ConMap: A Novel Framework for Optimizing Multicast Energyin Delay-constrained Mobile Wireless Networks

Xinzhe Fu1, Zhiying Xu2, Qianyang Peng1, Jie You2, Luoyi Fu1, Xinbing Wang1,2, Songwu Lu3Department of {Computer Science1, Electrical Engineering2}, Shanghai Jiao Tong University, China

Department of Computer Science, University of California, Los Angeles, USA3

Email: {fxz0114, xuzhiying, az950512, youjie9412, yiluofu, xwang8}@sjtu.edu.cn, slu@cs.ucla.edu

ABSTRACTThis paper studies the problem of optimizing multicast energy con-sumption in delay-constrained mobile wireless networks, whereinformation from the source needs to be delivered to all the k des-tinations within an imposed delay constraint. Most existing workssimply focus on deriving transmission schemes with the minimumtransmitting energy, overlooking the energy consumption at thereceiver side. Therefore, in this paper, we propose ConMap, anovel and general framework for efficient transmission schemedesign that jointly optimizes both the transmitting and receivingenergy. In doing so, we formulate our problem of designing min-imum energy transmission scheme, called DeMEM, as a combina-torial optimization one, and prove that the approximation ratio ofany polynomial time algorithm for DeMEM cannot be better than14 lnk . Aiming to provide more efficient approximation schemes,the proposed ConMap first converts DeMEM into an equivalent di-rected Steiner tree problem through creating auxiliary graph gad-gets to capture energy consumption, thenmaps the computed treeback into a transmission scheme. The advantages of ConMap arethreefolded: i) Generality– ConMap exhibits strong applicabilityto a wide range of energy models; ii) Flexibility– Any algorithmdesigned for the problem of directed Steiner tree can be embed-ded into our ConMap framework to achieve different performanceguarantees and complexities; iii) Efficiency– ConMap preservesthe approximation ratio of the embedded Steiner tree algorithm,to which only slight overhead will be incurred. The three featuresare then empirically validated, with ConMap also yielding near-optimal transmission schemes compared to a brute-force exact al-gorithm. To our best knowledge, this is the first work that jointlyconsiders both the transmitting and receiving energy in the designof multicast transmission schemes in mobile wireless networks.

1 INTRODUCTIONWith the surge of mobile data traffic, energy consumption in com-munication is rising radically in recent years. As reported by asurvey [1] where Telecom Italia is said to be the second largestconsumer in Italy, energy consumption of mobile communicationis already ranking the top. Hence, energy saving becomes a crucialissue that receives considerable attention in the design and imple-mentation of mobile wireless networks.

As the majority of nodes are battery powered, a key problem ofenergy cost reduction inmobile wireless networks is to find a routewith minimum total energy consumption for a given communica-tion session [2]. Under such circumstance, multicast turns out tobemore efficient due to the increasing demand of information shar-ing in those communication scenarios. It outperforms unicast byaggregating multiple flows from the same source so that network

resources such as capacity and energy can be saved. However, asa traffic pattern that generalizes both unicast and broadcast, theissue of minimum-energy multicast is far more complicated com-pared to that of minimum-energy unicast, which is essentially ashortest path problem in static network.

Among existing works that indeed study the minimum-energymulticast problem, most of them [3, 4] are limited to static net-works, which fail towell characterizemany realistic scenarioswhereusers are manifested to be moving. Consequently, the potentialenergy saving brought with mobility is also overlooked. Anotherconcern lies in that the state of art mostly focuses on optimizingthe energy consumed only at the transmitting side [5, 6], but do nottake into consideration the receiving energy, which, as suggestedby empirical studies [7, 8], is as significant as the transmission cost.Furthermore, due to the possible collisions and retransmissions inmulticast/broadcast protocols, the energy consumption is not syn-onymous at transmitting and receiving sides. Interpreted in amoredetailed way, the energy consumption of the transmitting side isusually determined by transmitting power, while at the receivingside, taking reliable MAC layer multicast as an example [9], theenergy consumption may depend on the number of intended re-cipients as each transmission involves waiting for feedback fromthe receivers and resending themissing packets. Such inherent het-erogeneity between transmitting and receiving energy can also befound in many other real applications such as wireless multihopnetworks [10] and full-duplex energy harvesting infrastructures[11]. Therefore, it is difficult to directly embed the receiving en-ergy into existing algorithms. These motivate us to propose a gen-eral framework that jointly optimizes both the transmitting andreceiving energy in mobile wireless networks.

In addition to the energy, our framework needs to incorporatethe latency, another crucial performance metric, of which the de-mand varies in different scenarios. Real time applications suchas video conferencing and traffic monitoring require the networklatency to be fairly small, whereas in other networks (e.g. satel-lites network), network delay is not so critical. Hence, it is rea-sonable to add to the network a delay constraint, meaning that allthe packages must be delivered within a time length of D. Basedon that, all the above cases can be well characterized by simplyadjusting the size of D. Following this, there emerge a flurry ofstudies that quest for the relationship between delay constraintand other network performances. For instance, Small et al. showthat relaxing delay constraint can bring about energy reduction orthroughput gain [12]. Ra et al. propose an online algorithm forachieving near optimal energy-delay tradeoff in smartphone appli-cations [13]. However, current literature on energy-delay tradeoffis limited to some specific applications, where the main goal is to

empirically optimize the tradeoff without the corresponding the-oretical exploration [13, 14]. Under such circumstance, two ques-tions still remain open:

• How hard it is to achieve the minimum multicast energyunder delay constraint in mobile wireless networks?

• How should we efficiently design such optimal multicasttransmission schemeswithminimumenergy consumptionin delay-constrained mobile wireless networks?

We therefore present a first look into the problem of multicast en-ergy consumption optimization in delay-constrained wireless mo-bile networks, with both transmitting and receiving energy consid-ered. Specifically, we consider a multicast session with k destina-tions selected in advance from all then nodes in the network. Withtime divided into equal slots, we impose a delay constraintD on themulticast session in the sense that packets from the source must bedelivered to all the k destinations within D time slots. We aim toefficiently design multicast routing schemes that achieve the min-imum transmitting and receiving energy. Unlike traditional multi-cast in wired network, which is essentially a minimum Steiner treeproblem, the issue becomes more complicated when it comes tomobile wireless mulitcast. First, network topology is time-varyingdue to the mobility of nodes. This makes the task of designingan optimal transmission scheme rather challenging since the sametree may correspond to many different transmission schemes. Sec-ond, the “broadcast nature” in wireless communication, to a largeextent, complicates the calculation of energy consumption sinceit cannot be obtained by simply taking the summation of edgeweights in a minimum Steiner tree problem.

We approach the problem of multicast energy optimization indelay-constrained wireless mobile networks by formulating it asa combinatorial optimization problem called the DeMEM problem.Upon theoretical analysis on the approximation hardness of theDe-MEM problem, we propose a novel energy optimizing frameworkcalled ConMap. Particularly, ConMap realizes multicast transmis-sion in three major steps. First, it constructs an intermediate graphresulted from snapshots of graph states taken under D differentslots and auxiliary gadgets for transmitting and receiving energy;Second, it builds a Steiner tree in that intermediate graph spanningall the destinations; Finally, it maps the obtained tree in that inter-mediate graph back into a multicast transmission scheme in theoriginal network. The proposed ConMap flexibly allows us to em-bed any directed Steiner tree algorithm into it, and turns out to beefficiently implementable with only a polynomial time complexityas well as preserving the approximation ratio of the underlying di-rected Steiner tree algorithms. Our main contributions are listedas follows:

• We prove that the approximation ratio of DeMEM prob-lem in mobile wireless networks cannot be better than14 lnk for any polynomial-time algorithm, which is an evenstronger result than demonstration of its NP-hardness.

• We propose ConMap, a novel framework that jointly opti-mizes the transmitting and receiving energy in the multi-cast session and harnesses the results of existing directedSteiner tree algorithms to achieve a good approximationfor DeMEM. Our framework does not rely on any specificcharacterization of energy consumption and is thus appli-cable to a broad range of applications.

• We empirically evaluate the performance of ConMap onreal datasets, where ConMap exhibits good applicabilityto real network traces and yields near optimal transmis-sion schemes compared to a brute-force exact algorithm.We also demonstrate the significance of jointly optimizingtransmitting and receiving cost by showing that incorpo-rating receiving energy into ConMap leads to transmis-sion schemes of lower energy consumption.

The rest of this paper is organized as follows: Section 2 is con-tributed to literature review and we introduce our network and en-ergy models in Section 3. We prove the approximation hardness ofDeMEM in Section 4. The approximation framework, ConMap, isproposed in Sections 5 and 6. We conduct simulations to evaluatethe performance of our framework in Section 7 and give conclud-ing remarks in Section 8.

2 RELATEDWORKIn the literature, the problem of transmitting energy consumptionoptimization is referred to as Minimum-Energy Multicast (MEM)problem, and has been extensively studied in static networks. Sinceboth MEM problem and Minimum-Energy Broadcast (MEB) prob-lem, a special case of MEM, are proven to be NP-hard [15–17], thecommunity focuses on designing efficient heuristics. Wieselthieret al. study the MEM problem using the same approach as theMEB problem and propose three greedy heuristics [3, 16]. Wanet al. prove that both MEM problem and MEB problem can beapproximated within constant ratio [4, 18]. Liang et al. proposeapproximation algorithms for MEB problem where the nodes havediscrete transmission power levels [17]. Existing works also relateto the extensions of MEM and MEB such as MEM and MEB prob-lem with reception cost [9, 19], duty-cycle-aware MEM [6] anddelay-constrained MEM [5]. Furthermore, energy efficient mul-ticast/broadcast algorithms for specific applications is also an ac-tive topic [20]. However, as noted earlier, prior literature only fo-cuses on the minimization of transmitting energy, relies on theassumption that the receiving energy is directly proportional tothe number of receivers [19] or considers the receiving energy inthe context of network-wide broadcast [9]. In contrast, the pro-posed framework in our work jointly minimizes the transmittingand receiving energy, with applicability to more general scenarios.

In spite of a flurry of works addressing the issue of MEM instatic networks, little is known about the energy consumption inmobile scenarios. As two exceptions, Wu et al. [21] study MEM inmobile ad hoc networks using network coding, but their focus ison energy gain from network coding rather than mobility. Guo etal. [2] propose an efficient Distributed Minimum Energy Multicastalgorithm, but how delay constraint affects the energy consump-tion is not considered. In many cases where the timely deliveryof messages is not of primary concern, relaxing delay constraintcan bring gain in many aspects such as capacity and energy [12].Considerable efforts have been made to investigate the theoreticaldelay-capacity relationship in large scale static andmobile wirelessnetworks [22–24].

Despite those dedications to multicast energy optimization, asfar as we know, there are no prior work, other than ours, that con-siders the problem of jointly optimizing transmitting and receivingenergy of multicast in mobile wireless networks.

3 MODELS AND ASSUMPTIONS3.1 Mobile Wireless Network ModelWe adopt the mobile wireless network model in [25] that can bereproduced as follows: time is divided into discrete time slots andthe nodes remain static within a time slot. With reasonable predic-tion of nodes’ futuremovements, amobile network can bemodeledas a sequence of static graphs G = {G1,G2, . . .}, with each staticgraph interpreted as a snapshot of nodes’ positions in a time slot.Specifically, denoted as Gt (V ,Et ,wt ) ∈ G, each static graph is aweighted directed graph withV = {v1,v2, . . . ,vn } correspondingto the n network nodes. Et ⊆ V × V denotes the edge set andwt : Et 7→ R+ assigns each edge a corresponding weight. Anedge e = (vi ,vj ) ∈ Et means that vi can send messages to vj intime slot t with minimum transmission powerwt (e).

3.2 Multicast ModelWe consider a wireless multicast session with a source s ∈ V anddestination nodes set T ⊆ V in mobile network G. We impose anintegral delay constraint D on the multicast session in the sensethat the message must be delivered to all nodes inT within D slots.Therefore, for a mulicast session, we only need to consider the net-work snapshots within the delay constraint. Hence, in the sequel,we truncate the mobile network G into D static graphs denoted as{G1,G2, . . . ,GD }.

Since we aim to design a minimum energy transmission schemefor a delay-constrained multicast session in a mobile wireless net-work, we next formally define the notion of transmission scheme.We refer to the process of some transmitter transmitting messageto the intended receivers as a transmission. A transmission schemespecifies the transmitter, intended receivers, timing and transmis-sion power of each transmission.

Definition 3.1. A transmission scheme π ⊆ V ×2V ×{1, . . . ,D}×R+ is a sequence of tuples {(u,V ′, t ,p)}, where a tupleτ = (uτ ,Vτ ,tτ ,pτ ) ∈ π denotes that node uτ should transmit to the nodes inVτ at time slot tτ with power pτ .

Also, in a transmission schemeπ , we define a node to be “reachedin time slot t” in a recursive way as follows: (i) the source s isreached in time slot 1, (ii) if a node is reached in time slot t , thenit is reached in time slot t ′ for all t ′ ≥ t , and (iii) if a node u isreached in time slot t and (u,V ′, t ,p) ∈ π , then all nodes inV ′ arereached in time slot t .

Now, we give the definition of reasonable transmission schemeand will restrict our consideration to the set of reasonable trans-mission schemes in the sequel. Intuitively, a scheme is said to bereasonable if it is feasible, without redundancy and qualified forthe delay constrained multicast session.

Definition 3.2. A transmission scheme π is reasonable if it ob-serves the following conditions: (i) For all τ = (u,V ′, t ,p) ∈ π , itsatisfies thatu is reached in t and for allv ∈ V ′, e = (u,v) ∈ Et andp ≥ maxe=(u,v)∈Et ,v ∈V ′ {wt (e)}. (ii) For all τ = (u,V ′, t ,p) ∈ π ,no node in V ′ is reached before t . (iii) All nodes in T are reachedin time slot D.

Figure 1 illustrates an example of a mobile wireless network,along with a reasonable transmission scheme proposed under amulticast session in the network.

time

space

1t = 2t = 3t = 4t = time

spacesourcedestination reached node

unreached node intended receiver

1v 1v1v

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1v 1v1v

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2v

3v3v

3v

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Figure 1: A mobile wireless network and a reasonable transmis-sion scheme for a multicast session in the network; (a) a mobilewireless network (a sequence of snapshots); (b) a reasonable trans-mission scheme for amulticast session in the networkwithv1 beingthe source, v4, v6 being the destinations and a delay constraint offour time slots.3.3 Energy ModelFor a transmission specified by tuple τ = (uτ ,Vτ , tτ ,pτ ), wemodelthe energy consumption of this transmission as

E(τ ) = pτ + f (|Vτ |),where the first part denotes the energy consumed by the transmit-ter side and f is a function of the number of intended receiversthat can represent the receiving energy. Note that our frameworkdoes not restrict to any specific f and the choice of f may dependon practical settings [7, 9], which will be specified in Section 6. Itfollows that the energy consumption of a transmission scheme πis given by:

E(π) =∑τ ∈πE(τ ).

Now we formulate the DeMEM problem as follows.Definition 3.3. (The DeMEM Problem) Given a mobile wireless

network G = {G1,G2, . . . ,GD }, a source node s , a setT of destina-tion nodes, and a delay constraintD, the goal is to find a reasonabletransmission scheme with minimum energy consumption.

We note that in the definitions above, we make the followingcharacterizations of communication in mobile wireless networks.First, we assume that the transmission rate is much faster than thatof the nodes’ mobility. In other words, it is feasible to transmitmultiple times to different receivers in a single time slot. This maynot be desirable if we only consider transmission cost. However, itcan bring possible gain when we take the receiving energy into ac-count. Second, we can utilize the wireless broadcast nature duringcommunications, i.e., different nodes can be reached within onetransmission as long as the transmission power is large enough.

4 APPROXIMATION HARDNESSIn this section we analyze the approximation hardness of DeMEMproblem. We show that even without receiving energy (i.e., f =0), the DeMEM problem cannot be approximated within a logarit-mic factor in polynomial time by reduction from acyclic directedSteiner tree problem.

Theorem 4.1. TheDeMEMproblem cannot be approximatedwithina factor better than 1

4 lnk unless NP⊆ DTIME(npolylogn ), where k isthe number of destinations.

Proof. Consider an instance of acyclic directed Steiner treeproblem defined by an acyclic directed graphG(V ,E) and an edgecost function d : E → R+, a root r ∈ V and a set of verticesS ⊆ V with |S | = k . The goal is to find a minimum cost Steinertree (arborescence) covering all vertices in S . Based on these, we

will construct an instance of the DeMEM problem by encoding thegraph information into a mobile wireless network.

To begin with, we do a topological sort on the graph and get asequence of nodes following the topological order. Without lossof generality, we only consider the nodes that do not lie before theroot r in the sequence, i.e., a truncated sequence {r = v0,v1, . . .}.Then, we sort the relevant edges according to the order of their out-going nodes in the truncated sequence into {e0, e1, . . . , el }. Fromthis sequence, we will build a mobile wireless network G.

First, we set V ′ as the set of nodes that appear in the truncatedsequence. Then, for any generic edge ei = (ui ,vi ) in the edgesequence, we add (ui ,vi ) into Gi (V

′,Ei ,wi ) and set its cost aswi (e) = d(e). The constructed network G consists of static graphs{G1,G2, . . . ,Gl } where each static graph has only one edge. Fur-ther, we designate r as the source, the nodes corresponding to thevertices in S as the destinations, and set the delay constraintD = l ,the receiving energy function as zero. By above procedures, wehave an instance of DeMEM problem and now we show that it isequivalent to the original acyclic directed Steiner tree instance.

For a Steiner tree in G, if it contains edges {ey1 , ey2 , . . . , eyj },then obviously the corresponding transmission scheme with tu-ples {(uy1 , {vy1 },y1,d(ey1)), . . . , (uyj , {vyj },yj ,d(eyj ))} is reason-able. Conversely, consider a reasonable transmission scheme forthe mobile wireless network. Since in each time slot there onlyexists one edge, each tuple in a reasonable transmission schemecorresponds to some edge in the acyclic graph G. And from therequirements for the scheme to be reasonable, it follows that thoseedges form a Steiner tree in G. Most importantly, the energy con-sumption of a reasonable transmission scheme equals to the cost ofits corresponding Steiner tree and all the reduction processes areimplementable in polynomial time. Therefore, the above reductionis an approximation preserving reduction. Since for a fixed k thereis no efficient 1

4 lnk approximation for acyclic directed Steiner treeproblem unless NP⊆ DTIME(npolylogn ) [26], we obtain the sameapproximation hardness for DeMEM problem. □

5 THE APPROXIMATION FRAMEWORK:CONMAP

Since by Theorem 4.1, even approximating the DeMEM problembetter than a logarithmic factor is NP-hard, it is unlikely that theDeMEM problem can be solved in polynomial time. Therefore, weneed to design efficient approximation scheme to trade complexityfor the optimality of the transmission scheme. In this section, wetherefore propose a novel approximation framework – ConMap,which effectively achieves such tradeoff by transforming our De-MEM problem into a directed Steiner tree problem.

Given a specific DeMEM problem instance, ConMap first con-verts the mobile wireless network graph to an intermediate graph,then adopts established algorithms to compute an approximateminimum Steiner tree in the intermediate graph, and finally usesthe computed Steiner tree to construct a near optimal transmissionscheme for the original DeMEMproblem. Aswewill disclose in thesequel, all the conversion processes are efficiently implementable,with a polynomial time complexity and the size of the intermedi-ate graph being polynomial to that of the original network graph.Also, we can embed different algorithms for directed Steiner treeto obtain various performance guarantees.

Here for clearance and convenience, we first consider DeMEMproblem without receiving energy and will demonstrate in Section6 how to integrate the receiving energy into the optimization.

5.1 Conversion into an Intermediate GraphTo derive an energy efficient transmission scheme, we firstly needto generate an intermediate graph so that the mobility of nodesand the broadcast nature of wireless environment can be captured.Note that the intermediate graph is a directed layered graph, witheach layer corresponding to the network at a certain time slot.

Given a mobile wireless network G = {G1,G2, . . . ,GD }, beforewe construct an intermediate graph, we adopt previous assump-tion [17] that there are pv adjustable transmitting power levelsfor a generic node v . Note that even for nodes with infinitely ad-justable transmitting power, there are at most (|V |−1) power levelsthat need to be taken into consideration. Upon the power level as-signment, we can now construct an intermediate graph at layer tby applying to all nodes the procedures listed as follows, with theresulting layer denoted as Lt :

(1) Create a vertex in Lt for each node in Gt and call thesevertices “original vertices”.

(2) For the pu power levels of node u, add a set of vertices de-noted as

{wu1,wu2, . . . ,wupu

}to the graph Lt . Hereby,

without ambiguity, we will also refer to these vertices as“power levels” and their labels as their corresponding trans-mission powers.

(3) Add a directed edge from u to each of its power levels andset the edge’s weight as the transmission power consumedby u when transmitting at that power level.

(4) Add edges fromwuj to all the vertices that can be coveredby u when transmitting at power level j and assign theirweights as zero. Formally, a vertex v can be covered byu at power level j if and only if ∃e = (u,v) ∈ Et andwt (e) ≤ wuj .

Upon the construction of D layers that correspond to D time slots,we connect them into the intermediate graph. Starting from L1,we establish the relation of adjacent layers by adding zero-weightedges between the original vertices in them that correspond to thesame node in different time slots. These edges resemble the tem-poral links in the model of [25], with their directions following thechronological order. With all the operations above, we completethe construction of the intermediate graph denoted as L, which,in the sequel, will be demonstrated to well capture both node mo-bility and the broadcast nature of wireless networks.

5.2 Computation of the Steiner TreeWe designate in L1 the vertex that corresponds to the source nodein the network as the root r and in LD the original vertices thatcorrespond to k destinations as terminals. Therefore, the DeMEMproblem is equivalent to establishing aminimumSteiner tree rootedat r spanning all k terminals in the intermediate graph L. Recallthat in the intermediate graph, the weight of edges from originalvertices to their power levels equals to the transmission power.And the weights of edges from power levels to the original ver-tices they cover are zero. It follows that the weight assignmentensures that the summation of the weights of edges in a Steinertree characterizes the energy consumption of a wireless multicast

session. In addition, the directed layering structure of the inter-mediate graph captures the nodes’ mobility and ensures that theSteiner tree in L will be generated following the chronological or-der. Hence, the intermediate graph L is an accurate abstraction ofmobile wireless network.

Since our task now becomes computing the minimum Steinertree in a directed graph, we can adopt existing efficient approxi-mate algorithms [27, 28] to fulfill our purpose. For completeness,we present one as the following lemma.

Lemma 5.1. There exists an algorithm that provides an l(l −1)k1l

approximation to Steiner tree problem in directed graph in timeO(nlkl+n2k + nm) where n is the number of vertices, k is the number of ter-minals and m is the number of edges. Specifically, set l = logk , weget an 2 log2 k approximation in time O((nk)logk ) [27, 29].

Through implementation of the Steiner tree algorithm, we ob-tain an approximate Steiner tree in our intermediate graph, whichwill serve as the basis for the construction of transmission scheme.

5.3 Mapping into the Transmission SchemeBased on the Steiner tree we computed, we proceed to design thecorresponding transmission scheme. First, we present a crucialproperty of the conversion process.

Lemma 5.2. For an instanceM of DeMEM problem, letM ′ be thedirected Steiner tree instance converted fromM by ConMap frame-work. Each reasonable transmission scheme inM corresponds to avalid Steiner tree inM ′, and the energy consumption of the schemeequals to the cost of its corresponding Steiner tree.

Proof. We describe a procedure that maps a reasonable trans-mission scheme π forM to an edges set ET that forms a Steinertree forM ′. For each tuple (u,V ′, t ,p) ∈ π , we add the edge fromu to the power level corresponding to p in layer Lt together withthe edges from the aforementioned power level to the original ver-tices of vertices in V ′ of Lt into ET . Finally, we add the neces-sary inter-layer edges into the set. The edge set constructed by theabove procedure will be a valid Steiner tree forM ′. In the sequel,we refer to the Steiner trees that have corresponding transmissionschemes as being in canonical form. □

From the proof of Lemma 5.2, we note that the energy con-sumption of any scheme is no less than the cost of the optimalSteiner tree in the constructed instance. Therefore, a transmissionstrategy mapped from an α-optimal Steiner tree is guaranteed tobe an α-optimal transmission scheme. However, the Steiner treecomputed in the intermediate graph may exhibit “aberrant phe-nomena” so that it has no corresponding reasonable transmissionscheme. Hence, in the final phase of ConMap, before we map ourcomputed Steiner tree back into a transmission scheme, we needto prune it into canonical form.

The possible aberrant phenomena that can cause a Steiner treeET not to have its corresponding reasonable transmission schemeare: (i) there is an edge from an original vertex to one of its powerlevels in ET , but no edge goes out of that power level, and (ii) thereare edges from different power levels to multiple original verticesthat correspond to the same network node, leading to redundanttransmissions. To deal with the first case, we delete those edgesfrom original vertices to their “dangling” power levels. For the sec-ond case, we keep the edge with the original vertex in the earliest

power level original node

sourcedestination

1t =

2t =

3t =

1( )v s

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12w

11w

21w22w23w

31w32w

51w

52w

Figure 2: Illustration of ConMap on a five-node mobile networkwith a delay constraint of three time slots.

time slot and delete other redundant edges. Then, we add neces-sary inter-layer edges to reconnect the resulting edge set into aSteiner tree. Note that the above two procedures do not increasethe cost of the resulting Steiner tree. Therefore, the pruned Steinertree can only be closer to optimal.

After describing the three main stages of the framework, wesummarize ConMap in Algorithm 1.

Algorithm 1: ConMap Steiner Framework for DeMEMInput: An instance of the DeMEM problemMOutput: A transmission scheme π forM

1 Based onM, construct its corresponding intermediate graphL and form an instance of directed Steiner treeM ′.

2 Compute a (approximate) minimum Steiner tree ET in L forM ′.

3 Prune ET into canonical form.4 Convert the pruned ET to its corresponding transmissionscheme π forM.

5.4 Illustration of ConMapNow we illustrate our proposed framework, ConMap, using an ex-ample of a five-node mobile network, where multicast is betweenone source and two destinations with a delay constraint of threetime slots. Figure 2 shows the intermediate graph of the network.Note thatwe omit some of the power levels due to space limitations.As shown in the figure, v1 corresponds to the source, v4 and v5correspond to the destinations. The solid lines denote the Steinertree the algorithm computes in the intermediate graph. Here, thetransmission scheme is {(v1, {v2}, 1,w11), (v2, {v3,v5}, 2,w22),(v3, {v4}, 3,w32)}, which can be interpreted as follows: In the firsttime slot, the source transmits to v2 using power w11; In the sec-ond time slot, v2 transmits to v3 and v5 using power w22; In thelast time slot, v3 transmits to v4 using powerw32.

5.5 Performance Analysis of ConMapWe proceed to provide analysis on the performance of the Con-Map framework in terms of running time and approximation ratio.Without loss of generality, we assume the approximation guaran-tee of embedded algorithms for directed Steiner tree only dependson the number of terminals in the graph [26, 27].

Theorem 5.3. For a mobile network with n nodes, let k be thenumber of destinations andD be the delay constraint. Suppose the di-rected Steiner tree algorithm embedded in ConMap runs inT (|V |, |E |,k)time and achieves an approximation ratio of д(k) on graph G(V ,E).Then ConMap returns a transmission scheme of which the energy costis less than д(k) times the optimal one in time O(T (Dn2,Dn3)).

Proof. First, we focus on the time complexity of ConMap. Inthe first phase, since there are at most (n − 1) power levels foreach node and the intermediate graph contains D layers, it takes atime ofO(Dn3) to construct such an intermediate graph. Note thatthe intermediate graph has at most Dn2 vertices and Dn3 edges.Hence, the time complexity of phase 2 isO(T (Dn2,Dn3)). As forphase 3, the pruning and the converting process can both be doneby traversing the tree, which takes O(Dn3) time. Hence, the totalrunning time is O(T (Dn2,Dn3)). Obviously the approximationratio of the whole framework is determined by the second phase.By Lemma 5.2 and the fact that the number of terminals in theintermediate graph equals to the number of destinations we con-clude that the approximation ratio of the proposed framework wellpreserves that obtained under the Steiner tree algorithm. □

6 INTEGRATION OF RECEIVING ENERGYIn this section, we illustrate how to integrate the optimization ofreceiving energy in our ConMap framework. Adopting the sameassumption in [9], we consider the receiving energy of a multicasttransmission grows sublinearly, linearly or superlinearly with re-spect to the number of intended receivers. The reasonability of thisassumption is supported by Johnson et al. [9] through a theoreticalabstraction of practical scenarios with different underlying proto-cols. Accordingly, we also divide the optimization technique intothree regimes where the receiving energy function f is sublinear,linear or superlinear respectively.6.1 Linear Receiving Energy FunctionsIn this case, the receiving energy of a transmission is directly pro-portional to the number of receivers. Hence, we can suppose f (k) =Ak . The optimization combined with receiving energy can be eas-ily integrated into the framework by setting the weights of all theedges from power levels to original vertices in the intermediategraph as A instead of zero and keeping the conversion proceduresand the pruning process unchanged. We can easily verify that thismodification does not influence the complexity and approximationratio of the framework.6.2 Sublinear and Superlinear Receiving

Energy FunctionsWhen the receiving energy is not linear with regard to the num-ber of receivers, the straightforward modification made in the caseof linearity is not applicable. We therefore present a new way totackle this and merge the consideration of sublinear and superlin-ear receiving energy into our ConMap framework.

Construction of intermediate graph: The main idea lies inthat instead of adding a simple edge from each power level of trans-mitters to their receivers, we create gadgets in the intermediategraph that charges the Steiner trees for the cost corresponding toreceiving energy.

For a power level wup (recall the definition in Section 5.1) oforiginal vertex u that covers k receiving nodes v1,v2, . . . ,vk , theconstruction of the gadget is as follows. First, we create k rows ofnew vertices, with each row containing k vertices, which, in thesequel, will be referred to as “virtual vertices”. We denote the ver-tices in row i as vi1,vi2, . . . ,vik . Second, we add a zero-weightedge from each virtual vertex to its corresponding receiving node(e.g, from v11,v21, . . . ,vk1 to v1). Then, for the virtual verticesin two adjacent rows, we insert edges between them such that

11v 12v 13v

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uu

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upw

upw

( )a ( )bFigure 3: Illustration of the gadgets for receiving energy: (a) theoriginal gadget in ConMap without receiving energy for a powerlevel of original vertex u and the nodes v1, v2, v3 that u can covertransmitting at power p . (b) the corresponding gadget ConMap cre-ates for this power level ofu when considering the receiving energy.

the subgraph formed by the nodes in each two adjacent rows isa complete bipartite graph. The direction of these edges is fromthe row with lower index to the row with higher index. And weset the weights of all the edges between row i and row i + 1 asf (i + 1) − f (i). Finally, we add edges from vp to all the virtualvertices in the first row and set their weights as f (1). An exampleof the gadget is shown in Figure 3.

We justify the above construction by proving a similar argu-ment as Lemma 5.2 that each reasonable transmission scheme cor-responds to a valid Steiner tree in the intermediate graph with thecost of the tree equal to its total energy consumption. Indeed, for atuple τ = {uτ , {v1,v2, . . . ,vm },pτ , tτ }, it can be encoded as a sub-graph consisting of an edge from the original vertex correspondingto uτ to its power level pτ in Lt , edges that form a path traversingone of each receiver node’s corresponding virtual nodes once inthe gadget on some row (e.g. v11,v22, . . . ,vmm ) and edges fromthe traversed virtual vertices in the gadget to their correspondingreceiver nodes (e.g. (v11,v1), (v22,v2), . . . , (vmm ,vm)). It is easyto verify that the total weight of this subgraph equals to the en-ergy consumption of the transmission. And also as in the proofof Lemma 5.2, we add necessary edges between different layers,relating a reasonable transmission scheme to a Steiner tree.

After the computation of a Steiner tree in the intermediate graph,however, the existence of the gadgets complicates the pruning pro-cess. In addition to the pruning procedures mentioned in the pre-vious section, we also need to consider the aberrant phenomenacaused by the edges in the gadgets. The pruning process is demon-strated in Figure 4 using two illustrative cases. Taking Figure 4(b)for example, the constructed steiner tree entails two receivers v1andv2 in the transmission but the edges that the algorithm choosesin the gadget are (v11,v1), (v11,v22), (v22,v2), (v11,v23), (v23,v3)instead of the path (v11,v22), (v22,v33) and edges (v11,v1), (v22,v2),(v33,v3), whereas the latter are the ones that correspond to a legit-imate transmission scheme. Therefore, we incorporate searchingsuch aberrant subgraphs in the gadgets and correct them to thelegitimate ones in the pruning process of the framework.

Performance Analysis of ConMap with receiving energy:In terms of the change of cost of the tree brought by the pruningprocess, since when f is sublinear, f (i + 2) − f (i + 1) ≤ f (i +1)− f (i) for all i and when f is superlinear, f (i +2)− f (i +1) ≥f (i + 1) − f (i) for all i , the pruning process does not increasethe cost of the resulting tree when f is sublinear and increases thecost to a factor of at most f (k)

kf (1) when f is superlinear. For com-plexity issues, anm-receiver gadget contains at mostm2 vertices

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p

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Figure 4: Illustration of two pruning processes of converting thecomputed Steiner tree in the intermediate graph into canonicalform. (a) and (b) are two cases of the conversion.

and m3 edges. Hence, all the gadgets in the intermediate graphcontain at most Dn4 vertices and Dn5 edges. Hence, if we use aSteiner tree algorithm with a time complexity of T (|V |, |E |,k) andan approximation ratio of д(k), our Conmap framework will yielda д(k)-approximation of the optimal transmission scheme for sub-linear cost functions and a д(k)f (k)

kf (1) -approximation for superlinearcost functions in a time complexity of O(T (Dn4,Dn5,k)).

6.3 An Alternative Gadget ConstructionAsmentioned in the previous section, ConMap framework increasesthe approximation ratio of the embedding Steiner tree algorithmby a factor up to f (k)

kf (1) when the receiving energy function is su-perlinear. Therefore, we propose a second method of constructinggadgets in ConMap to capture the receiving energy that can elim-inate this factor for superlinear receiving energy functions. Themethod produces polynomial number of additional nodes whenthe maximum number of neighbors of a node in the network islogarithmic to the total number of nodes.

The construction works as follows. Starting from the originalintermediate graph constructed in Section 5, for each power level,we add a new vertex for each possible combination of receivingnodes and refer these vertices as receiving levels. Then, we addedges from each power level to all their corresponding receivinglevels and from the receiving levels to all their corresponding re-ceivers. The first set of edges are assigned with weights equal tothe receiving energy, i.e, f (k) for a receiving level with k receivers.And the second set of edges are zero-weighted ones.

Now, we justify the validity of the construction. In the gadgetsconstructed as above, a tuple (u,V ′, t ,p) in a transmission schemecorresponds to a subgraph consisting of an edge from the originalvertex associated with u to its power level corresponding to p inLt , an edge from the aforementioned power level to the receivinglevel corresponding toV ′ as the intended receivers and edges fromthe receiving level to all the vertices associated with nodes in V ′

at time slot t . For the pruning procedure, apart from the first onementioned in the previous section, we also need to guarantee thatthe receiving levels included in the computed Steiner tree is con-nected to all its receivers. If not, we can replace this receiving levelwith the receiving level that corresponds to the set of connected re-ceivers only. This, again, will not increase the cost of the resultingSteiner tree, and thereby the approximation ratio can be preserved.

In terms of the time complexity of the framework, suppose theembedded Steiner tree algorithm runs in T (|V |, |E |,k) time. Inthis case, the number of additional vertices in a gadget is ofO(2∆).

Hence, the total time complexity of ConMap that applies this alter-native gadget constructionmethod becomesO(T (2∆Dn2, 2∆Dn3)).Therefore, this method is suitable for implementation only whenthe maximum degree of a node is no larger than logarithmic to thetotal number of nodes in the network. An example of this alterna-tive construction is illustrated in Figure 5.

power level

original nodereceiving level

(1)(2)(3)

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fff

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u

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Figure 5: An alternative construction of receiving gadgets.

7 EXPERIMENTSIn this section, we empirically evaluate the performance of Con-Map framework. We discuss our experimental settings in Section7.1 and present detailed empirical results in subsequent sections.

7.1 Experimental SetupWe validate the effectiveness of our ConMap framework based onthree real datasets. The basic descriptions of the three data sets arepresented as follows.

• Brightkite SocialNetwork [30]: Brightkite is a location-based networking service providerwhere users share theirlocations by checking-in. We consider the users as net-work nodes and use their location information at differenttimestamps to create mobility trace.

• Gowalla Social Network [30]: Gowalla is a location-based social networking website where users share theirlocations by checking-in. Similarly, we consider users asnetwork nodes and use their location information to cre-ate mobility trace.

• Roma Taxi [31]: This dataset contains mobility traces oftaxi cabs in Rome, Italy. It contains GPS coordinates ofapproximately 320 taxis collected over 30 days.

Parameter setting: For each dataset, we extract ten groupsdata of 50 nodes in 100 time slots and normalize the distance be-tween nodes into the range of 10 to 5000 for consistency. Wedesignate one source and four/six destinations and set the delayconstraint of the multicast session from 10 to 100 time slots. Dueto space limitations, we defer all the graphical representation ofresults in the scenarios of four destinations to Appendix A. Fur-thermore, for the energy model, we adopt a widely used assump-tion that the transmission energy is α power of the transmissionrange [3, 4]. And for receiving energy, we select f (k) = 50k ,f (k) = 100k

12 and f (k) = 20k2 where k is the number of in-

tended receivers in a transmission as the representatives of linear,sublinear and superlinear functions of receiving energy, respec-tively. Note that we are interested in the relative quantity of en-ergy consumption, and thus we do not relate the amount of energyconsumption to specific physical quantity.

Algorithms used for performance comparison: To justifythe performance and flexibility of our ConMap framework, weembed three different algorithms in ConMap for computing the

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Figure 6: The energy consumption of the transmission schemes yielded by the three/four algorithms under different scenarios with sixdestinations. The dataset and receiving energy function are labeled in the figures.

Steiner tree in the intermediate graph. The first one is the SPTheuristic [3, 4]. It constructs the Steiner tree by finding the short-est path from the source to all the destinations. The second one isthe MST heuristic [3, 4] that computes the Steiner tree by firstlyestablishing a minimum spanning arborescence and then pruningthe unnecessary edges. And the last one is the approximation al-gorithm for directed Steiner tree problem proposed by Charikar etal. [27] and further improved in [29]. We also implement a brute-force algorithm to obtain the exact solutions for linear receivingenergy function as the baseline. However, for the cases of sublin-ear and suplinear settings, the brute-force algorithm exhibits pro-hibitively large time complexity, and we thus do not present thecorresponding exact results here.

7.2 Evaluation of ConMap FrameworkFirst, we investigate the performance of the proposed ConMapFramework. For each group of data under three different receiv-ing energy functions, we apply the ConMap framework embed-ded with three directed Steiner tree algorithms mentioned aboveand name them ConMap-SPT, ConMap-MST, ConMap-CHArespectively. Imposing a delay constraint from 10 to 100 time slots,we compute the average energy consumption for the transmissionschemes designed for the ten groups of data by each algorithm. Ad-ditionally, we run a brute-force algorithm on each group of dataunder linear receiving energy function to present the cost of theoptimal transmission scheme for comparison. The results are plot-ted in Figure 6 (and Figure 8 in Appendix A).

As demonstrated in Figures 6(a), 6(d) and 6(g), when embed-ded with the Charikar’s approximation algorithm [27] that has

a logarithmic approximation ratio and the receiving energy func-tion is linear, the energy consumption of the transmission schemesyielded by ConMap is only 9.1% and 2.6% higher than the optimalones derived by the brute force algorithm in Brightkite, Gowallaand Roma Taxi datasets respectively. This verifies the conclusionthat the ConMap framework indeed preserves the approximationratio of the embedded directed Steiner tree algorithms, as we haveproved theoretically in Theorem 5.3. We conjecture that the sim-ulation results under superlinear and sublinear receiving energyfunctions still enjoy such property. However, as deriving the opti-mal transmission scheme in such cases is too computationally in-tensive, we leave the comparisons between ConMap and the bruteforce algorithm under superlinear and sublinear receiving energyfunctions as future work.

Apart from the Charikar’s approximation algorithm which hasrelatively high time complexity, we can also embed faster heuris-tics into the framework to achieve the tradeoff between the opti-mality of the derived transmission schemes and the complexity ofthe algorithms. Figure 6 also shows this flexibility of ConMap, aswe can more efficiently obtain transmission schemes that lead tomore energy consumption by embedding SPT or MST heuristics inthe framework. Furthermore, it turns out that although the approx-imation ratio of MST and SPT heuristic can be proportional to thenumber of nodes [4] in worst cases, the performance of ConMap-MST and ConMap-SPT are significantly better when applied onreal life datasets under differnet settings. The simulation resultsindicate that in each scenario, the average approximation ratio isless than 1.4 for ConMap-MST and is less than 1.6 for ConMap-SPT.This demonstrates the generality of the ConMap framework.

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Figure 7: Comparisons of the performance ofConMapwith (white)and without (black) consideration for receiving energy in differentscenarios with six destinations. The ratios between the average en-ergy achieved by ConMap with receiving energy and that by Con-Map without receiving energy is labeled on the figure.

7.3 Significance of Receiving EnergyIn order to capture the importance of jointly optimizing the trans-mitting energy and receiving energy for multicast in mobile wire-less networks, we embed the Charikar’s approximation algorithminto the original ConMap framework that only considers the trans-mitting energy (as depicted in Section 5) and apply it to the afore-mentioned scenarios. We compare it with the ConMap frameworkintegrated with receiving energy (ConMap-CHA) and summarizein Figure 7 (and Figure 9 in Appendix A) the average energy con-sumption of the schemes they derive under different delay con-straints. For the ConMap without receiving energy consideration,the optimization is completed with respect to transmitting energyonly, but the energy consumption of the schemes derived by Con-Map is calculated for the sum of transmitting and receiving energy.

From Figure 7, we can see that in the three datasets, when theenergy receiving function is superlinear or sublinear, the consid-eration of receiving energy can bring a gain of more than 15% interms of energy consumption of the derived schemes. The gain forlinear energy model is comparatively smaller as linear receivingenergy function is more homogeneous to transmitting energy.7.4 Characterization of Delay-Energy TradeoffOur empirical results also delineate the delay-energy tradeoff exist-ing in mobile wireless networks. As shown in Figure 6, the achiev-able energy consumption of transmission schemes decreases withthe relaxation of delay constraint on the multicast session. Morespecifically, by relaxing the delay constraint from 10 time slots to100 time slots, the minimum achievable energy decreases by about50%. Interpreted from the perspective of mobile wireless networks,this is attributed to the fact that as the delay constraint becomeslooser, nodes can store and carry the message until there are betterchances to forward it, therefore it leads to the feasibility of trans-mission schemes with smaller energy consumption.

8 CONCLUSIONIn this paper, we studied the problem of jointly optimizing mul-ticast energy consumption in delay-constrained mobile wirelessnetworks. We first formulated our problem , called DeMEM, as acombinatorial optimization problem, and then proved that the ap-proximation ratio of any polynomial time algorithms for DeMEMcannot be better than 1

4 lnk . As the approximation hardness of De-MEM implies its NP-hardness, aiming to provide efficient approx-imation schemes, we proposed a novel approximation framework

– ConMap. The proposed ConMap first converts DeMEM into anequivalent directed Steiner tree problem through creating auxil-iary graph gadgets to capture energy consumption, then maps thecomputed tree back into a transmission scheme. We have demon-strated the generality, flexibility and efficiency of ConMap frame-work by theoretical analysis and extensive simulations on real lifenetwork datasets. Our experiments also reveal useful insights asthe significance of the consideration of receiving energy and thedelay-energy tradeoff in mobile wireless networks

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Figure 8: The energy consumption of the transmission schemes yielded by the three/four algorithms under different scenarios with fourdestinations. The dataset and receiving energy function are labeled in the figures.

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A GRAPHICAL RESULTS OF SCENARIOSWITH FOUR DESTINATIONS

In this section, we present the figures (Figures 8 and 9) that demon-strate our simulation results on the settings with four multicastdestinations.

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500

700

900

1100

1300

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Trans-CHA Joint-CHA

Dataset: Brightkite:1-3; Gowalla: 4-6; Roma Taxi: 7-9Receiving Energy:Superlinear: 1,4,7; Sublinear: 2,5,8; Linear: 3,6,9

0.799 0.849

0.9030.869

0.875

0.919

0.821

0.844

0.899

Figure 9: Comparisons of the performance ofConMapwith (white)and without (black) consideration for receiving energy in differentscenarios with four destinations. The ratios between the averageenergy achieved by ConMap with receiving energy and that by Con-Map without receiving energy is labeled on the figure.