Energy efficient all-to-all broadcast in all-wireless networks

Information Sciences 180 (2010) 1781–1792

Contents lists available at ScienceDirect

Information Sciences journal homepage: www.elsevier.com/locate/ins

Energy efficient all-to-all broadcast in all-wireless networks Doina Bein *, S.Q. Zheng Department of Computer Science, University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083-0688, USA

a r t i c l e

i n f o

Article history: Received 20 November 2008 Received in revised form 1 August 2009 Accepted 6 November 2009

Keywords: All-to-all broadcast Broadcast tree Convergecast Energy efficiency Minimum power Wireless network

a b s t r a c t All-to-all broadcast is a communication pattern in which every node initiates a broadcast. In this paper, we investigate the problem of building a unique cast tree of minimum total energy, which we call Minimum Unique Cast (MUC) tree, to be used for all-to-all broadcast. The MUC tree is unoriented and unrooted. We study three known heuristics for the minimum-energy broadcast problem: the Broadcast Incremental Power (BIP) algorithm, the Wireless Multicast Advantage-conforming Minimum Spanning Tree (WMA-conforming MST) algorithm, and the Iterative Maximum-Branch Minimization (IMBM) algorithm. Experimental results conducted on various types of networks are reported. We show that neither of these methods is best overall for building all-to-all broadcast trees. Ó 2009 Published by Elsevier Inc.

1. Introduction In wireless ad hoc networks basic tradeoffs exist between energy and information, and their time critical effect on operations have been studied extensively. Architectural design issues with regards to the number of nodes needed to cover a certain region, placement of such nodes, clustering, and routing are generally treated as emergent behaviors. Worst-case scenarios are studied in order to obtain either exact or approximate solutions. One calls a communication the process of sending a message in a single transmission. Broadcast (one-to-all), multicast (one-to-some), convergecast (all-to-one) and anycast (all-to-all) are important communication mechanisms for diffusing information in the network. A broadcast is the mechanism through which a message sent by a fixed node, called initiator or source, reaches the rest of the nodes. A multicast is similar to a broadcast, with the restriction that the message needs to reach only a subset of nodes. Many routing protocols for wireless networks need a broadcast or multicast mechanism to update the routing tables in order to maintain routes between nodes. Given a fixed node called sink, a convergecast is the mechanism through which each node sends a message to that sink. Thus a convergecast is the dual of a broadcast since the data flows back to a single node. For an example of convergecast, consider acquiring data in a wireless sensor network from the leaf nodes (i.e. the sensors), to the root node (referred to as the ‘‘source” or ‘‘initiator”) for collection and analysis. Indeed, this is the most common type of communication in a wireless sensor network. If the convergecast to a certain node r uses as communication paths a broadcast tree rooted at r, then the total cost of this convergecast can be different from the total cost of the broadcast initiated by r. The reason is that during a convergecast, each node has to send the data to its parent and not to its children. Thus the total energy spent during a convergecast, which uses communication paths arranged as a tree rooted at the sink node, is the sum of the energy spent by each node to send that packet to its parent. In all-to-all broadcast, every node is an initiator for broadcasting messages in the network. But constructing and maintaining n individual broadcast trees for an n-node network where each is rooted in an initiator is unfeasible since it requires

* Corresponding author. Tel.: +1 972 883 2184; fax: +1 972 883 2349. E-mail addresses: [email protected] (D. Bein), [email protected] (S.Q. Zheng). 0020-0255/$ - see front matter Ó 2009 Published by Elsevier Inc. doi:10.1016/j.ins.2009.11.013

1782

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

Cast Tree

Tree

Network Fig. 1. Minimum cast tree for all-to-all broadcast.

large memory space and/or large and fast processing capabilities at each node. Large bandwidth is needed when such broadcast trees are computed. Additionally, significant bandwidth is needed when a network topology change occurs. This is because in the worst case, all nodes have to recompute their broadcast trees when a single topology change occurs. A simple solution, albeit the most power-consuming, is to allow a node to send its data directly to all the nodes in the network. Then the transmission range of any node is proportional to the maximum distance between a pair of two nodes. In a wireless network, a node equipped with an omnidirectional antenna can simultaneously communicate with multiple nodes located within some distance which is called the transmission range. A message sent by some node v with transmission range r v is received simultaneously by all nodes situated at a distance no more than r v . We call these nodes siblings or neighboring nodes. The power required to support a communication link of length d between two nodes is divided into two components: the power to send a packet and the power to receive a packet. To send a packet at distance d, a node uses the power a

P e ¼ d þ ce :

ð1Þ

Here a is a constant parameter which depends on the medium and typically has a value between 2 and 4. The value ce is a parameter representing an overhead due to signal processing. For any two nodes u and v, let duv be the physical distance between u and v. We use notation Puv to denote the minimum energy that node u has to use for the transmission of one message in order to ensure that the message will reach node v. Consider an example of four nodes, a; b; c, and d, with arbitrary coordinates chosen in such a way that dab < dac < dad . To communicate with the nodes b; c, and d, node a would spend at least P ab ; P ac , and P ad energy to reach each individual node. But a transmission of power P 1 ¼ maxfPab ; P ac ; P ad g at node a with an omnidirectional antenna will reach the nodes b; c, and d simultaneously, while a transmission of power P 2 ¼ maxfPab ; P ac g at node a with an omnidirectional antenna will reach only nodes b and c. There is a tradeoff between the energy spent and the connectivity among nodes in the network. In order to save energy, a node tends to reduce its transmission range, in this way reducing also the number of possible neighbors. But in order to keep the network connected, some nodes may have to spend more energy to reach other nodes beyond the nearest ones. If that is the case, a node will attempt to minimize its transmission range and thus its power level, while some nodes will have to increase their power level in order to maintain a connected network. In this paper, we address the issue of minimizing total energy for the all-to-all broadcast when all nodes need to broadcast one message each. We propose a power assignment for each node with the purpose of creating communication paths form an unrooted spanning tree, with the additional requirement that the tree has the minimum total energy among all such possible trees. We call this tree T the minimum unique cast (MUC) tree (see Fig. 1). We propose a heuristics, called MUCT, which determines for each node i in the network the minimum-power broadcast tree rooted at i using some approximation algorithm to build broadcast trees. (In our experiments we use the algorithms proposed in [28,13,4].) Then MUCT selects the tree T which achieves the minimum value for the total power, powerT , given by Eq. (3). The paper is organized as follows. In Section 2 we present related work. In Section 3 we define the MUC problem. We propose an approximation algorithm for the MUC problem in Section 4. In Section 5 we show the relationship between the broadcast, the convergecast, and the MUC problem, and we give upper and lower bounds for the total power required by the MUC problem. Experimental results are presented in Section 6. We conclude in Section 7. 2. Related work Minimizing total energy consumption – the sum of the energy spent by each node – for broadcast and multicast has been then extensively studied in the literature under the assumption that the transmission range of a node can be adjusted (see [28,23,29,13,25,31,4,9,26,30,24,3]).

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

1783

The work of Wieselthier et al. [28] as well as Stojmenovic et al. [23] has given a new orientation in designing broadcast and multicast trees for wireless networks. Wieselthier et al. [28] introduced the notion of wireless multicast advantage and proposed the Broadcast Incremental Power (BIP) algorithm. Starting from the source node, BIP constructs the broadcast tree incrementally, by adding nodes to the tree based on the minimum additional cost. Empirical results showed the advantage of BIP over shortest-path tree, minimum-spanning tree, and variations, but no performance analysis was provided. Stojmenovic et al. [23] proposed the concept of internal nodes as alternatives to clusterheads, to reduce the communication overhead of broadcasting. A yardstick metric was proposed in [29] to measure the efficiency of approximation algorithms, in terms of how many destinations are reached per unit energy; the purpose is to reach a large fraction of the number of desired destination (for a multicast-tree). Li and Nikolaidis [13] proved that the minimum-energy broadcast problem (i.e. building a broadcast tree rooted at some given node) is NP-hard, and proposed the Iterative Maximum-Branch Minimization (IMBM) algorithm. Independently, Cagali et al. [4] and Liang [14] proved that the problem is NP-complete. Maric and Yates [17] proposed a new strategy, called accumulative broadcast, in which a node collects energy from unreliable reception. Das et al. [7] proposed three integer programming models for the broadcast/multicast tree problem, which can be solved optimally using branch-and-bound (exponential time), but also gave approximations using genetic algorithms. Montemanni and Gambardella [18,19] as well as Montemanni et al. [20] studied the problem of assigning transmission power to the nodes of a static wireless network in such a way that all nodes are connected by bidirectional links and the total power is minimized. Once the broadcast tree is constructed, it needs to adapt to unreliable communication from interference and to link failures due to energy depletion of some nodes. Wang and Chao [27] proposed a backup mechanism at each node for quick recovery of the routing data due to link failures. The algorithm proposed, called Dynamic Backup Routes Routing Protocol (DBR2P), creates alternative routes on-demand. Chang et al. [5] addressed the problem of degraded performance of an broadcast tree due to node mobility or diminished energy, and proposed to search for alternative routes to compensate for the broken links. For building a broadcast tree of minimum total energy, the source node and the network are the instance of the problem. The total power used by the nodes in the tree rooted at some source node and spanning the entire network is to be minimized among all such trees. Hence, for an n-node wireless network, each node has to keep track of n rooted broadcast trees, including the one rooted at itself. Papadimitriou and Georgiadis [21] constructed a single broadcast tree for the entire network; broadcasting initiated by any source node takes place in a predetermined manner. They proposed a polynomial-time algorithm for building such a tree, in which the total power consumed for broadcasting by any node is at most 2Hðn  1Þ Pn times the optimal power. (The function HðnÞ is the harmonic function, HðnÞ ¼ k¼1 1k). The algorithm works for any type of weighted, general networks. They established that the minimum-spanning tree (MST) for Euclidean graphs is not always the broadcast tree of minimum total power. They also showed that the MST is within D times the optimal total power of any tree, where D is the maximum node degree in the network. The convergecast problem proposed by Chlamtac and Kutten [6] has been studied from the point of view of collision detection [1,11,33], latency (total steps needed to collect the data) [32,12,8], and network lifetime (the time interval until the first node depletes its battery) [10]. For wireless sensor networks, Huang and Zhang [11] proposed a coordinated convergecast protocol to solve the collision problem. Zhang and Huang [33] studied data aggregation and duplication, whereas [32,22] focused on channel contention and packet collision, when a high-volume traffic occurs in a short period of time. In this context they designed a window-less block acknowledgment scheme. Gandham et al. [8] and Kesselman and Kowalski [12] studied the problem of minimizing the total time necessary to complete a convergecast. They proposed distributed convergecast scheduling algorithms as well as randomized distributed algorithms for the TDMA problem. He et al. [10] presented an aggregation protocol with the objective of maximizing the network lifetime and minimizing the error of sensed data. The authors proposed to periodically modify a filter threshold for each sensor in a way that is optimal, by translating the problem into a mathematical programming formulation with constraints. Their work took into consideration the objective of the user, the current characteristics of the sensor network (namely the power remained at each node and the connectivity) and the characteristics of the sensed data. All-to-all broadcast consumes much energy. Few results exist for this type of communication. Existing solutions for single [21] or source-dependent broadcast trees [28] are not optimal for all-to-all broadcast, since back-communication to the initiator is not considered. The effect on energy efficiency was studied by [15,16]. An earlier approach [15] is to select a central node that is the closest to all the nodes, let the central node collect the data and send then to everyone else. This approach is not very expensive in power, but requires central coordination. It is extended to a distributed scheme ([16]), in which the network is partitioned into clusters and some selected nodes (called clusterheads) play the role of a central node for their clusters. Bauer et al. [2] proposed a data structure, called legend, which gathers and shares its contents with visited nodes; several traversal methods have been explored.

3. Models A static n-node wireless network can be modeled as a pair ðV; wÞ where V is the set of nodes, jVj ¼ n, and w is a non-negative function, which is defined over V  V, and which measures the distance between the nodes. If the nodes have assigned certain power values then we model the network as a weighted digraph G ¼ ðV; E; wÞ, where E is the set of arcs that represent unidirectional communication links: The power level of some node decides to which nodes it is connected. We note that if

1784

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

the power value of some node changes, then the set E may change. Given any two nodes i and j in G, we define the function cost to be costij ¼ waij if ði; jÞ 2 E and 1 otherwise (where a is the constant defined in Eq. (1)). Nodes do not necessarily have the identical transmission ranges. The transmission range of each node dictates whether there is an arc to some other node. Thus between two nodes u and v, there can be an arc ðu; v Þ 2 E, or ðv ; uÞ 2 E, or both, or neither. Consider the example given in Fig. 2. The power values of each node are in written in the box under the node IDs. The power level of node a has is high enough for a transmission to cover the distance to node c. The power level of node b is enough to cover the distance to node d, and for node d to cover the distance to node c. But node c does not have enough power to reach any node. If we consider the function w to be the Euclidean distances between nodes, then there is a bidirectional communication between nodes b and d since wbd < wcd , and a unidirectional communication from a to b, as wbd < wab . Node c is isolated, since there is no node situated at a distance less than wcd . For the rest of the paper we assume the Euclidean distance as the metric distance among the nodes. By selecting (n  1) edges to connect the nodes in G, we are able to obtain an unoriented (or unrooted) tree T ¼ ðV; ET Þ. The total number of possible unoriented trees buildable from an n-node network is no more than ðn þ 1Þn1 , by the Cayley’s tree enumeration. In an unoriented tree, there is no notion of parent or child for a node, only the notion of siblings – the set of nodes to which a node is connected. For unoriented tree T and for any node i, let Si be the set of siblings of i in T: Si ¼ fj 2 V : ði; jÞ 2 ET g. We propose to select the transmitting energy at a node based on the distance to the farthest sibling. Given a tree T ¼ ðV; ET Þ rooted at some node r which spans the underlying graph G ¼ ðV; EÞ, we propose a new measure for the power used by a node i, powerTi , to be

powerTi ¼ maxj2Si costij :

ð2Þ

We denote by powerT the total power of all the nodes in the tree T:

powerT ¼

X

powerTi :

ð3Þ

i2V

Different power assignments at a node generate different network topologies. In Fig. 3, let jxyj be the distance between nodes x and y, for x; y 2 fa; b; c; dg, in such a way that jcdj < jacj < jabj < jbcj < jbdj; jacj < jabj < jadj, and jcdj < jadj. In Fig. 3a, if node c has power P ca (enough to reach nodes a and d but not node b), then the only possible spanning tree for the entire network is T 1 , which is the tree rooted at node b and with the set of arcs fðb; aÞ; ða; cÞ; ðc; dÞg. The total power used for broadcasting with node b as the source node is given by P ba þ P ac þ P cd . If the nodes set their power levels as indicated by Eq. (2), then powerT 1 ¼ P ab þ P ba þ P ca þ P dc .

b

d

Pbd

Pcd

a

c

Pac

< Pcd

Fig. 2. Asymmetric communication between nodes.

a

a

Pac

c P ca b Pba

b

d a

Pdc

Pac

c P cb b Pba

d P dc

Fig. 3. Selecting the transmission power.

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

1785

In Fig. 3b, if node c has power P cb (enough to reach all the nodes a, b, and d), then another tree T 2 , which is rooted at c and with the arcs fðc; aÞ; ðc; bÞ; ðc; dÞg, can be considered. The total power used for broadcasting with node c as the source node is given by P cb . If the nodes set their power levels as indicated by Eq. (2), then powerT 2 ¼ P ac þ Pbc þ Pcb þ Pdc . One can observe that T 2 has total broadcast power smaller than T 1 since jbcj < jbaj þ jacj (triangle inequality), jbaj þ jacj < jbaj þ jacj þ jcdj, and power at a node is defined as in Eq. (1). At the same time, powerT 1 < powerT 2 since we assume jbaj < jbcj. Thus selecting the tree with the minimum total broadcast power is not necessarily a good heuristic for selecting the tree with the minimum total power for all-to-all broadcast. During a single-source broadcast a node may receive the same message more than once. Therefore, the broadcast does not induce an oriented tree, but a connected graph which contains an oriented tree. Then, by eliminating the redundant messages, we can consider the induced graph to be a tree. We formulate the minimum unique cast (MUC) graph problem as follows: MUC: Given a wireless network ðV; wÞ, assign power levels to the nodes such that the corresponding unidirectional links (arcs) formed between nodes induce a strongly connected graph which can be used for all-to-all broadcast and such that the sum of power of all nodes is minimum. By choosing some node in the network as the root (or source node), an unoriented tree can become oriented: select the parent of each node as the neighboring node with the shortest distance to the root. 4. Minimizing total energy for all-to-all broadcast Since the value powerT is an upper bound for the total power to be spent during any single-source broadcast or any singlesink convergecast in T (Theorem 1), we need to select the tree with the minimum such value. We also show that for any tree T, in case of a broadcast followed by convergecast towards the same node, we cannot do better than powerT . In other words, the total energy spent is more than power T . Thus powerT is a lower bound for any broadcast followed by a convergecast (Lemma 5)). Consider the following algorithm MUCT for a given wireless network ðV; wÞ. For each node r in V, one can construct a single-source broadcast tree T r rooted at r using some approximation algorithm. (See [28,23,29,4,17,7,18–20] for approximation algorithms). For each tree T r , consider at the nodes the power levels given by Eq. (2) and the total power given by Eq. (3). We apply the following heuristic: select the tree T o that has the minimum value for the total power, power T o ¼ r min powerT . 8r2V

Algorithm 4.1. Algorithm MUCT (Minimum Unique Cast Tree)) Read the input: ðV; wÞ Initialization: Let powero ¼ 1 and OPT ¼ ;. Main Procedure: Forall r 2 V do Build a broadcast tree rooted at r; BT, using an approximation algorithm A. Let T to be the unoriented tree obtained by ignoring the orientation in BT. Define the power level of some node i as in Eq. (2). Compute powerT as in Eq. (3). If powerT < powero then set OPT to T and powero to powerT . Endfor

The power level of any node in the tree T o ensures a bidirectional communication with its siblings. Thus the tree can be used for all-to-all broadcast, and Algorithm MUCT is an approximation for the MUC problem. We note that when a node sends data to its parent, all its children receive the same data since they are within the range: when node i sends its data back to its parent pi , the data is also automatically received by all i’s children at no additional cost. When the data is then forwarded to the parent of pi , all pi ’s children receive it at no additional cost, and so on. These nodes could also send the packet along their own subtrees, eventually piggybacking with other data. 5. Proofs Let T be an oriented tree spanning the underlying graph of G, rooted at some node r and constructed by some approximation algorithm. We choose the power of some node i to be the maximum cost for reaching the siblings (Eq. (2)). We show that between any two siblings in T there is a bidirectional communication link (Lemma 1). We also show that during a broadcast from any node in the tree, the total power spent is less than the value of powerT given in Eq. (3) (Lemma 2). Furthermore, during a convergecast back to any node in the tree the total power spent is less than the value of powerT (Lemma 3). This concludes a lower bound for power T (Theorem 1).

1786

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

Then, we show that the total power spent during a convergecast is greater than half of powerT (Lemma 4) and during a broadcast followed by a convergecast, the total power spent is at least powerT (Lemma 5). This concludes an upper bound for powerT (Theorem 2). Lemma 1. For any node i in the tree, if the power level at i is defined as in Eq. (2) then there is a communication path between any two nodes in the tree. Proof. Recall that the power level of a node decides whether there is a unidirectional, bidirectional, or no communication between the node and some other node. We show that there will be a bidirectional communication between any two neighboring nodes in T if we select the power level of some node to be the value given in Eq. (2). Let i be some node in T and j be some sibling of i in T, j 2 Si . From Eq. (2) it follows that powerTi P costij thus there is an unidirectional communication link from i to j. Similarly, powerTj P cost ji for node j. Therefore, there is a unidirectional communication link from j to i. Thus between any two sibling nodes in T there is a bidirectional communication. This implies that there is a communication path between any two nodes in the tree. h Since between any two neighboring nodes there is a bidirectional communication link, we can consider T to be a graph instead of a digraph. When we consider the orientation of the tree T with respect to some node r as the root, the ancestor/descendant relationship between nodes induces a partial order: i
pr;T ¼ max costij : i

ð4Þ

8j2Si r6i
The total power spent for r-broadcast is pr;T

pr;T ¼

X

pr;T i :

ð5Þ

i2V

Obviously, pir;T 6 power Ti , for any nodes i and r. Moreover, if i is a leaf node of the oriented tree T rooted at r, then pir;T ¼ 0, thus pir;T < powerTi , for any node r. It follows then that the total power spent for r-broadcast, pr;T , given by Eq. (5), is strictly smaller than the value of powerT . h Lemma 3. If any node i in the tree T has the power level as defined in Eq. (2) , then the total power used for a convergecast towards some node r along the tree T is less than the value of powerT given by Eq. (3). Proof. Some node i other than r uses during the r-convergecast an amount of power, ppir;T , equal to

ppr;T ¼ costik ; i

ð6Þ ppr;T r

where k 2 Si ; r 6 k < i, and ¼ 0. The total power spent for r-convergecast is ppr;T

ppr;T ¼

X i2V

ppr;T ¼ i

X

costij :

ð7Þ

ði;jÞ2T;i
Obviously, ppir;T 6 powerTi , for any nodes i and r. Moreover, if i is a leaf node in the oriented tree T rooted at r then T ¼ powerTi . Note also that ppr;T r ¼ 0 < power r . It follows then that the total power spent for r-convergecast, ppr;T , given by Eq. (7), is strictly smaller than the value of powerT . h

ppir;T

We can then conclude: Theorem 1. For any node r in the network G and any tree T which spans G, if the power level at r is the one defined in Eq. (2) then

powerT P maxðpr;T ; ppr;T Þ: Lemma 4. The total power used for convergecast towards some node r in the tree given by Eq. (7) is greater or equal to half of powerT given by Eq. (3), for any node r.

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

1787

P P P Proof. Note that 2ppr;T ¼ 2 ði;jÞ2T cost ij ¼ i2V j2SðiÞ costij . For any set A, the sum of elements in A is greater or equal to the maximum element in A. It follows that, for any node P i; j2SðiÞ costij P maxj2SðiÞ costij . In this inequality, the right expression is powerTi . Thus 2ppr;T P powerT . h Lemma 5. If for any node i in the tree T, the power level at i is the one defined in Eq. (2), then the total power used for a broadcast initiated by some node r followed by a convergecast towards r along the tree T is greater or equal to the value of powerT given by Eq. (3). Proof. The power spent by a node i during r-broadcast and r-convergecast is pir;T þ ppir;T . The power spent by all nodes during the r-broadcast and r-convergecast is

X r;T r;T ðpi þ ppr;T þ ppr;T : i Þ ¼ p i2V

Let SðiÞ be the set of neighboring nodes, which includes the parent of i and its children. Call the parent of i as the node k. Using Eq. (2) it follows that

powerTi ¼ max costij ¼ maxðcostik ; max cost ij Þ: 8j2Si

8j2Si r6i
For any two non-negative values a and b; maxða; bÞ 6 a þ b; it follows that

maxðcostik ; max costij Þ 6 costik þ max costij : 8j2Si r6i
8j2Si r6i
In this inequality, the left expression is powerTi (Eq. (3)), and the right expression is the sum of pr;T (Eq. (4)) and ppir;T (Eq. (6)). i Thus for any node i, powerTi 6 pir;T þ ppir;T . It follows then directly that powerT 6 pr;T þ ppr;T . h We can then conclude: Theorem 2. For any node r in the network G and any tree T that spans G, if the power level at r is the one defined as in Eq. (2) then

powerT 6 ppr;T þ minðppr;T ; pr;T Þ: From Theorems 1 and 2 it follows that maxðpr;T ; ppr;T Þ 6 powerT 6 pr;T þ ppr;T . 6. Comparative analysis and results To validate the effectiveness of the proposed MUCT scheme, we have conducted extensive experiments to compare the minimum power used for all-to-all broadcast based on the broadcast trees generated by algorithms BIP, IMBM, WMA-conforming MST, in terms of total energy. These experiments took into account different types of data sets and different network sizes. Experimental results conducted on various types of networks using BIP, IMBM, and WMA-conforming MST showed that both BIP and WMA-conforming MST are good for building all-to-all broadcast trees, while IMBM produces trees of very high energy cost. We have used four sets of networks. The first set included general networks with arbitrary distance among nodes randomly generated in the range [0.01..4.99], and a randomly chosen source node. We call this set of networks random general networks. The second set included Euclidean networks where the coordinates of the nodes were generated randomly in the range [0.01..4.99] and a source node was randomly chosen. We call this set of networks random Euclidean networks. The third set included Euclidean networks where nodes were placed on a square grid with distances between rows and columns randomly generated in the range of [0.05, 1.04] and the source node was randomly chosen from grid nodes. We call this set of networks random Euclidean grid networks. The fourth set included Euclidean networks with nodes placed on a square grid where the row and column distances were the same, and the source node was randomly chosen from grid nodes. We call this set of networks perfect Euclidean grid networks. Since computing the optimal broadcast tree for a network with more than 10 nodes with randomly generated node locations would have taken too long, we measured the performance of BIP, IMBM, and WMA-conforming MST in terms of minimum energy used for all-to-all broadcast based on the broadcast trees generated by the corresponding method. For each network instance m, let P BIP ; P IMBM , and PWMA-MST be the minimum total energy used for all-to-all broadcast using the broadcast trees generated by algorithms BIP, IMBM, and WMA-conforming MST, respectively. We select the minimum energy tree among them,

Pmin ¼ minfPBIP ; P IMBM ; P WMA-MST g: GT GT Then we normalize P BIP ; P IMBM ; P WMA-MST against P min : pðBIPÞ ¼ PPmin ; pðIMBMÞ ¼ PPmin and pðWMA  MSTÞ ¼ PWMA-MST . The normalPmin ized energy associated with the tree generated by an algorithm is independent on the size of distance scaling factor.

1788

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

6.1. Experiments with a low node density We considered first networks of low node density, within the range 4  105 to 16  104 . The number of nodes for the first, second, third, and fourth set of networks was in the range of [5, 400], [5, 400], [9, 400], and [9, 400], respectively. Similar to [28,13], we ran 1000 simulations for each setup consisting of a network m of a specified size and node distances, parameter a, and an algorithm. For random general networks, BIP outperformed WMA-conforming MST when the number of nodes was greater than 10, and both outperformed IMBM. Fig. 4 summarizes the performance of the three algorithms. (The results were obtained by sampling 1000 such networks of 5, 10, 20, 30, . . ., and 400 nodes, respectively.) The reason is that BIP selected a smaller number of internal nodes than WMA-conforming MST, of relatively small transmission radius. IMBM, even though it selected an even smaller number of internal nodes, the transmission range of these internal nodes was relatively large. For random Euclidean networks, WMA-conforming MST outperformed BIP very slightly, and both outperformed IMBM. Fig. 5 summarizes the performance of the three algorithms. (The results were obtained by sampling 1000 such networks of 5, 10, 20, 30, . . ., and 400 nodes, respectively.) The reason is that WMA-conforming MST selected a smaller number of internal nodes than BIP, of relatively small transmission radius. As for random general networks, IMBM, even though it selected an even smaller number of internal nodes, the transmission range of these internal nodes was relatively large. For random Euclidean grid networks, WMA-conforming MST slightly outperformed BIP and both outperformed IMBM. Fig. 6 summarizes the performance of the three algorithms. (The results were obtained by sampling 1000 such networks of 9, 16, 25, 36, . . ., and 400 nodes, respectively.) The reason is the same as for random Euclidean networks: WMA-conforming MST selected a smaller number of internal nodes than BIP, of relatively small transmission radius, while the transmission radius of the internal nodes selected by IMBM was relatively large.

Fig. 4. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for random general networks, a = 2.

Fig. 5. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for random Euclidean networks, a = 2.

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

1789

Fig. 6. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for random Euclidean grid networks, a = 2.

Fig. 7. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for perfect Euclidean grid networks, a = 2.

Fig. 8. Normalized total energy for all-to-all broadcast using BIP and WMA-conforming MST for random general networks, a = 2.

1790

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

For perfect Euclidean grid networks, WMA-conforming MST always outperformed BIP, and both outperformed IMBM. Fig. 7 summarizes the performance of the three algorithms. (The results were obtained by sampling 1000 such networks of 9, 16, 25, 36, . . ., and 400 nodes, respectively.) The reason is the same as for random Euclidean networks and random Euclidean grid networks: WMA-conforming MST selected a smaller number of internal nodes than BIP, of relatively small transmission radius, while the transmission radius of the internal nodes selected by IMBM was relatively large. 6.2. Experiments with a higher node density We considered then networks of higher node density, within the range 16  104 to 4  103 . The number of nodes for the first, second, third, and fourth set of networks was in the range of [400, 2400], [400, 2400], [400, 2500], and [400, 2500], respectively. We ran 100 simulations for each setup consisting of a network m of a specified size and node distances, parameter a, and an algorithm. We measured the performance of BIP, IMBM, and WMA-conforming MST in terms of minimum energy used for all-to-all broadcast based on the broadcast trees generated by the corresponding method. The trees obtained by Algorithm IMBM had a very large normalized total power (extremely large for general random networks), so we did not include them in the charts. Thus the charts contain only a comparison between the BIP and WMA-MST trees. We note that neither BIP nor WMA-MST generates the lowest minimum total energy trees overall. For random general networks, BIP always outperformed WMA-conforming MST. Fig. 8 summarizes the performance of the two algorithms. (The results were obtained by sampling 100 such networks of 400, 450, up to 2400 nodes, respectively.) The reason is that BIP selected a smaller number of internal nodes than WMA-conforming MST, of smaller transmission radius. For random Euclidean networks, WMA-conforming MST always outperformed BIP. Fig. 9 summarizes the performance of the two algorithms. (The results were obtained by sampling 100 such networks of 400, 450, up to 2400 nodes, respectively.)

Fig. 9. Normalized total energy for all-to-all broadcast using BIP and WMA-conforming MST for random Euclidean networks, a = 2.

Fig. 10. Normalized total energy for all-to-all broadcast using BIP and WMA-conforming MST for random Euclidean grid networks, a = 2.

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

1791

Fig. 11. Normalized total energy for all-to-all braodcast using BIP and WMA-conforming MST for perfect Euclidean grid networks, a = 2.

The reason is that WMA-conforming MST selected a smaller number of internal nodes than BIP, of smaller transmission radius. For random Euclidean grid networks, BIP outperformed WMA-conforming MST for most instances. Fig. 10 summarizes the performance of the two algorithms. (The results were obtained by sampling 100 such networks of 400, 441, up to 2500 nodes, respectively.) The reason is the same as for random Euclidean networks: BIP selected a smaller number of internal nodes than WMA-conforming MST, of relatively small transmission radius, while the transmission radius of the internal nodes selected by WMA-conforming MST was slightly larger. For perfect Euclidean grid networks, WMA-conforming MST always outperformed BIP. Fig. 11 summarizes the performance of the two algorithms. (The results were obtained by sampling 100 such networks of 400, 441, up to 2500 nodes, respectively.) The reason is the same as for random Euclidean networks and random Euclidean grid networks: WMA-conforming MST selected a smaller number of internal nodes than BIP, of relatively small transmission radius, while the transmission radius of the internal nodes selected by BIP were slightly larger. 7. Conclusion We propose an approximation algorithm MUCT which builds a unique cast tree to be used for all-to-all broadcast; the tree is unoriented and has minimum total power. To validate the effectiveness of Algorithm MUCT, we conducted extensive experiments. Experimental results conducted on various types of networks using BIP, IMBM, and WMA-conforming MST show that neither of these methods is overall the best for building all-to-all broadcast trees. In all the experiments, IMBM produces trees of much higher energy cost, compared with the trees produced by BIP and WMA-conforming MST. We also give a number of lower and upper bounds for the total energy of any tree that can be used for all-to-all broadcast. Our proposed algorithm provides a sufficient approximation for the MUC problem, but not a necessary one. It is an interesting open problem to find a tree of minimum total power for all-to-all broadcast which does not use a broadcast tree as a preprocessing step. Another future line of work would be to construct broadcast trees of minimum energy in which certain nodes can only be leaf nodes. This problem is motivated by the following aspect. In order to preserves its energy when is idle, namely it does not sense any event, a sensor node can go in a ‘‘listening” mode. In this mode, a sensor node only receives packets, but it does not send or forward packets. Thus, it can be part of the broadcast tree only as a leaf node, but not as an internal node. Acknowledgment This work is supported in part by National Science Foundation, Grant NSF-0714057. References [1] V. Annamalai, S. Gupta, L. Schwiebert, On tree-based convergecasting in wireless sensor networks, in: Proceedings of The IEEE Wireless Communications and Networking Conference, vol. 3, 2003, p. 1942. [2] N. Bauer, M. Colagrosso, T. Camp, Efficient implementations of all-to-all broadcasting in mobile ad hoc networks, Pervasive and Mobile Computing (2005) 311–342. [3] D. Bein, S.Q. Zheng, An effective algorithm for computing energy-efficient broadcasting trees in all-wireless networks, in: Proceedings of The Workshop on Wireless Ad Hoc and Sensor Networks (WWASN), 2008, pp. 273–278. [4] M. Cagali, J.P. Hubaux, C. Enz, Minimum-energy broadcast in all-wireless networks: NP-completeness and distribution issues, in: Proceedings of The ACM International Conference on Mobile Computing and Networking (MOBICOM), 2002, pp. 172–182.

1792

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

[5] B.-J. Chang, Y.-H. Liang, Y.-M. Lin, Distributed route repair for increasing reliability and reducing control overhead for multicasting in wireless MANET, Information Sciences 179 (11) (2009) 1705–1723. [6] I. Chlamtac, S. Kutten, Tree-based broadcasting in multihop radio networks, IEEE Transactions on Computers 36 (10) (1987) 1209–1223. [7] A.K. Das, R.J. Marks, M. El-Sharkawi, P. Arabshahi, A. Gray, Minimum power broadcast trees for wireless networks: integer programming formulations, in: Proceedings of The IEEE Conference on Computer Communications (INFOCOM), vol. 2, 2003, pp. 1001–1010. [8] S. Gandham, Y. Zhang, Q. Huang, Distributed minimal time convergecast scheduling in wireless sensor networks, in: Proceedings of The IEEE International Conference on Distributed Computing Systems (ICDCS), 2006, p. 50. [9] Z.J. Haas, J.Y. Halpern, L. Li, Gossip-based ad hoc routing, in: Proceedings of The IEEE Conference on Computer Communications (INFOCOM), vol. 3, 2002, pp. 1707–1716. [10] Z. He, B.S. Lee, X.S. Wang, Aggregation in sensor networks with a user-provided quality of service goal, Information Sciences 178 (9) (2008) 2128–2149. [11] Q. Huang, Y. Zhang, Radial coordination for convergecast in wireless sensor networks, in: Proceedings of the IEEE Workshop on Embedded Networked Sensors, 2004, pp. 542–549. [12] A. Kesselman, D.R. Kowalski, Fast distributed algorithm for convergecast in ad hoc geometric radio networks, Journal of Parallel and Distributed Computing 66 (4) (2006) 578–585. [13] F. Li, I. Nikolaidis, On minimum-energy broadcasting in all-wireless networks, in: Proceedings of The IEEE Conference on Local Computer Networks (LCN), 2001, pp. 193–202. [14] W. Liang, Constructing minimum-energy broadcast trees in wireless ad hoc networks, in: Proceedings of The ACM International Conference on Mobile Computing and Networking (MOBICOM), 2002, pp. 112–122. [15] S. Lindsey, C. Raghavendra, Energy efficient broadcasting for situation awareness in ad hoc networks, in: Proceedings of The International Conference on Parallel Processing, 2001, pp. 149–155. [16] S. Lindsey, C. Raghavendra, Energy efficiency all-to-all broadcasting for situation awareness in ad hoc networks, Journal of Parallel and Distributed Computing 63 (1) (2003) 15–21. [17] I. Maric, R. Yates, Efficient multihop broadcast for wideband systems, in: DIMACS Workshop on Signal Processing for Wireless Transmission, 2002, pp. 285–299. [18] R. Montemanni, L.M. Gambardella, Minimum power symmetric connectivity problem in wireless networks: a new approach, Proceedings of the IFIP IEEE International Conference on Mobile and Wireless Communication Networks, vol. 162, Springer-Verlag, 2004, pp. 497–508. [19] R. Montemanni, L.M. Gambardella, Exact algorithms for the minimum power symmetric connectivity problem in wireless networks, Computers and Operations Research 32 (11) (2005) 2891–2904. [20] R. Montemanni, L.M. Gambardella, A. Das, Mathematical models and exact algorithms for the min-power symmetric connectivity problem: an overview, in: Jie Wu (Ed.), Handbook on Theoretical and Algorithmic Aspects of Sensor, Ad Hoc Wireless, and Peer-to-Peer Networks, CRC Press, 2006, pp. 133–146. [21] I. Papadimitriou, L. Georgiadis, Minimum-energy broadcasting in multi-hop wireless networks using a single broadcast tree, Mobile Networks and Applications 11 (2006) 361–375. [22] T.-L. Sheu, Y.-J. Wu, A preemptive channel allocation scheme for multimedia traffic in mobile wireless networks, Information Sciences 176 (2006) 217– 236. [23] I. Stojmenovic, M. Seddigh, J. Zunic, Internal nodes based broadcasting in wireless networks, in: Proceedings of the Hawaii International Conference on System Sciences (HICSS), 2001, p. 10. [24] I. Stojmenovic, M. Seddigh, J. Zunic, Dominating sets and neighbor elimination-based broadcasting algorithm in wireless networks, IEEE Transactions on Parallel and Distributed Systems 13 (2002) 14–25. [25] P.-J. Wan, G. Calinescu, X.-Y. Li, O. Frieder, Minimum Energy Broadcast Routing in Static Ad Hoc Wireless Networks, in: Proceedings of The IEEE Conference on Computer Communications (INFOCOM), 2001, pp. 1162–1171. [26] P.-J. Wan, G. Calinescu, X.-Y. Li, O. Frieder, Minimum energy broadcast routing in static Ad Hoc wireless networks, ACM Wireless Networking (WINET) 8 (6) (2002) 607–617. [27] Y.-H. Wang, C.-F. Chao, Dynamic backup routes routing protocol for mobile ad hoc networks, Information Sciences 176 (2) (2004) 161–185. [28] J.E. Wieselthier, G.D. Nguyen, A. Ephremides, On the construction of energy-efficient broadcast and multicast trees in wireless networks, in: Proceedings of The IEEE Conference on Computer Communications (INFOCOM), 2000, pp. 585–594. [29] J.E. Wieselthier, G.D. Nguyen, A. Ephremides, Algorithms for energy-efficient multicasting in static ad hoc wireless networks, Mobile Networks and Applications 6 (3) (2000) 251–263. [30] J.E. Wieselthier, G.D. Nguyen, A. Ephremides, The energy-efficiency of distributed algorithms for broadcasting in ad hoc networks, in: Proceedings of IEEE International Symposium on Wireless Personal Multimedia and Communication, 2002, pp. 499–503. [31] J.E. Wieselthier, G.D. Nguyen, A. Ephremides, Energy-efficient broadcast and multicast trees in wireless networks, ACM Journal of Mobile Networks and Applications (MONET) 7 (6) (2002) 481–492. [32] H. Zhang, A. Arora, Y.R. Choi, M.G. Gouda, Reliable bursty convergecast in wireless sensor networks, in: Proceedings of the ACM International Symposium on Mobile Ad Hoc Networking and Computing, 2005, pp. 266–276. [33] Y. Zhang, Q. Huang. Coordinated convergecast in wireless sensor networks, in: Proceedings of the Military Communication Conference (MILCOM), vol. 2, 2005, pp. 1152–1158.