The investigation of delay-constrained multicasting

Int. J. Ad Hoc and Ubiquitous Computing, Vol. 4, Nos. 3/4, 2009

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The investigation of delay-constrained multicasting with minimum-energy consumption in static ad hoc wireless networks Wen-Lin Yang∗ Department of Computer Science and Information Engineering, National University of Tainan, 33, Sec. 2, Su-Lin Street, 700 Tainan City, Taiwan E-mail: [email protected] ∗ Corresponding author

Lung-Jen Wang Department of Information Technology, National Pingtung Institute of Commerce, 51, Ming-Sheng East Road, 900 Pingtung City, Taiwan E-mail: [email protected] Abstract: In this paper, we focus on the problem concerning how to construct a delay-constrained multicast tree with minimum-energy consumption in ad-hoc wireless networks. Based on link-replacing strategies, an algorithm called Energy-based Link Replacement (ELR) is presented. The ELR algorithm begins with a two-level multicast tree and then iteratively replaces the high-power links with a few of lower-power links in order to reduce the total power expended by the multicast tree. Unlike the prior approaches, the ELR can compute desired multicast trees directly. The simulation results show that our ELR outperforms the previously published method in terms of solution-quality and efficiency. Keywords: ad hoc wireless networks; delay constraint; energy; multicast tree; routing; QoS; quality-of-service. Reference to this paper should be made as follows: Yang, W-L. and Wang, L-J. (2009) ‘The investigation of delay-constrained multicasting with minimum-energy consumption in static ad hoc wireless networks’, Int. J. Ad Hoc and Ubiquitous Computing, Vol. 4, Nos. 3/4, pp.237–250. Biographical notes: Wen-Lin Yang received the PhD Degree in Computer Science from the Pennsylvania State University, University Park, USA, 1993. He is currently a Professor at the Department of Computer Science and Information Engineering in the National University of Tainan, Tainan, Taiwan. His primary research interests include routing protocols, quality of service, ad hoc networks and distributed computing. Lung-Jen Wang received the PhD Degree in Computer Science and Engineering from the National Sun Yat-sen University, Taiwan, 2001. Currently, he is a Chairman and Associate Professor at the Department of Information Technology in the National Pingtung Institute of Commerce, Pingtung, Taiwan. His primary research interests are in the areas of multimedia systems, computer networks and real-time systems.

1 Introduction Multicasting is a one-to-many communication mechanism that can transmit messages from a given source to multiple destinations. The unicasting and broadcasting are considered to be two special cases of multicasting. For most multicast routing protocols, a multicast tree must be determined before it is used to route data to destinations. Since a message package can be duplicated at the intermediate nodes, only one copy of messages is transmitted from the source to the destinations.

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This mechanism saves a large amount of bandwidths especially when the transmitted data are multimedia streams (Yang, 2002). Hence, for wired and wireless networks, multicast is one of the most important techniques for implementing multimedia applications, which usually require the underlying networks to provide end-users with Quality-of-Service (QoS) guarantees, such as delay, delay jitter, bandwidth, and so on (Perkins, 2001). Recent advances in wireless communication technologies and portable computing devices have led to active research in the development of protocols and

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real-time services for ad hoc networks (Bheemarjuna et al., 2005; Chakrabarti and Mishra, 2001). Unlike Wireless Local Area Networks (WLANs) and cellular networks, no fixed infrastructure is installed in ad hoc networks, where multiple hops may be needed for communication between two distant nodes, due to the limited range of radio transmission. The sensor network is one of applications of static ad hoc networks where wireless hosts are immobile (Akyildiz et al., 2002). Since ad hoc networks can be flexibly and quickly deployed and reconfigured without any base stations, they are useful for implementing real-time applications such as remote monitoring, video/audio conferencing, and emergency services. The real-time applications usually require QoS guarantees provided by the underlying networks. One of the most important QoS requirements is the packet delay, which specifies the upper bound on the end-to-end delay for all the packets. For most of real-time applications, the packets missed the delay bounds are discarded (Chaporkar and Sarkar, 2002). For example, in battle fields or disaster areas, the detected signals of targets must be routed back to the control centre within a pre-specified time interval so that a proper action can be invoked in time (Akkaya and Younis, 2003). The distributed database systems deployed on the ad hoc networks is another example (Hara, 2004), where a delay-constrained multicast tree can be used to simultaneously update multiple copies of data stored in remote wireless hosts. Data inconsistency may occur if any routing path in the multicast tree violates the given delay bound. Similar situations are also required for wireless video/audio conferencing. However, providing QoS routing in the ad hoc networks is in general more difficult than in the cellular networks since no centralised coordinators exist (Choi and Shin, 1998). In ad hoc networks, nodes are portable devices and powered by stand-alone batteries, which can not be replaced or recharged during network operation. Since the lifetime of ad hoc networks is limited by the battery power, it becomes imperative to take energy consumption into account when designing a wireless network. In particular, energy-efficiency can be treated as a QoS constraint like path delay or packet loss rate in ad hoc networks (Chakrabarti and Mishra, 2001). Furthermore, the studies show that energy consumption is not just affected by the types and operations of hardware such as battery, amplifier, antenna, etc, but also affected by the choice of higher-level protocols (Ephremides, 2002). Thus, for multicasting (or broadcasting) in energy-constrained wireless networks, an energy-efficient design paradigm for the network protocols is to take the assignments of transmitted power-levels into accounts during the formation of multicast (or broadcast) routing trees. Such cooperative considerations on topology connectivity and routing mechanism may lead to improved energy efficiency. Two essential problems close-related this cross-layered design methodology are under extensive investigation recently (Wieselthier et al., 2002; Yuan et al., 2008). They are

often referred as the Minimum-Energy Multicast (MEM) problem and the MEB problem respectively. Based on the similar communication model considered in Wieselthier et al. (2002), in this paper we study the problem concerning the determination of MEB trees subject to delay constraints for ad hoc wireless networks. Different from the problem studied in Wieselthier et al. (2002), an additional QoS constraint is introduced into our problem, which is then referred as the Minimum-Energy and Delay-constrained Multicast (MEDM) problem. The MEDM is a NP-complete problem, since the MEB problem, which has been proved to be NP-complete (Cagali et al., 2002; Liang, 2002), is a special case of the MEDM problem. In the source-initiated multicast tree, a packet experiences three different kinds of delay: propagation delay, processing delay and queuing delay, at every hop along the path from the source to a destination. The end-to-end delay for a packet is the sum of the delays it experiences at every hop. For the sake of simplicity, the path delay is assumed to be approximately proportional to the number of hops. The delay bound given for a multicast tree in our MEDM problem is then represented by the number of hops. In other words, the maximum number of hops of each multicast path is not greater than the given bound in order to support real-time applications. In addition, only one session of multicasting is considered here to avoid managing the resources like bandwidths or frequencies, so that the complex trade-offs between these three constraints: power, routing and delay, can be evaluated more precisely. As far as we know, this is the first study on the energy-efficient multicast problem that is required to simultaneously satisfy the above three constraints. Based on a number of link-replacing strategies developed in this study, we present a heuristic called ELR algorithm for determining a multicast tree for the MEDM problem. Our ELR method begins with a two-level multicast tree and then iteratively replaces the high-power links with lower-power paths, so that the total power expended by the multicast tree can be gradually reduced. Those replacements are carried according to five link-replacing strategies proposed in this paper. Furthermore, we give a proof to show that the replacing strategies can cover all the situations, if the searching area for substituting paths is limited to be the neighbourhood of nodes of the highest-power links. With this restriction, the simulations show that the efficiency of ELR algorithm can be maintained without losing the solution quality. An important feature of our ELR algorithm is that the desired multicast trees can be constructed directly, which is very different from the prior published method named Multicast International Power (MIP) presented in Wieselthier et al. (2002). The simulation results show that ELR outperforms MIP (Wieselthier et al., 2002) in terms of both solution-quality and executing efficiency, when the same delay constraint is given for a multicast tree. Although our ELR heuristic like MIP method is

The investigation of delay-constrained multicasting a centralised algorithm, a distributed implementation of ELR algorithm is also suggested in this paper. The rest of the paper is organised as follows. In Section 2, prior related works are described. A formal definition of MEDM problem is given in Section 3. Our ELR algorithm is developed in Section 4. A distributed implementation of ELR algorithm is described in Section 5. Section 6 compares the performance and efficiency of ELR against MIP on a set of random networks. The conclusion is given in Section 7.

2 Related works In the literature, at least two cost objectives can be found for determining energy-efficient broadcast and multicast trees. The first one is to minimise total transmitted power consumed by the network (Li et al., 2004; Li and Nikolaidis, 2001; Liang, 2002; Rodoplu and Meng, 1999; Wan and Calinesscu, 2002; Wiselthier et al., 2000a, 2001, 2002); and the second one is to maximise the lifetime of each individual node in the network (Floreen et al., 2005; Maric and Yates, 2005; Papadimitriou and Georgiadis, 2004; Wieselthier et al., 2002). These two objectives have been compared based on the performance (the total volume of bits delivered to the destination nodes of a multicast tree) and network lifetime in literature (Ephremides, 2002). For most of cases, their studies show that the performance gained based on the first objective tends to be better, although the maximum network lifetime may not be guaranteed by it. Thus, the objective of minimising the total power consumption of the entire network is chosen as a cost function for our MEDM problem studied in this paper. For the MEM problem, the simplest approach is to collect and merge all the least-power unicast paths to each individual destination from the source node. This heuristic is referred as the Shortest Path Tree (STP) Method (Wieselthier et al., 2000b). Another simple heuristic is to find the Minimum Spanning Tree (MST) first by applying the Prim’s algorithm, and then to prune and convert it as a tree rooted at the source node (Wieselthier et al., 2000b). These two methods are all developed based on the link cost (the power required to maintain the link) associated with each pair of nodes. That is, they are link-based approaches. The studies conducted in Wieselthier et al. (2002) have been shown that they are not appropriate for wireless communications. In Wieselthier et al. (2002), two heuristics called Broadcast Incremental Power (BIP) and MIP are proposed for MEB and MEM problems. An important feature of them is that they are developed based on the node-based wireless communications. The BIP is similar in principle to the Prim’s algorithm in forming the minimum spanning tree. It begins with a single-node spanning tree rooted at the source node. Based on the principle of increasing the least power to the spanning tree, all nodes outside the spanning tree are compared. A node is then selected to become a new member of the spanning tree, and the transmitting

239 power levels of nodes in the spanning tree are adjusted accordingly. This operation is repeated until all the nodes are included in the tree. If the destination node set is only a subset of nodes, the broadcast tree obtained from the BIP method is then pruned by recursively removing all the leaf nodes that are not destinations. This procedure is called the MIP algorithm. Since MIP uses a broadcast tree to generate the embedded multicast tree, it could be very inefficient for the wireless networks where only a small set of nodes are the destinations. In addition, our experimental results show that the multicast trees constructed by MIP may be highly unbalanced, since QoS constraints like the delay or delay variation are not considered during the formation of multicast trees. The obtained multicast trees may not be suitable for the real-time applications as the delay or the delay jitter of multicast paths tends to be very large. Another studies on MEM and MEB problems based on integer programming models can be found in Guo and Yang (2006) and Yuan et al. (2008). Based on an integer programming approach proposed in Yuan et al. (2008), they establish the lower bound of the optimum for MEB/MEM problems. A heuristic called Success Power Adjustment (SPA) is also reported to be able to find near-optimal broadcast and multicast trees. In Cagali et al. (2005), they provide a proof that the MEB problem can not be approximated better than O(log N ). They also propose a heuristic called Embedded Wireless Multicast Advantage (EWMA) which can take full advantage of the broadcast nature of the wireless channel. The EWMA method begins with link-based MST. Then, the solution quality is improved by exchanging some existing branches in the initial tree for new branches so that the total energy necessary to maintain the broadcast tree is reduced. Unfortunately, EWMA can not be applied to MEM problem directly. Another contributions for MEM problem are from (Liang, 2006), they provide an approximate algorithm which is within 4lnK times of the optimum, where K is the number of destination nodes in the multicast group. Their technique is to reduce MEM problem into a minimum node-weighted Steiner tree problem in a node-weighted auxiliary, undirected graph. The Steiner tree embedded in the auxiliary graph is then transformed into a multicast tree. In Li and Nikolaidis (2001), a heuristic called IMBM is proposed for the MEB problem. They starts from a two-level broadcast tree rooted at the source node, and iteratively reduce the maximum energy required for each transmitting node. Two operations named MBR and ROC are provided by the IMBM method in Li and Nikolaidis (2001). They are used to replace the link expended the maximum energy with one or two less-energy links for each transmitting node. The operations are executed only if the resulting broadcast tree with less energy expended. However, the link-replacing strategy considered the IMBM method is too simple. Furthermore, solutions found by the IMBM heuristic may be trapped in the local minimum, since the IMBM has no mechanism to accept worse solutions temporarily during the searching process.

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It is worth noting that energy is only QoS parameter to be considered in the studies reported above. In this paper, energy and delay are taken into account simultaneously during the formation of multicast trees. Five link-replacing strategies are presented in Section 4 for solving MEDM problem, where strategy #1 is similar to the MBR operation in IMBM algorithm. One feature, which makes our ELR different from IMBM, is that the link replacements in ELR are accepted as long as the delay constraints are satisfied, even though the total energy consumption of the resulting multicast tree is not reduced. As a result, the searching mechanism implemented in our ELR method can be prevented from being trapped in the local minimum.

3 Minimum-Energy and Delay-constrained Multicast (MEDM) problem

•

for each node u in M , the delay on the path from s to u is bounded by a given delay constraint ∆

•

the total power expended by all the transmitting nodes in T is minimised.

A certain number of nodes in the set V − M − {s} are selected as relay nodes and added in T to reduce the total transmission power expended. Formally, for each node u in M , if p(s, u) is a simple path from s to u, then


A directed graph G = (V, E) is used to denote an ad hoc wireless network, where V represents the node set, V  = n, and E is a set of directed edges between two nodes in V . In this paper, a directed edge from node i to node j is defined as a link (i, j) ∈ E, where i, j ∈ V and i is the transmitting node and j is one of many receiving nodes in the vicinity of node i. However, the link (i, j) ∈ E does not imply that the link (j, i) is also in E. Based on the same communication model presented in Wieselthier et al. (2002), the following assumptions are applied in our MEDM problem: no mobility, unlimited bandwidth, the power consumption dominated by the transmission energy, and the omni-directional antennas used by all nodes to transmit and receive signals. Furthermore, we assume that each node can dynamically adjust its transmitting power based on the distance to the receiving nodes and the background noise. Since the packet transmission energy is much larger than the packet reception energy and the idle energy in traditional ad hoc networks (Oyman and Ersoy, 2004), only transmission energy is considered in this study. According to the power-attenuation model proposed in Rappaport (1996), the received signal power Preceive is equal to w × r−α , where w is the transmission power, r is the distance and α is a parameter between 2 and 4, depending on the wireless environment. Preceive must be greater than a power threshold so that it can be detected by receivers. We assume that the power threshold is one for signal detection for all receivers. As a result, w × r−α ≥ 1. Hence, the minimum transmitting power required to support a link l = (i, j) can be defined as follows (Wieselthier et al., 2002): α wi,j = ri,j , where ri,j is the distance between node i and node j, and 2 ≤ α ≤ 4.

The main goal of this study is to construct an energyefficient and delay-constrained multicast tree T rooted at s, that spans all the destinations nodes in M , where T must satisfy the following two conditions simultaneously:

(1)

On this network, we assume that a source node s and a set of destination nodes M are given to consist of the multicast group. Furthermore, the delay of each link l in E is assumed to be one. Hence, for a given path p, the path delay is the number of hops between its two end-points.

delay(l) ≤ ∆, where delay(l)

l∈p(s,u)

= 1, and ∆ is an integer on the number of hops.

(2)

In the multicast tree T = (VT , ET ), all the internal nodes should be the transmitting nodes, and the leaf-nodes must be destination nodes in M . For an internal node i ∈ VT , the power required by i to communicate its direct downstream nodes is computed based on the equation: α }. W (i) = Max{wi,j | link (i, j) ∈ ET , wi,j = ri,j

(3)

Since leaf-nodes do not transmit any messages, no power is required for them. The total power associated with the multicast tree T is simply the sum of power expended at all transmitting nodes. Hence, an energy-efficient and delayconstrained multicast tree T must simultaneously satisfy equation (2) and the following cost function:
W (i). (4) Ψ = Minimise i∈VT

4 The Energy-based Link Replacement (ELR) algorithm In this section, we propose a heuristic algorithm to determine a minimum-power and delay-constrained multicast tree deployed on static ad hoc wireless networks. This heuristic is referred as the ELR algorithm in this paper.

4.1 Basic idea Initially, we set the emission energy of source node s to be maximum, so that the signal sent out by the s can be received directly by all the destination nodes including the farthest one. As a result, an initial multicast tree T is formed. Since the two-level multicast tree T consists of source node s and the destination nodes only, total power expended by T is determined by the distance between s and the farthest destination node. Based on this multicast tree T , the high-power links are then iteratively replaced with the goal on minimising the total energy

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The investigation of delay-constrained multicasting required by the tree T . Obviously, the link-replacing procedure determines the performance and efficiency of our ELR heuristic algorithm, which are presented in the following. 1

A two-level multicast tree T rooted at source node s can be constructed as follows: T = (VT , ET ), where VT = {s} ∪ M

and

ET = {(s, u) | u ∈ M }. Since all the destination nodes are direct downstream nodes of s and only s is the transmitting node, the total power associated with T is then computed as α follows: W (s) = Max{rs,u | u ∈ M }. Although some nodes not in set M may receive signals from source node s, they are not included in the multicast tree T . That is, there are no relay nodes in the tree T . 2

3

Based on the two-level multicast tree T constructed above, we iteratively substitute the highest-power link with a simple path that consists of a certain number of lower-power links. For each iteration, the highest-power link, say l, which can satisfy the following two conditions simultaneously, has priority to be selected for replacement. a

A link l selected for replacement must be based on one of five link-replacing strategies presented in Subsection 4.2.

b

After link l being substituted, the longest delay in the resulting multicast tree must be not greater than the given delay constraint ∆. In order to fulfill this requirement, a procedure for checking delay constraints is developed in Subsection 4.4 for our ELR algorithm.

The iteration loop proceeds until no such link can be found. For one single link replacement, the total transmission power associated with the resulting multicast tree may be not decreased. However, according to our simulation results presented in the following section, the total transmission power required by the final multicast trees is much less than the power required by the initial two-level multicast trees for most of the benchmarks.

Case 1: Initially, node q is not a member of the multicast tree T in this case. Hence, we need to add node q into T to act as a relay node for node r. For example, in the graph on the left side of Figure 1, the highest-power link (p, r) can be replaced by a substituting path Φ with lower power. In Figure 1, the replacement is done by setting Φ = 2. In Figure 2, the replacement is done by setting Φ = 3, where we assume that relay nodes w, q are not in T . The link-replacing strategy is shown as follows. a

in Figure 1, (p, r) − (p, q) + (q, r), since wp,r ≥ wp,q + wq,r (i.e., 6α ≥ 3α + 4α , where α ≥ 2)

b

in Figure 2, (p, r) − (p, w) + (w, q) + (q, r), since wp,r ≥ wp,w + ww,q + wq,r (i.e., 6α ≥ 2α + 2α + 1, where α ≥ 2).

For the case that Φ ≥ 3 and some of the upstream nodes of q are in T , the link-replacing strategy is similar to above. After the path with the least energy is found, reconnection procedure is then invoked to repair the multicast tree. For example, in Figure 1 where wp,q + wq,r < wp,r , a new multicast tree T = (VT , ET ) is reconstructed in which VT = VT + {q} and ET = ET − (p, r) + (p, q) + (q, r). Figure 1 Link-replacing strategy #1 when relay node q
∈ T and length of Φ equals 2

Figure 2 Link-replacing strategy #1 when relay nodes w, q
∈ T and length of Φ equals 3

4.2 Link-replacing strategies When a high-power link (p, r) is selected for deletion, we may need to reconnect node r and its subtree back the current multicast tree T to keep all destination nodes in T . Note that the subtree rooted at node r contains at least one leaf which is a destination node in M . Assume that the reconnection is done by making node q be the parent node of r and its subtree. Dependent on whether the relay node q is in the current multicast tree T , the following two cases have to be considered:

Case 2: In this case, after deleting the highest-power link (p, r), we intend to reconnect destination node r back to the multicast tree T by making node r to be a direct son of a node q, where node q is already in the tree T . Based on the relative location of link (p, r), there are at most four situations for node q inside tree T , which are: a

node q is in one of siblings of subtree rooted at node r. Hence, nodes q and r have a common ancestor node p

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b

node q is in one of siblings of subtree rooted at node p. Hence, nodes q and p have a common ancestor node h

c

node q is an ancestor node of node p

d

node q is in the subtree rooted at node r.

The topologies corresponding to the above four situations are shown in Figure 3(a)–(d), respectively. Based on the topologies given in Figure 3, four distinct link-replacing strategies can be developed for our ELR algorithm. For the sake of simplicity, four link-replacing strategies shown in Figure 4 are derived by restricting the searching area for node q to be the neighbourhood of link (p, r). Only nodes which are in two links away from node p or r are examined to find the best position for reconnecting node r. In fact, strategies presented in Figure 4 are still valid by substituting node w with a subtree of height greater than 2. It means that the radius of searching area is allowed to be greater than 2 links. Figure 3 Four different topologies of node q inside tree T based on the relative location of link (p, r)

In general, there may exist many ways to substitute the highest-power link (p, r) with a path Φ with lower energy consumption, where Φ is an alternate simple path from node p to node r. Obviously, we need an exhaustive method to explore all such lower-power paths. In order to avoid such expensive searching, we restrict the length of path Φ to be no more than 3 in our ELR heuristic algorithm. In fact, according to our experiments, it does not lead to great improvement to the quality of obtained results by making the length of path Φ to be large. Therefore, we only consider all the possible paths of length not greater 3 for the cases given in Figures 2 and 4. In the next subsection, we prove that five link-replacing strategies developed in this subsection cover all the link-replacing situations under the constraint that the searching radius for reconnecting point q is limited to be two links away from node p or r.

4.3 Completeness Proposition 1: Assume that the highest-power link is (p, r), and the radius of searching area for reconnecting point q is no more than two links away from node p or r. The five strategies given in the above subsection cover all the link-replacing situations. Proof: Suppose link (p, r) is found and deleted, the subtree rooted at r is required to be reconnected back to the tree T . Assume the connecting point is node q, the following two cases are required to be considered:

After all possible paths are considered for all strategies in Figures 3 and 4, a node q which can lead to the least energy-consumption and satisfy a given delay constraint is then determined. Since the node q is selected from the vertex-set V of multicast tree T , only edge-set E is needed to be updated when the new multicast tree T is reconstructed. The updating procedure is given as follows. •

for the strategy #2, ET = ET − (p, r) + (q, r), if wq,r < wp,r

•

for the strategy #3, ET = ET − (p, r) − (h, q) + (p, q) + (q, r), if (wp,q + wq,r ) < wp,r

•

for the strategy #4, ET = ET − (p, r) + (q, r), if wq,r < wp,r

•

for the strategy #5, ET = ET − (p, r) − (w, q) + (p, q) + (q, r), if (wp,q + wq,r ) < (wp,r + ww,q ).

•

Node q is not in T initially. Assume that the substituting path started from node p is found to be either (p, w, q) or (p, q), the reconstructing path is then either (p, w, q, r) or (p, q, r) respectively. These cases are covered by strategy #1 shown in Figures 1 and 2.

•

Node q is in T initially. Since all possible topologies concerning location of node q relative to link (p, r) are shown in Figure 3, the strategies developed based on them and presented in Figure 4 must cover all the link-replacing situations under the constraint that the searching radius for reconnecting point q is limited to be two links away from node p or r.

4.4 Delay-constraint checking Five link replacing strategies described in the above sub-sections are used to find a node q such that (wq,r + wp,q ) or wq,r is minimal and the height of the resulting multicast tree satisfies the given delay constraint ∆. However, we need a procedure for checking tree height during the process of applying link replacing strategies for finding a good node q. In order to support such function, each tree node q maintains two attributes: level and height. For example, in Figure 5(C), both c → level and c → height are 1 for node c, while d → level is 2 and d → height is 0 for node d. For each node q in a delay-constrained multicast tree, the following equation

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The investigation of delay-constrained multicasting Figure 4 Four link-replacing strategies when relay node q ∈ T

must be hold: q → level + q → height ≤ ∆. With this equation, a procedure called height_checking() is invoked to see whether a new multicast tree can be built. Five main functions of height_checking() are shown as follows: Case(a): When q ∈ / T && wq,r + wp,q < wp,r , check p → level + 2 + r → height ≤ ∆. Case(b): When q ∈ T && wq,r < wp,r && Ancestor (p, q) = YES && Ancestor(r, q) = NO, check q → level + 1 + r → height ≤ ∆. Case(c): When q ∈ T && wq,r + wp,q < wp,r && Ancestor (p, q) = NO && Ancestor (q, p) = NO, check p → level + 2 + r → height ≤ ∆ && p → level + 1 + q → height ≤ ∆ && p → level + 2 + w → height ≤ ∆, where w is one of direct sons of q.

check p → level + 2 + r → height ≤ ∆ && r → level + 1 + r → height ≤ ∆ && w → level + 1 + w → height ≤ ∆, where w is one of direct sons of r. The above five cases are derived from the five link-replacing strategies shown in Figures 1 and 4. For example, in case(a), the location of node r in the new multicast tree new_T is changed since it becomes a grandson-node of node p after a new node q is inserted into the tree. In the new_T, r → level = p → level + 2. Hence, the sum of level and height in node r is the height of the new_T , and it must be not greater than the given delay constraint ∆. As for the situation in case(c), two Ancestor() functions are used to check whether nodes p and q are in two different subtrees. Since the locations of nodes q, r and w in the new_T are all changed, we once again use p → level to compute the new levels of nodes q, r and w, which are than used to for checking the height of the new_T . The other height-checking cases are all derived by applying similar approaches.

Case(d): When q ∈ T && wq,r < wp,r && Ancestor (q, p) = YES, no check is needed.

4.5 An example

Case(e): When q ∈ T && wq,r + wp,q < wp,r + ww,q &&Ancestor (q, p) = YES,

An example of five-node network is embedded in a Euclidean graph shown in Figure 5(A), where distances of

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any two nodes are given. In Figure 5(A), source node is s and destination nodes are a and d. Initially, a two-level multicast tree is established in Figure 5(B), where total power Ψ0 equals to W (s) which is 36 = M ax{62 , 52 }, when α is set to be 2. In Figure 5(C), based on the strategy #1 shown in Figure 1, link (s, d) is replaced by two links: (s, c) and (c, d), and the total power consumed is 26 (W (s) + W (c) = 25 + 1). In Figure 5(D), link (s, a) is replaced by two links: (s, c) and (c, a). This replacement is matched the strategy #2 shown in Figure 4. The total power required by the final multicast tree is then reduced to 5 (W (s) + W (c) = 4 + 1).

and transmission power associated with them. These data structures are described as follows. •

Energy table Transmitting nodes

Maximum power

Receiving nodes

Dominant link

s

W (s) = Max v0 , . . ., vh−1 {ws,v0 , . . ., ws,vh−1}

(s, vi )

u

v0 , . . ., vk−1 W (u) = Max {wu,v0 , . . ., wu,vk−1}

(u, vj )

At each row of the ‘Energy Table’, ‘Maximum power’ is the energy consumed by the dominant link, and total number of rows equals to the number of transmitting nodes in the multicast tree T . Hence, the total power Ψ associated with the multicast tree can be computed by summing up all the ‘Maximum power’ values in the table. The ‘Maximum power’ value of a transmitting node may need to be updated, when a receiving node is removed or added due to the link-replacing operations. At line 4 of Figure 6, an ‘Energy Table’ named ETAB is initiated for storing the transmitting nodes and power consumed by them.

Figure 5 A network example for ELR algorithm

•

Index list The ‘Index List’ is created for storing the address (or index) of each row in ‘Energy Table’. The ‘Index List’ is sorted in decreasing order based on the ‘Maximum power’ values stored in ‘Energy Table’. At line 10 of Figure 6, the first element η of ‘Index List’ Q, which corresponds to the link with the largest ‘Maximum power’ value in ‘Energy Table’ ETAB, has the highest priority to be selected for link-replacing.

•

Distance list We maintain a linked-list Li associated with each node vi in the network. The list is defined as follows: vi → (u0 , d0 ), . . . , (un−2 , dn−2 ), where uj ∈ V and uj = vi , dj is the distance between vi and uj ; These lists are sorted in increasing order based on distances between two nodes. They are used at line 2 of Figure 7 by link_replacing() procedure for finding desired links for substitution.

4.6 The ELR algorithm Based on the above idea, our ELR algorithm is designed and presented in Figure 6, where three special data structures: ‘Energy Table’, ‘Index List’ and ‘Distance List’ are created for recording all the transmitting nodes

At line 12 of Figure 6, the link_replacing() procedure is called to find a node q such that the link (p, r) with the most expended energy can be replaced by a few of energy-economic links related to q. At line 4 and 9 of the Figure 7, the height_checking() function is implemented to predict whether the height of the resulting tree satisfies the given delay constraint. If the test is successful, the multicast tree is updated at line 14 of Figure 6, and all the information stored in ‘Index List’ Q and ‘Energy

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The investigation of delay-constrained multicasting Figure 6 The ELR algorithm

Table’ ETAB must also be updated accordingly. Since the status of Q is changed after a successful replacement, at line 8 the pointer Qhead is reset back to the first element of Q by setting done_f lag = 0 at line 20. At line 19, the new multicast tree with the less-power expended is saved. When all the dominant links stored in Q have been processed without incurring any replacement, the first ‘while’ loop at line 7 stops and the minimum-power multicast tree is obtained. However, the obtained multicast tree may contain some leaf nodes which are not in the destination-node set M . Therefore, a leaf-pruning process is required at line 24 for recursively removing unwanted leaves. In Li and Nikolaidis (2001), an operation named Recursive Omni-directional Check (ROC) was proposed to check the neighbouring nodes within the area covered by a transmitting node. The similar method is also implemented at line 15 of Figure 6. This Omni_directional_Check() method can be illustrated by an example shown in Figure 8, where α is assumed to be two. Based on link_replacing strategy #1 given in Figure 1, the dominant link (s, a) shown in Figure 8(A) can be replaced by two less-power links (s, k) and (k, a) shown in Figure 8(B).

Figure 7 The link_replacing() procedure

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We assume that the distance between nodes k and b is two. Since node b is within the transmitting territory of node k and (s, b) is the dominant link of node s, the total power expended by the multicast tree can be reduced by replacing (s, b) with (k, b).

4.7 Time complexity For the link_replacing() procedure in Figure 7, the time complexity is O(n) which is determined by the ‘for’ loop at line 2, where n is the number of nodes in the network. As for the height_checking() executed at line 4, its running time is bounded by O(∆), which is the height of the multicast tree and is required by the Ancestor() where the tree-transversing process is needed. Since ∆ is a small constant, the time complexity of link_replacing() is then bounded by O(n). As for the ELR algorithm in Figure 6, the nested ‘while’ loops take O(|E|) steps at the worst case, where |E| is the number of links in the network and is the maximum number of elements available in the ‘Index List’ Q. Inside the inner ‘while’ loop, it takes at most O(n) steps for each operation executed at line 14–19. Hence, the time complexity of our ELR algorithm is then bounded by O(n3 ) at the worst case.

5 Distributed implementation of ELR algorithm Although the ELR algorithm described above is centralised, however, the link-replacing strategies proposed in the algorithm is well suited for a distributed implementation. Such implementation is important for some practical situations where global topology knowledge is not immediately available or topology may be changing frequently. In this section, we give a brief description of a distributed ELR algorithm based on ant algorithm (Subing and Zemin, 2000). Assume that each node can adjust its energy-level to a maximum value, such that the transmitted signal can be received by all destination nodes. Initially, source node can adjust its energy-level to a maximum value so that all the destinations can be reached to setup a two-level multicast tree. The overall energy consumption of the multicast tree can be iteratively reduced by an ant-based distributed algorithm, which consists of two stages described as follows. •

Exploration stage. The source node sends to every direct son-node a forward-ant. Each intermediate node duplicates the forward-ant it receives, and sends each direct son-node a copy of the forward-ant. When a forward-ant is duplicated, the information stored at it is also copied to new ants. Every forward-ant records all the links it passes during its trip to a destination node. Finally, each destination node receives a forward-ant. The longest link can be determined and stored at a new type of ant called the backward ant. Each destination node then sends a backward-ant to upstream nodes using the reverse path recorded in the forward-ant. When a backward-ant arrives an intermediate node, it can only continue its trip up to source node, if the longest link stored at it is larger than the link stored at the intermediate node. Otherwise, the backward-ant is deleted. Based on the backward-ants received by the source node, the link associated with the highest power in the current multicast tree can be determined. The source node then informs the node which owns this link.

•

Reconstruction stage. Assume node p is informed by source node that the link (p, r) is the highest-power link and should be replaced. Based on the five link-replacing strategies described above, the reconnecting point q is determined by searching the neighbourhood’s area of nodes p and r. The radius of searching area is limited to be two links away from node p or r to save the computation time. According to the best alternate path found, all relative nodes adjust their energy-level so that tree edges can be reconstructed. The information about modifications are sent back to source node by node p.

Figure 8 Omni-directional check

After source node receives a modification-message from node p, a cycle is completed, and total energy required by the multicast tree is then recomputed. The distributed

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The investigation of delay-constrained multicasting ELR algorithm consists of a number of cycle. Each cycle has two stages: exploration and reconstruction. In the exploration stage, a link with high-power is determined. In the reconstruction stage, an alternate path is found to substitute the high-power link. After a number of cycles, the total energy expenditure by the multicast tree can be gradually reduced.

Figure 9 The performance comparisons when α = 2

6 Simulation results In this section, we have several sets of experiments to compare performance and efficiency for our centralised ELR algorithm and the MIP algorithm (Wieselthier et al., 2002). The MIP algorithm is derived from the BIP algorithm with sweep() operation presented in Wieselthier et al. (2002). In this study, the BIP method has been modified to take the given delay constraint into consideration during the spanning-tree construction phase, where a selected node can be added into the tree only if the resulting tree satisfies the given delay constraint. All the simulations are done with the following experimental environment: P4 3.0 GHz CPU, 256 MB RAM, Windows XP OS. The simulation programs were developed by Visual C++. Networks with 100 nodes are randomly generated within a 100 × 100 square region. The number of nodes in a multicast group is in the range of ten nodes to 50 nodes. The delay constraints for 100-node networks are set to be around ten hops. It is not practical to assign large delay constraints to wireless real-time applications. Each data reported in this study is the average value of 100 sets of data, which are measured based on 100 different random networks.

6.1 Performance

Figure 10 The performance comparisons when α = 3

Figure 11 The performance comparisons when α = 4

Recall that the wireless environments are reflected by the α values in equation (1), and the larger α value represents the higher interference in radio transmissions. Hence, our simulations of performance comparisons are carried out based on three different α values which are 2, 3, 4, respectively. The experimental results are shown in Figures 9–11, where a term called ‘power ratio’ is defined as follows: power ratio = (total power required by ELR)/ (total power required by MIP) × 100%. For each benchmark reported in Figures 9–11, the multicast tree found by ELR requires less power than the one determined by MIP. For example, the power ratios decrease from 80% to around 30%, when the delay constraint is set to be 12 hops and the α values have changed from 2 to 4. It indicates that ELR performs much better than MIP when the interference of radio transmissions becomes high. We also notice that the power ratios decrease as the numbers of delay hops are reduced. It implies that the performance of our ELR is superior to MIP when the delay constraint becomes strict.

In Figure 12, a term named ‘superior ratio’ is used to show the superiority of the ELR over the MIP. Its definition is shown as follows: superior ratio = (number of runs in which the ELR performs better than the MIP)/ (total number of runs) × 100.

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For each data shown in Figure 12, it is an average value of five sets of simulations, where each set of simulations is carried out based on different sizes of multicasting groups for 100 distinct 100-node networks. The sizes of multicasting groups are set to be 10∼50% for 100-node networks. The experimental results show that the probability of that ELR performs better than MIP is higher than 90% for all of three various wireless environments.

tree-construction phase. This is an important advantage for managing large-scale networks, especially when only small amount of nodes participate in the multicasting. Figure 13 The efficiency comparisons when α = 2

Figure 12 The superior ratio for ELR-to-MIP

Figure 14 The efficiency comparisons when α = 3

The above phenomenon is due to that our ELR method is designed by taking both delay-constraint and power-consumption into considerations at the same time, while MIP is designed only for minimising the total power expended. As a result, the multicast trees built by MIP tend to be highly unbalanced. Therefore, ELR heuristic is more suitable than MIP algorithm for the real-time applications.

6.2 Efficiency Based on three different α values in Figures 13–15, it is shown that the ELR is much more efficient than the MIP. For example, when the delay constraint is 12 hops in Figure 13, the CPU time required by ELR is only 18.4% of the CPU time required by MIP for 100-node networks with 50 nodes participating multicasting. However, the cpu-time ratio tends to slightly increase as the α value increases. It is also noticed that the cpu-time ratios decrease when the numbers of delay hops are reduced. It has once again shown that the executing efficiency of our ELR is superior to MIP when the delay constraint becomes stringent. Since the multicast tree found by MIP is reduced from a broadcast tree (i.e., a spanning tree), which is determined by BIP method presented in literature (Wieselthier et al., 2002), the running time required by MIP tends to be large when the large-size networks are considered. Hence, although the time complexity required by both ELR and MIP algorithms is O(n3 ) at worst case, ELR constantly outperforms MIP in terms of cpu-time ratio as the results shown in Figures 13–15. High efficiency of ELR is due to that only destination nodes and a small set of relay nodes are needed to be considered during the

Figure 15 The efficiency comparisons when α = 4

For example, in Table 1, the cpu time required by MIP does not depend on the number of nodes in the multicast group,

The investigation of delay-constrained multicasting whereas the cpu time required by the ELR decreases as the size of multicast group decreases. In particular, it only takes 0.16 sec on the average for ELR to determine the multicast tree when 50% of nodes participating in the multicasting. Therefore, it is concluded that ELR is a practical approach for the MEDM problem in large-scale networks. Table 1 The running time of ELR and MIP algorithms for 100 distinct networks ∗

Network size = 100 nodes, ∆ = 12 hops, α = 4 # of nodes in the multicast group 10 20 30 40 ELR cpu-time (s) 2.28 4.86 8.19 11.82 MIP cpu-time (s) 43.73 43.34 43.22 43.61

50 16.74 43.08

7 Conclusions The MEDM problem concerning the determination of a delay-constrained multicast tree with minimum-power consumption is studied in this paper for ad hoc wireless networks. Based on a number of link-replacing strategies, a centralised heuristic called ELR algorithm is developed for constructing a multicast tree for the MEDM problem. These strategies has been proved that they can cover all the link-replacing situations if only the local areas two links away from the substituted links are allowed for finding the substituting paths. A distributed implementation for ELR is also provided. Based on various wireless environments, the simulation results show that our centralised ELR consistently outperforms the famous MIP method in terms of both solution-quality and executing efficiency, when the same delay constraint is given for a multicast tree. It concludes that the ELR is a suitable method for determining energy-efficient multicast trees subject to QoS supports for real-time multimedia systems deployed on wireless networks. In future work we intend to implement and evaluate the distributed version of ELR algorithm based on static and mobile wireless networks.

Acknowledgements This work was supported by the National Science Council, ROC (NSC-94-2213-E-251-003).

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