WiOpt’04: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks March 24-26, 2004, University of Cambridge, UK Session 2 : Energy Management Paper : Minimum-Energy Broadcasting in Wireless Networks Using a Single Broadcast Tree Ioannis Papadimitriou Co-Author : Prof. Leonidas Georgiadis ARISTOTLE UNIVERSITY OF THESSALONIKI, GREECE FACULTY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Division of Telecommunications

WiOpt'04March 24-26, 2004, University of Cambridge, UK2 Presentation Plan 1.Introduction 2.Definitions and Problem Formulation 3.Broadcasting using a Single Broadcast Tree 4.Numerical Results 5.Conclusions – Issues for Further Study

WiOpt'04March 24-26, 2004, University of Cambridge, UK3 1. Introduction Energy-Efficient Broadcasting in Wireless Networks Assumptions : Omnidirectional antennas Node-based environment Bidirectional transmit powers General undirected graph model Common approach : Min-sum (of node powers) criterion minimum- energy broadcast problem depending on a specific source node (NP-complete) Our setup : Minimum-energy broadcasting using a Single Broadcast Tree Advantages : General networks (not unit disk graphs or other geometric properties) Independence of the source node – considerable simplification, scaling Approximation ratio close to best achievable bound in polynomial time

WiOpt'04March 24-26, 2004, University of Cambridge, UK4 2. Definitions and Problem Formulation A.Model for Wireless Broadcasting Undirected graph G (N, L), power for transmission over link l (link cost) c l > 0 If node i transmits with power p, it can reach any node j for which c (i, j) ≤ p s-rooted directed spanning tree induced by undirected tree T Node i transmits with power, where if i is a leaf Example : T : {(A,B), (A,C), (B,D)} (undirected) T A and T D (directed) are induced by T, D is a leaf node in T A, (D,B) is outgoing link of D in T D

WiOpt'04March 24-26, 2004, University of Cambridge, UK5 2. Definitions and Problem Formulation B.The Minimum-Energy Broadcast Problem  : total power consumed for broadcasting from source node s  In general, for different source nodes, the trees that minimize the sum of node powers are different (|N| broadcast trees, one for every possible source)  Objective : Find a single (undirected) spanning tree T to be used by all nodes for broadcasting, such that the sum of consumed node powers P(T s ) is minimized for any source node s.  A node needs to store only a small set of links that belong to tree T  Simplifies considerably the tree maintenance problem (similar to CDS)  Processing of broadcast information is minimal (scaling to larger networks)

WiOpt'04March 24-26, 2004, University of Cambridge, UK6 2. Definitions and Problem Formulation B.The Minimum-Energy Broadcast Problem (cont.) Two open issues : If all broadcasts take place on the same tree, then Issue 1 : Certain broadcasts may need much more total power than others, depending on the source node (widely varying total power consumption for different source nodes). Issue 2 : If one attempts to find a tree for which the total powers consumed for broadcasting initiated by different source nodes are approximately the same, then, for a given source node, the resulting total power may be far away from the optimal. We address both issues and provide satisfactory answers in the sequel

WiOpt'04March 24-26, 2004, University of Cambridge, UK7 3. Broadcasting using a Single Broadcast Tree Addressing Issue 1 : We prove that,  If the same spanning tree T is used for broadcasting by all nodes, then the total broadcast power consumption for source node s is at most twice the total broadcast power consumption for any other source node s ΄, P( T s ) ≤ 2P( T s ΄ ). Addressing Issue 2 : We propose a polynomial time approximation algorithm for the construction of a single broadcast tree, such that,  For any source node s, the total power consumed for broadcasting using tree has an approximation ratio 2H(n-1) with respect to the optimal power. Approximation ratio close to the best achievable bound in polynomial time (n=|N| is the number of nodes in the network and H(n) is the harmonic function)

WiOpt'04March 24-26, 2004, University of Cambridge, UK8 3. Broadcasting using a Single Broadcast Tree Single Broadcast Tree (SBT) algorithm :  At every iteration, SBT maintains a forest of trees in the network, such that each node belongs to a forest tree.  Initially, each node constitutes a forest tree.  The forest is expanded by joining trees through nodes, so that the “incremental power consumed per joined tree” is minimal.  This is achieved by examining the adjacent links of every node i in the network that terminate outside the tree to which node i belongs.  The algorithm terminates when the forest consists of a single (undirected) spanning tree.

WiOpt'04March 24-26, 2004, University of Cambridge, UK9 3. Broadcasting using a Single Broadcast Tree Example of SBT algorithm : Node i min is selected to be joined with the forest tress T F1 and T F2. Link l min joins tree T Fmin with T F1. Only one of the links (i min, m), (i min, n) must be selected to join tree T Fmin with T F2 to avoid the creation of cycle. Broadcasting using a Minimum Spanning Tree :  For any source node s, the total power consumed for broadcasting using a minimum spanning tree, is at most Δ times the optimal power, where Δ is the maximum node degree in the network. Hence, an MST may be a good candidate for broadcasting in sparse networks.

WiOpt'04March 24-26, 2004, University of Cambridge, UK10 4. Numerical Results Algorithms compared : 1) “BIP” 2) “SBT” 3) “MST” Networks created : 100 randomly generated networks for a given |N| 1) (20,40,…,100) nodes in a rectangular grid of 100×100 points (networks represented by unit disk graphs) 2) “Special” nodes added to the grid – 3-dimensional network (instances of general networks) Performance metric : Average total broadcast power consumption

WiOpt'04March 24-26, 2004, University of Cambridge, UK11 4. Numerical Results Networks represented by unit disk graphs : link costs a = 2, complete networks a = 4, complete networks Note : BIP determines a different broadcast tree for every possible source node, while SBT algorithm constructs a single tree used by all nodes for broadcasting.  Average tree power of SBT is slightly larger than that of BIP.  The difference in performance of the algorithms vanishes for larger values of a. The “penalty” of using longer links increases and all algorithms converge to MST.

WiOpt'04March 24-26, 2004, University of Cambridge, UK12 4. Numerical Results Instances of general networks : model a physical environment in 3-dimensional space Ratio of avg. tree power of SBT to BIP, a = 2, 100-node sparse networks + 1 “special” node a = 2, 1 “special” node added to the sparse networks, factor f = 0.1 Note : The power of a link between the “special” node and any other node on the grid at distance d is f  d 2, where f is a factor 0 < f ≤ 1(less hostile communication channel).  There is a range of values of f for which SBT significantly outperforms BIP.  SBT succeeds in selecting links of “special” node when they are more cost efficient.

WiOpt'04March 24-26, 2004, University of Cambridge, UK13 4. Numerical Results Main observations :  SBT algorithm performs fairly well, compared to BIP algorithm, for networks represented by unit disk graphs, while using a single broadcast tree.  There are interesting instances of general networks, for which SBT algorithm significantly outperforms BIP and MST.  MST algorithm performs worse for most of the network instances considered. Conclusion : SBT algorithm presents a good compromise between simplicity and achieved performance.

WiOpt'04March 24-26, 2004, University of Cambridge, UK14 5. Conclusions – Issues for Further Study Distributed Implementation :  SBT algorithm is applicable in networks where at least partial information of network topology is proactively maintained at each node.  Similarities with Kruskal’s algorithm for determining an MST. Distributed implementation possible (further study is needed). Other :  Multicast extensions (new heuristics must be developed).  Energy-limited and resource-limited environment, Lifetime maximization.  Dynamic power assignments (periodic updates of broadcast tree).

WiOpt'04March 24-26, 2004, University of Cambridge, UK15 End of Presentation Thank you for your attention Paper : Minimum-Energy Broadcasting in Wireless Networks Using a Single Broadcast Tree Ioannis Papadimitriou Co-Author : Prof. Leonidas Georgiadis ARISTOTLE UNIVERSITY OF THESSALONIKI, GREECE FACULTY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Division of Telecommunications