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Minimum-Power Multicast Routing in Static Ad
Hoc Wireless Networks
Peng-Jun Wan∗^ Gruia C˘alinescu∗^ Chih-Wei Yi∗
Abstract — Wieselthier et al. [16] proposed three greedy heuristics for M IN -P OWER A SYMMETRIC B ROADCAST R OUTING : SPT (shortest-path tree), MST (minimum span- ning tree), and BIP (broadcasting incremental power). Wan et al. [15] proved that SPT has an approximation ratio of at least n 2 where n is the total number of nodes, and both MST and BIP have constant approximation ratios. Based on the approach of pruning, Wieselthier et al. [16] also proposed three greedy heuristics for M IN -P OWER A SYMMETRIC M ULTICAST R OUTING : P- SPT (pruned shortest-path tree), P-MST (pruned min- imum spanning tree), and P-BIP (pruned broadcasting incremental power). In this paper, we first prove that the approximation ratios of these three heuristics are at least n− 2 1 , n − 1 , and n − 2 − o (1) respectively. We then present constant-approxiation algorithms for M IN -P OWER A SYMMETRIC M ULTICAST R OUTING. We show that any ρ -approximation Steiner tree algorithm gives rise to a cρ - approximation heuristic for M IN -P OWER A SYMMETRIC M ULTICAST R OUTING , where c is a constant between 6 and 12. In particular, the Takahashi-Matsuyama Steiner tree heuristic [14] leads to a heuristic called SPF (shortest- path first), which has an approximation ratio of at most 2 c. We also present another heuristic, called MIPF (minimum incremental path first), for M IN -P OWER A SYMMETRIC M ULTICAST R OUTING and show that its approximation ratio is between^133 and 2 c. Both SPF and MIPF can be regarded as an adaptation of MST and BIP respectively in a different manner than pruning. Finally, we prove that any ρ -approximation Steiner tree algorithm also gives rise to a 2 ρ -approximation algorithm for M IN^ -P^ OWER S YMMETRIC M ULTICAST R OUTING.
I. I NTRODUCTION
The recent advances in the development of afford- able and portable wireless communication and com- putation devices has fostered a tremendous amount of research on ad hoc wireless networks. Ad hoc networks can be used wherever a fixed backbone
∗Department of Computer Science, Illinois Institute of Tech- nology, Chicago, IL 60616. Emails: {wan, calinescu}@cs.iit.edu, yichihw@iit.edu.
infrastructure is not viable, e.g. in battlefield and emergency disaster relief. Typically, omnidirectional antennas are used by all nodes to transmit and re- ceive signals. Thus, a transmission made by a node can be received by all nodes within its transmission range. A communication session is achieved either through single-hop transmission if the recipient is within the transmission range of the source node, or by relaying through intermediate nodes otherwise. For this reason, ad hoc wireless networks are also called multi-hop packet radio networks.
One of the major concerns in ad hoc wireless networks is reducing node energy consumption. In fact, nodes are usually powered by batteries of limited capacity. Once the nodes are deployed, it is very difficult or even impossible to recharge or replace their batteries in many application scenarios. Hence, reducing power consumption is often the only way to extend network lifetime. As demon- strated by [7], the power consumption is dominated by communications. For the purpose of energy conservation, each node can (possibly dynamically) adjust its transmitting power, based on the distance to the receiving node and the background noise. In the most common power-attenuation model [10], the signal power falls as (^) d^1 κ where d is the Euclidean distance from the transmitter antenna and κ is the pass-loss exponent of the wireless environment–a real constant typically between 2 and 5. Assume that all receivers have the same power threshold for signal detection, which is typically normalized to one. Then, the power required to cover a transmis- sion range of radius d is dκ.
Due to the nonlinear power attenuation, the total power consumption required by a communication session can be potentially reduced by relaying signal through intermediate nodes [12] [16]. We assume throughout this paper that the set V of nodes in a given wireless ad hoc network are located in a
two-dimensional plane. A communication session is specified by a pair (s, D) where s ∈ V is the source and D ⊆ V \ {s} is the set of destinations. If D consists of a single node, the communication session is a unicast. If D consists of all nodes other than the source s, the session is a broadcast. For general D, the communication session is a multicast. Depend- ing on whether the communications is unidirectional or bidirectional, there are two variations on routing and power consumptions.
A. Min-Power Asymmetric Routing
In the case of unidirectional communications, a routing for a communication session (s, D) is an arborescence (a directed tree) T rooted at s which reaches all nodes in D. An arborescence T determines the transmission power of every node in T as follows: the transmission power of each sink is zero, and the transmission power of each node v of T other than sinks is equal to the κ- th power of the longest distance between u and its children in T. The total power required by T is then the sum of the transmission power of all nodes in T. The problem M IN -P OWER A SYMMETRIC ROUTING then seeks, for any given communication session (s, D), an arborescence of minimum total power which is rooted at s and reaches all nodes in D.
There are two special cases of M IN -P OWER A SYMMETRIC ROUTING. The special case in which all communication sessions are restricted to uni- casts is referred to as M IN -P OWER A SYMMET- RIC U NICAST ROUTING , and the special case in which all communication sessions are restricted to broadcasts is referred to as M IN -P OWER A SYM - METRIC B ROADCAST ROUTING. While the former is easily solved in polynomial time by shortest- path algorithms, the latter is NP-hard as shown by Clementi et al. [5]. Wieselthier et al. [16] presented three greedy broadcasting heuristics, namely MST (minimum spanning tree), SPT (shortest-path tree), and BIP (broadcasting incremental power), and eval- uated them by simulations. Wan et al. [15] provided a theoretical analysis on the approximation ratios of these three heuristics. They showed that MST has an approximation ratio between 6 and 12 , BIP has an approximation ratio between 143 and 12 , and SPT
has an approximation ratio as large as n 2 − o (1), where n is the number of nodes.
M IN -P OWER A SYMMETRIC M ULTICAST ROUT- ING is the most general form of M IN -P OWER A SYMMETRIC ROUTING where the communication sessions can be arbitrary multicasts. As a generaliza- tion of M IN -P OWER A SYMMETRIC B ROADCAST ROUTING , M IN -P OWER A SYMMETRIC M ULTI - CAST ROUTING is also NP-hard. Wieselthier et al. [16] adapted their three broadcasting heuristics to three multicasting heuristics by a technique of pruning, which we refer to as P-MST (pruned min- imum spanning tree), P-SPT (pruned shortest-path tree), and P-BIP (pruned broadcasting incremental power) respectively. The idea is to first obtain a spanning arborescence rooted at the source of a given multicast session by applying any of the three broadcasting heuristics, and then eliminate from the spanning arborescence all nodes which do not have any descendant in the given multicast session.
Wieselthier et al. [16] had evaluated their three multicasting heuristics through simulations, but their performance in terms of approximation ratios re- mained unsolved. The analysis in [15] on the per- formance of SPT can be extended to P-SPT, which leads to the same lower bound on the approxima- tion ratio of P-SPT. However, the analysis in [15] on the constant approximation ratios of MST and BIP cannot be extended P-MST and P-BIP. One might expect that both P-MST and P-BIP have the same approximation ratios as their broadcasting counterparts. Surprisingly, as will be demonstrated in this paper, neither P-MST nor P-BIP has constant approximation ratio. We show that P-MST has an approximation ratio of at least n − 1 , and P-BIP has an approximation ratio of at least n − 2 − o (1).
Since none of the three heuristics proposed by Wieselthier et al. [16] have constant approximation ratios, we are motivated to address the existence of heuristics with constant approximation ratio. We first prove that any heuristic of approximation ra- tio ρ for Steiner minimum tree (SMT) [8][9] in graphs gives rise to a heuristic for M INIMUM - E NERGY A SYMMETRIC M ULTICAST ROUTING of approximation ratio at most cρ, where c is a con- stant between 6 and 12. In particular, the Robins- Zelikovsky Steiner tree heuristic [11] leads to an
large as Ω (n), where n is the number of nodes in the network. We begin with the P-SPT algorithm.
Lemma 1: The approximation ratio of P-SPT is at least n− 2 1.
1−ε 1
2
3
1
2
3
v
u u v v v m
u m
ε (^) w^ u
Fig. 1. A bad instance for P-SPT.
Proof: Let be a sufficiently small positive number. Consider m nodes u 1 , u 2 , · · · , um evenly distributed on a circle of radius 1 centered at node w (see Figure 1). For 1 ≤ i ≤ m, let vi be the point in the line segment wui with ‖wvi‖ = . We consider a network consisting of n = 2m + 1 nodes
w,u 1 , u 2 , · · · , um, v 1 , v 2 , · · · , vm,
and a multicast session from node w to the m nodes u 1 , u 2 , · · · , um. The shortest-path tree is the superposition of the paths wviui. Since no node can be removed from this shortest-path tree, it is also the output by P-SPT. Its total power is
^2 + m (1 − )^2.
On the other hand, if the transmission power of node w is 1 , then the signal can reach all points u 1 , u 2 , · · · , um. Thus the minimum power is at most
- So the approximation ratio of P-SPT is at least ^2 + m (1 − )^2. As −→ 0 , this ratio converges to m = n− 2 1.
Next, we construct bad instances for P-MST.
Lemma 2: The approximation ratio of P-MST is at least n − 1.
2m+k
v
v v 0 1 2 m-1 m
m+
m+
m+
m+k- m+k- m+k v m+k+ v 2m+k- 2m+k-
v
......
v v v v
v
v
v
v v
Fig. 2. A bad instance for P-MST.
Proof: Fix a circle of radius 1 centered at node v 0 (see Figure 2), and choose a sufficiently large integer m. Let v 1 , · · · , vm be the m points on a radius of the circle with
‖vivi+1‖ =
m for 1 ≤ i < m. Then vm is on the circle. Let vm+1, · · · , vm+k be the k points distributed on the circle counterclockwise with
‖vivi+1‖ =
m for all m ≤ i < m + k and the angle vmv 0 vm+k is between π 2 and π. Let be a sufficiently small positive number which is less than (^) m^1. Let vm+k+1, · · · , v 2 m+k− 1 be the m − 1 points on the radius v 0 vm+k with
‖vivi+1‖ =
m for m + k ≤ i < 2 m + k − 2 and
‖v 2 m+k− 2 v 2 m+k− 1 ‖ =
m
Then ‖v 0 v 2 m+k− 1 ‖ =
m
Finally let v 2 m+k be the node lying on the circle of radius (^) m^1 + centered at node v 0 with
‖v 2 m+k− 1 v 2 m+k ‖ =
m
Now we consider the network consisting of these n = 2m + k + 1 nodes and a multicast session from
node v 0 to the two nodes v 2 m+k− 1 and v 2 m+k. As the angle vm− 1 vmvm+1 is greater than π 3 , we have
‖vm− 1 vm+1‖ > ‖vm− 1 vm‖ = ‖vmvm+1‖ =
m
Similarly,
‖vm+k− 1 vm+k+1‖ >
m
Thus the execution of Prim’s algorithm for MST outputs the path
v 0 v 1 · · · vm− 1 vmvm+1 · · · vm+k− 1 vm+k vm+k+1 · · · v 2 m+k− 2 v 2 m+k− 1 v 2 m+k
as the MST. Notice that except the link v 2 m+k− 2 v 2 m+k− 1 which has length of (^) m^1 − , all other 2 m + k − 1 links have length of (^) m^1. In addition, no node can be pruned from this path. Thus this path is output by P-MST. It’s total power consumption is
(2m + k − 1)
m^2
m
On the other hand, a direct transmission from node v 0 at the power of
m +^
can reach the two destination nodes v 2 m+k− 1 and v 2 m+k. Thus the minimum power is at most
m +^
. Therefore, the approximation ratio of P-MST is at least
(2m + k − 1) (^) m^12 +^
m −^
m +^
As −→ 0 , this ratio converges to 2 m + k = n − 1.
Finally, we construct bad instances for P-BIP. Its construction is similar to that in the above proof but the analysis is more subtle.
Lemma 3: The approximation ratio of P-BIP is at least n − 2 − o (1).
Proof: In the instance constructed in the proof of Lemma 3, we add a point u in the line segment v 0 v 1 with ‖v 0 u‖ = . Then ‖uv 1 ‖ = (^) m^1 − . We consider the network consisting of these n = 2m + k + 2 nodes and a multicast session from node v 0 to the two nodes v 2 m+k− 1 and v 2 m+k. Let’s first examine the execution of the BIP al- gorithm. Initially, the arborescence consists of only v 0. The first node to be added is u by increasing
1
v
2 m-1^ m
m+
m+
m+
m+k- m+k- m+k v m+k+ v 2m+k- 2m+k-
v 2m+k
v 0 uv
......
v v v
v
v
v
v v v
Fig. 3. A bad instance for P-BIP.
the transmission power of node v 0 from 0 to ^2. The incremental power to reach v 2 m+k− 1 or v 2 m+k from node v 0 is ( 1 m
− ^2 =
m^2
m
m^2
Thus the second node to be added is v 1 by in- creasing of the transmission power of node u from 0 to
m −^
. Subsequently, the m − 1 nodes v 2 , · · · , vm are added sequentially and the m − 1 nodes v 1 , · · · , vm− 1 all transmit at the power level of (^) m^12. Notice that the incremental power to add nodes v 2 m+k− 1 or v 2 m+k is always (^) m^12 + (^2) m > (^) m^12 , and thus neither v 2 m+k− 1 nor v 2 m+k can be added by this time. In the next step, node vm+1 is added. However, it is reached not from node vm by increasing the transmission power of node vm from 0 to (^) m^12 , but from node vm− 1 whose transmission power is increased from
‖vm− 1 vm‖^2 =
m^2 to
‖vm− 1 vm+1‖^2 =
m^2
(1 − cos ∠vm− 1 vmvm+1)
m^2
2 m
m^2
m^3
Thus node vm does not relay the signal, and the incremental power to add node vm+1 is ( 2 m^2
m^3
m^2
m^2
m^3
asymmetric multicasting of a given communication session.
Lemma 4: Any asymmetric routing for a com- munication session (s, D) requires total power of at least (^1) c times the weight of the Steiner minimum tree for {s} ∪ D.
Proof: Let OP T be a min-power arborescence for (s, D) with total power opt. For any none-sink node u in OP T , let Tu be an Euclidean MST of the point set consisting u and all children of u in T. Suppose that the longest Euclidean distance between u and its children is ru. Then the transmission power of node u is rκu. Since all children of u lie in the disk centered at u with radius ru, and the length of each edge in is at most ru, by the definition of c we have ∑
e∈Tu
‖e‖ ru
)κ ≤
e∈Tu
‖e‖ ru
≤ c,
which implies that
rκu ≥
c
∑^ κ
e∈Tu
‖e‖κ^ =
c
ω (Tu).
Let OP T ′^ denote the tree obtained by superposing of all Tu’s for non-sink nodes u of OP T. Then OP T ′^ is a Steiner tree for {s} ∪ D. Since opt is equal to the summation of rκu over all non-sink nodes u of OP T , we have
opt ≥
c
ω (OP T ′)
As ω (OP T ′) is at least the weight of the Steiner minimum tree for {s} ∪ D, the lemma follows.
Let A be any polynomial-time approximation algorithm for Steiner minimum tree in graphs. For any given communication session (s, D), we first apply A on G to obtain a Steiner tree for {s} ∪ D, and then orient it to an arborescence rooted at the source, which is used for the asymmetric routing of (s, D). We use Aa to denote this heuristic for M IN - P OWER A SYMMETRIC M ULTICAST ROUTING.
Theorem 5: For any ρ-approximation algorithm A for Steiner minimum tree, the approximation ratio of the algorithm Aa is at most cρ.
Proof: Fix a communication session (s, D) and let opt be the minimum power required by its
asymmetric routing. Let T ∗^ be a Steiner minimum tree for {s} ∪ D in G, and T be the Steiner tree for {s} ∪ D in G output by the algorithm A. Then ω (T ) ≤ ρ · ω (T ∗).
On the other hand, by Lemma 4,
opt ≥
c
ω (T ∗).
which implies that ω (T ∗) ≤ c · opt.
Thus, ω (T ) ≤ ρ · ω (T ∗) ≤ cρ · opt. Since the total power required by the arborescence oriented from T is at most ω (T ), the theorem follows.
A number of constant-approximation algorithms for Steiner minimum trees in graphs have been proposed in literature [8][9]. Robins and Zelikovsky [11] gave an approximation algorithm with the best known approximation ratio approaching 1 + ln 3 2 ≈^1.^55. Thus it can give rise to a heuris- tic for M IN -P OWER A SYMMETRIC M ULTICAST ROUTING with approximation ratio approaching c
1 + ln 23
1 + ln 23
≈ 18. 6. However, this heuristic may be not practical for ad hoc wireless networks due to its implementation complexity.
Takahashi and Matsuyama [14] gave a simple 2- approximation algorithm for Steiner minimum tree in graphs. A heuristic for M IN -P OWER A SYMMET- RIC M ULTICAST ROUTING based on Takahashi- Matsuyama Steiner tree heuristic can be imple- mented easily as follows. Throughout the execution, we maintain an arborescence T rooted at the source node. Initially, the arborescence T contains only the source node. At each iterative step, the arborescence T is grown by one path from T with least total power that can reach a destination not yet in T. This path can be found by collapsing the entire arborescence T into one artificial node and then applying the single-source shortest-path algorithm. This procedure is repeated until all required nodes are included in T. This heuristic is referred to as Shortest Path First (SPF). It can be regarded as an adaptation of MST for M IN -P OWER A SYMMETRIC B ROADCAST R OUTING. Indeed, when the commu- nication session is a broadcast, it acts the same way
as MST. From Theorem 5, we have the following performance of SPF.
Corollary 6: The approximation ratio of SPF is between 6 and 2 c, which is at most 24.
In the next, we present an adaptation of BIP for M IN -P OWER A SYMMETRIC B ROADCAST ROUT- ING to a heuristic, which we refer to as Minimum Incremental Path First (MIPF), for M IN -P OWER A SYMMETRIC M ULTICAST ROUTING. MIPF is im- plemented as follows. Throughout the execution we maintain an arborescence T rooted at the source node. Initially, the arborescence T contains only the source node. At each iterative step, we first find a path from T with the least incremental power that can reach a destination not yet in T , where the incremental power of a path is defined as its total power minus the transmission power in T of its first node. To find this path, we contract T into an artificial node t and construct a weighted complete graph over t and the rest nodes not in T as follows. The weight of the edge between any pair of nodes u 1 , u 2 ∈/ T is still ‖u 1 u 2 ‖κ. The weight of an edge between a node u /∈ T and the node t is given by
max
0 , min v∈T (‖uv‖κ^ − pT (v))
where pT (v) is the transmission power in T of node v. We then apply the single-source shortest- path algorithm on this constructed graph to find the shortest paths from t to all destination nodes not in T and pick up the one which has the smallest weight. The desired path is then obtained by replacing t with the appropriate node in T. After this path is found, the entire path is attached to T. This procedure is repeated until all destinations are included in T.
Note that if the given communication session is a broadcast, the algorithm MIPF acts in the same way as BIP. As the approximation ratio of BIP for broadcasting is at least 133 [15], so must be the approximation ratio of MIPF for multicasting. In the next, we will derive upper bounds on its approximation ratio.
Theorem 7: The approximation ratio of MIPF is between 133 and 2 c, which is at most 24.
Proof: Fix a communication session (s, D) and let opt be the minimum power required by its
asymmetric routing. Let T ∗^ be a Steiner minimum tree for {s} ∪ D in G, and T be the Steiner tree for {s} ∪ D in G output by the algorithm A. Let H be the weighted complete graph over {s} ∪ D in which the weight of each edge is equal to the weight of the shortest-path in G between the endpoints of this edge. Let mst (H) denote the minimum spanning tree of H. The weight of mst (H) in H is exactly the weight of the Steiner tree for {s} ∪ D in G produced by Takahashi- Matsuyama Steiner tree heuristic [14]. Thus, the weight of mst (H) in H is at most 2 ω (T ∗). Now we build another weighted complete graph H′^ over {s}∪D as follows. Suppose that during the execution of MIPF the nodes in {s} ∪ D are added in the order u 0 , u 1 , · · · , um where u 0 = s. Let Ti be the arborescence just after node ui is added to T. In H′, the weight of the edge uiui+1 is equal to the incremental power of the path from Ti to ui+1; and the weight of any other edge is the same as that in H. Then H′^ has the following properties:
- The weight of any minimum spanning tree in H′^ is no more than the weight of mst (H) in H. This is because each edge uiui+1 has the same or less weight in H ′^ than in H and every other edge has the same weight in H′ as in H.
- The path u 0 u 1 · · · um is a minimum spanning tree of H′. Indeed, the execution of Prim’s algorithm on H′^ will emulate the algorithm MIPF on G in the sense that it will add the nodes in {s} ∪ D in the same order.
- The total power required by T is exactly the weight of the path u 0 u 1 · · · um in H′. This can be easily proven by induction on i for 1 ≤ i ≤ m. From these three properties, we conclude that the total power required by T is at most the weight of mst (H) in H, which is in turn at most 2 ω (T ∗). The theorem then follows from Lemma 4.
B. Symmetric Multicasting
In this section, we present constant- approximation algorithms for M IN -P OWER S YMMETRIC M ULTICAST ROUTING. Let A be any polynomial-time approximation algorithm for Steiner minimum tree in graphs. For any given
a topic of continued research, is the development of efficient distributed implementation of some of the heuristics proposed in the literature and in this paper.
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Peng-Jun Wan Dr. Peng-Jun Wan received his PhD degree from University of Minnesota, MS degree from The Chinese Academy of Science, and BS degree from Qsinghua Uni- versity. He is currently an Associate Profes- sor in Computer Science at Illinois Institute of Technology. His research interests include wireless networks and optical networks.
Gruia C˘alinescu Dr. Gruia Calinescu received his PhD from Georgia Insitute of Technology, and Diploma from University of Bucharest. He is currently an Assistant Professor of Com- puter Science at the Illinois Institute of Tech- nology. His research interests are in the area of algorithms.
Chih-Wei Yi Mr. Chih-Wei Yi received his MS and BS degrees from National Taiwan University. He is currently a PhD candidate at the Illinois Institute of Technology. His dissertational research focuses on wireless ad hoc networks. He is expected to graduate in
