WiOpt’04: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks

1. 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

2. Presentation Plan • Introduction • Definitions and Problem Formulation • Broadcasting using a Single Broadcast Tree • Numerical Results • Conclusions – Issues for Further Study March 24-26, 2004, University of Cambridge, UK

3. 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 March 24-26, 2004, University of Cambridge, UK

4. 2. Definitions and Problem Formulation • Model for Wireless Broadcasting • Undirected graph G (N , L), power for transmission over link l (link cost) cl > 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) • TAand TD(directed) are induced by T • , D is a leaf node in TA • , (D,B) is outgoing link of D in TD March 24-26, 2004, University of Cambridge, UK

5. 2. Definitions and Problem Formulation • 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(Ts) 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) March 24-26, 2004, University of Cambridge, UK

6. 2. Definitions and Problem Formulation • 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 March 24-26, 2004, University of Cambridge, UK

7. 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(Ts)≤ 2P(Ts΄). 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) March 24-26, 2004, University of Cambridge, UK

8. 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. March 24-26, 2004, University of Cambridge, UK

9. 3. Broadcasting using a Single Broadcast Tree • Example of SBT algorithm : • Node imin is selected to be joined with the forest tress TF1 and TF2. • Link lmin joins tree TFmin with TF1. • Only one of the links (imin , m) , (imin , n) must be selected to join tree TFmin with TF2 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. March 24-26, 2004, University of Cambridge, UK

10. 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 March 24-26, 2004, University of Cambridge, UK

11. 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. March 24-26, 2004, University of Cambridge, UK

12. 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 d2, 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. March 24-26, 2004, University of Cambridge, UK

13. 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. March 24-26, 2004, University of Cambridge, UK

14. 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). March 24-26, 2004, University of Cambridge, UK

15. 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 March 24-26, 2004, University of Cambridge, UK