The Cost Advatage of Networ Codig i Uiform Combiatorial Networs Adrew Smith, Bryce Evas, Zogpeg Li, Baochu Li Departmet of Computer Sciece, Uiversity of Calgary Departmet of Electrical ad Computer Egieerig, Uiversity of Toroto Abstract Codig advatage refers to the potetial for etwor codig to improve ed-to-ed throughput or reduce routig cost. How large ca the codig advatage be? We ivestigate this fudametal questio i the classic udirected etwor model. So far, all ow etwors where such potetial exists are based o a special class of topologies ow as combiatorial etwors. We try to prove a rather small upper-boud close to 1 for the codig advatage for the class of combiatorial etwors ad its variatios. Such a result, iterestigly, will lead us to the followig dilemma: either we are still missig the most appropriate etwor topologies for demostratig the power of etwor codig, or we have bee igorig a very effective perspective for efficietly approximatig the miimum Steier tree problem, after a few decades of research i Steier trees. We elaborate o the above argumets ad preset the early stage results of our research: the codig advatage i terms of routig cost is upper-bouded by 1.15 i the class of uiform combiatorial etwors. I. INTRODUCTION Networ Codig [1], [] is a field of iformatio ad codig theory which allows the expasio of the fuctio of idividual odes i a computer etwor beyod the stadard operatios used i routig. It does this by allowig odes to perform ecodig operatios o icomig iformatio flows, which provides more flexible solutios for propagatig iformatio throughout the etwor. The beefits of etwor codig have bee show to be maifold. Amog them two saliet oes are icreasig the ed-to-ed throughput ad reducig the routig cost, especially for multicast trasmissios or i wireless ad hoc etwors. I the cotext of throughput, the codig advatage is defied as the ratio of the maximum edto-ed throughput with ad without the use of etwor codig; i the cotext of routig cost, the codig advatage is defied as the ratio of the miimum cost ecessary to achieve a certai throughput, without ad with etwor codig. I the latter case we also refer to the codig advatage as the cost advatage of etwor codig. Whe the codig advatage is 1, etwor codig does ot mae a differece; whe the codig advatage is large, etwor codig provides a substatial boost i throughput or a dramatic reductio i cost. I the classic udirected etwor model that we study here, the codig advatage is best show whe the commuicatio sessio is i the form of oe-to-may multicast. Previous research has show that a for directed etwors, the codig advatage is essetially ubouded ad ca grow at rate Θ, where is the size of a multicast etwor [3]; b for udirected etwors, the codig advatage is fiite ad its best upper-boud prove so far is [4], [5]. Note that wireless ad hoc etwors with uiform commuicatio radius, i.e., the uit-dis graphs, ca be modelled usig a udirected etwor sice i reachability betwee eighborig odes are always symmetrical, ad ii data trasmissio i the two directios betwee a pair of eighbors share the total available chael capacity. A more accurate model, however, would tae ito accout details i wireless iterferece ad the wireless broadcast advatage. Although the best theoretical boud so far is, the largest value of the codig advatage observed i practice, icludig both cotrived ad radom etwors, is oly 1.15 or 9 8 for fiite etwors, ad 1.143 or 8 7 for ifiite etwors. Therefore it is probable that the tight upper-boud of the codig advatage is much closer to 1 tha to. I this paper, we prove that the cost advatage of etwor codig is tightly upper-bouded by 1.15 i the class of uiform combiatorial etwors. This class of etwors is particularly importat sice almost all ow examples which show codig advatage a larger tha 1 belog to it or ca be reduced to a istace of it. Defiitio A Combiatorial Networ with parameters,, deoted C,, is a udirected graph G V, E. C, cosists of a seder, relay odes, ad re-
Fig. 1. A example of a combiatorial etwor, C 4,3. Here the codig advatage is 1.15, i terms of both throughput ad cost. Fig.. The Butterfly Networ, with a topology isomorphic to Fig. 3. ceivers. Each of the relay odes are coected to the seder directly, while each of the receivers is coected to a uique subset of size of the relay odes. Further, if all edges i E have uiform capacity ad cost, we refer to the etwor as a uiform combiatorial etwor. While the ultimate goal remais to provide a improved boud o codig advatage i geeral udirected etwors, we iitially restrict our attetio to combiatorial etwors for several reasos. First, most udirected etwor topologies exhibit a codig advatage of 1, which is equivalet to o improvemet from the use of etwor codig, while combiatorial etwors geerally exhibit a o-trivial codig advatage. Secod, the largest codig advatage yet demostrated was show i two combiatorial etwors, specifically C 4,3 show i Fig. 1 ad C 4,. Furthermore, almost all 1 udirected topologies with o-trivial codig advatage cotai simple variats of combiatorial etwors, suggestig that the structure of these etwors is fudametally related to the potetial of etwor codig to improve multicast routig solutios. For example, the celebrated butterfly etwor example show i Fig., frequetly cited ad through which etwor codig is itroduced to the public [1], has a topology that is isomorphic to C 3,. The example topology depicted by Jaggi et al. [3] whe showig that the codig advatage is ubouded i directed etwors, is exactly C 6,3. The cotrived topologies tested by Li et al. [6] for the codig advatage icluded C 3,, C 4,, C 4,3, C 5, ad C 5,3. The reaso that larger combiatorial etwors were ot tested is computatioal: the geeral Steier tree algorithm employed requires examiig 1 The oly exceptio we ow of is the ifiite etwor topology that leads to a codig advatage of 1.143, as metioed earlier i the paper. Fig. 3. The combiatorial etwor C 3,, a possible drawig. early 50 millio differet trees already for C 5,3. We develop a closed-form formula for the cost advatage i this paper istead, by exploitig the specific structure of uiform combiatorial etwors, thereby avoidig expesive computatios. We are worig to prove that for all combiatorial etwors ad their certai variatios, the codig advatage is upper-bouded by a umber close to 1. Sice provig a tight boud i fact aythig smaller tha for the codig advatage for geeral etwor topologies has prove difficult, worig o the combiatorial class seems to be a reasoable first step, especially sice all ow topologies where etwor codig maes a improvemet are variats of this class. Below we discuss aother importat motivatio for our research o the codig advatage, which is related to efficietly approximatig the miimum Steier tree ad Steier tree pacig problems. Both variats of the Steier tree problems have bee show to be NP-hard, ad it is also ow that a α-approximatio exists for oe of them i polyomial time if ad oly if the same is true for the other [7]. O the other had, both multicast throughput ad cost with etwor codig ca be computed i polyomial time [6], [8]. Therefore, a boud of α o the codig advatage i terms of throughput or cost, directly implies the existece of
3 a polyomial-time α-approximatio algorithm for the Steier tree pacig ad miimum Steier tree problems, respectively. Note that the Steier tree pacig ad miimum Steier tree problems correspod to multicast routig without etwor codig. If we accomplish the proof of a small upper-boud α o the codig advatage for combiatorial etwors ad their variats, the the followig questio ca be ased: Does there exist aother fudametally differet class of etwor topologies, for which a larger codig advatage ca be observed? If the aswer is Yes, the it implies that after a decade of research i etwor codig, we are still missig the essece of it, i that the most appropriate etwors for etwor codig to show its power have ot bee discovered ad studied yet. If the aswer is No, the that leads to a much better approximatio algorithm for Steier trees tha the state-of-the-art 1.55- approximatio [9], achieved after a few decades of research i the area of Steier trees. Note that the result preseted i this paper provides a exact formula for the cost of the miimum Steier tree, i the special class of uiform combiatorial etwors. We defie our etwor model i the ext sectio, ad preset a early stage result i Sec. III: the cost advatage of etwor codig is tightly upper-bouded by 1.15 for uiform combiatorial etwors. I Sec. IV, we describe how we pla to exted this boud from uiform combiatorial etwors to geeral combiatorial etwors with heterogeeous costs ad capacities o lis. Cocludig remars are i Sec. V. II. NOTATION AND NETWORK MODEL We deote the topology of a multicast etwor usig a udirected graph G V, E. M {S, T 1,..., T } V cotais termial odes i the multicast group, of which S is the multicast seder. The desired multicast throughput rate is d. Each li e E has a capacity Ce ad a cost we that are both positive ratioal umbers. I this papers we focus o uiform combiatorial etwors with uiform li capacities ad costs, ad assume that Ce ad we are always 1 for all lis. Without loss of geerality, we also scale the throughput d to 1 the importat poit is that li capacities are o less tha the desired throughput ad therefore will ot become a limitig factor i routig. Now, for a give multicast problem P defied upo a certai etwor cofiguratio topology, throughput, li costs ad capacities, defie π P to be the miimum cost required to achieve the throughput without etwor codig, ad defie χ P to be the miimum cost required whe employig etwor codig. The, the cost advatage of etwor codig is give as π P /χ P. III. A TIGHT UPPER-BOUND FOR UNIFORM COMBINATORIAL NETWORKS Now focus o a multicast problem P that is defied upo a uiform combiatorial etwor C,, with desired multicast throughput 1 ad uiform li cost of 1. We prove a tight upper-boud of 1.15 for the cost advatage of etwor codig. Theorem III.1. The cost advatage of etwor codig for the uiform combiatorial etwor C, has a closed-form represetatio of + + 1 + Proof: Theorem III.1 follows from lemmas III. ad III.3, which prove the miimum cost of multicast i combiatorial etwors with ad without etwor codig respectively. Lemma III.. The miimum multicast cost of rate 1 i a uiform combiatorial etwor C, with etwor codig is +. We begi by provig the existece of a multicast solutio of the give cost, followed by a proof of its miimality. Proof of existece: First ote that the combiatorial etwor C, cotais lis to each of the receivers ad a sigle li to each of the relays for a total of + lis. Clearly, a flow assigmet of 1 o every li results i the desired throughput. I additio, every relay ode has a icomig li with a flow of 1 ad ca thus provide the ecessary flow of 1 o each of its outgoig edges. Proof of miimality: Cosider the followig statemet of the MAX-FLOW MIN-CUT theorem [10]. A flow rate d from ode S to ode T is achievable iff every cut betwee S ad T has size at least d. Further, ote that a multicast rate d is achievable if ad obviously oly if a uicast rate d is feasible from
4 the seder to each receiver, i a directed etwor [1], []. This allows us to cosider each flow from the seder ode S to some receiver T i idepedetly. Oe cut that separates S from T i is to remove all lis eterig T i. I order to sustai a throughput of 1, the sum of the flows assiged to these lis must be at least 1. By applyig the same reasoig to each of the receivers, the total flow o all lis coectig the receivers must be oticig that for each Ti there is o overlap of lis icluded i the cut with ay other receivig ode. Every subset of relay odes must also have a icomig flow that sums to 1, also by the MAX-FLOW MIN-CUT theorem. Because there are of these subsets, that would imply a miimum additioal flow of if they shared o lis. However, each relay ode ad its icomig lis participates i 1 1 of these subsets ad so the flow o icomig lis to that particular relay is shared by those subsets over couted 1 1 times. The resultig miimum flow implied by these cuts is thus 1 1!!! 1! 1! 1 1!! 1!! 1!!! Therefore, by the MAX-FLOW MIN-CUT theorem, the miimum possible cost for uit flow i C, is + Lemma III.3. The miimum multicast cost of rate 1 i a uiform combiatorial etwor C, without etwor codig is + + 1. Proof: It is has bee show that the miimum cost of multicast without etwor codig is equivalet to the cost of the miimum Steier tree. We demostrate that i ay combiatorial etwor, a Steier tree of ca be costructed with o fewer tha + + 1 lis. To coect each of the receivers, each receiver must have a icomig li from a relay ode. This gives the first lis. We wat to coect the miimum umber of relay odes while still havig a coected relay available to each receiver. If we coect r of the relay odes to the seder, the receivers that remai without coectio are those that choose their available lis from the r discoected relays. By the symmetric costructio of the etwor, idepedet of which r odes are coected, there remai r discoected receivers. Thus whe r there remais 1 receiver discoected. Hece there must be at least + 1 relays coected to coect the Fig. 4. Codig Advatage for Uiform Combiatorial Networs Base Case Steier tree to all receiver odes. The total umber of lis i this miimum Steier tree is + + 1. So we ow have a formula for computig codig advatage for ay combiatorial etwor with ubouded capacity ad uiform cost. The ext result shows that this formula, ad hece the codig advatage, is bouded iclusively by 9 8. Theorem III.4. The etwor codig cost advatage for the uiform combiatorial etwor C, is bouded iclusively by 9 8. Proof: It is easy to verify that the value of this formula for ad up to 16 is o greater tha 9 8. This is demostrated i Figure 4, which displays the Codig Advatage as a height map for values of ad up to 16. This forms the base case for our iductio o for all C,. We ow demostrate that this upper boud holds for all values of ad. First observe that a proof that the above formula is bouded by 9 8 ca be achieved by
5 provig the followig series of equivalet expressios: + + 1 + 9 8 + + 1 + + + 9 8 1 + 1 + 1 8 8 + 1 + 1 8 + 1 + 8 8 + 8 8 8 8 9 + 8 Now, treatig separately the cases where 1 or 1, we assume that. Begiig with the right had side, otice that:! 1!! Usig this as a boud, we show that 8 8 9 + 8 1. Still worig uder the assumptio that : 8 8 9 + 8 8 16 9 + 8 8 8 9 8 8 8 1 16 1 1, 16 Thus, for 16 ad for all,, we have established a boud of 9 8 for the formula. For the case where 1 or 1 we have the followig: So we show that 8 8 9 8 8 9 1 1 + 8. For 1: + 8 8 8 9 + 8 ad for 1 8 8 9 + 8 16 9 1 16 which shows that the boud also holds i this case whe 16. Hece we have the desired upper-boud of 1.15 for the class of uiform combiatorial etwors. Sice a cost advatage of 1.15 is achieved i both C 4, ad C 4,3, we ow the boud is tight. IV. NEXT STEPS As discussed earlier i Sec. I, our evetual goal is to prove a small upper-boud for the codig advatage for both throughput ad cost i all etwors that are either combiatorial etwors or their variatios. A simple variatio ca be obtaied, for example, by replacig a li i C, with two sequetial lis, each with the origial capacity ad cost. Such a variatio is modelled with heterogeeous li costs doublig the cost o the origial li has the same effect. We are curretly worig o further geeralizig the boud towards all combiatorial etwors, with both heterogeeous li costs ad heterogeeous li capacities. After that, we shall leverage the results obtaied for cost advatage towards provig a similar boud o the throughput advatage. V. CONCLUSIONS At this poit the result is prove for the subclass of uiform combiatorial etwors where li capacities are o-limitig ad the cost is equal across all lis. We also expect to prove that havig heterogeeous costs with or without li capacity costraits will still lead to combiatorial etwors with codig advatage less tha or equal to 9 8. The result i this paper improves upo the upperboud established by Li et al. [6] for geeral udirected etwors. While the class the boud is idetified with is highly structured, it is oe of the few topologies that demostrate a o-trivial codig advatage. Hece, boudaries o this class of structures may have further reachig cosequeces with respect to the maximum possible codig advatage for multicast i udirected etwors. Aother motivatio for our research o boudig the codig advatage, as discussed i the itroductio, has bee show to relate to the topic of desigig
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