Improvng Ntwork Throughput, n IEEE Informaton Thory Workshop, Octobr 004. [17] C. Fragoul and E. Soljann, Ntwork Codng Fundamntals, Now Publshrs, 007. [18] S. M. S. T. Yazd, S. A. Savar, and G. Kramr, Ntwork Codng n Star Ntworks, n Proc. of IEEE Intrnatonal Symposum on Informaton Thory (ISIT), 008. IV. CONCLUSION AND FUTURE DIRECTIONS Ths work s among th frst that studs th problm of spac nformaton flow, wth a focus on th cas of multpl uncast sssons. W provd that for multpl uncast n a Eucldan spac, ntwork codng s quvalnt to routng. Ths partally vrfs th wll-known multpl uncast conjctur