1. David S. L. Wei Dept of Computer and Information Sciences Fordham University Bronx, New York Szu-Chi Wang and Sy-Yen Kuo Dept of Electrical Engineering National Taiwan University Taipei, Taiwan Joint work with

2. Introduction  Wireless ad hoc networks are - Characterized by scarce resources - Prone to topology changes - Lack of physical infrastructure  The flexibility and mobility of wireless ad hoc networks make them suitable for applications such as automated battlefields and disaster rescues

3. Introduction (cont.)  A wireless ad hoc network can be modeled by an undirected/directed graph G = (V, E)  Power conservation has been widely used as a primary control parameter in the design of protocols for wireless ad hoc networks A B C D E F AC F B DE

4. Introduction (cont.) AC F B DE A B C D E F AC FB DE A B C D E F

5. Motivations  Each node in a wireless ad hoc network can potentially change the network topology by adjusting its transmission range  The primary goal of topology control is to design power-efficient algorithms that - Maintain network connectivity - Optimize performance metrics (network lifetime, throughput,…)

6. Preliminaries  In the most common power-attenuation model - The transmission power between node u and v is denoted as - All receivers have the same power threshold for signal detection  We assume that - Each mobile host has a low-power GPS receiver - Initially all the nodes are operated at full transmitter power ─ the resulted graph G is a unit-disk graph (denoted as UDG (V)) t  ||uv||  + rp(u,v) transmitter power receiver power

7. Preliminaries (cont.)  Hereinafter we use G to present a wireless ad hoc network - The edge weight is defined as w(u, v) = t  ||uv||  + rp(u,v) - We also call G the transmission graph  A path from node u to node v is denoted as  The total transmission power of this path is  (u, v) = v 0 v 1 …v h-1 v h where u = v 0 and v = v h

8. Transmission Power Assignment  A transmission power assignment on the vertices is a function f from V into real numbers  Given a graph H = (V’, E’), the transmission power assignment f is induced by H if for each node v  V’,  The total transmission power of f is defined as  A transmission power assignment f is complete if the associated graph G f is strongly connected

9. Minimum-Energy Path  Given a communication graph H  G, the minimum-energy path between node u and node v, denoted by  H min (u, v), is a path whose total transmission power is the minimum among all paths that connect (u, v) in H  Let p H (u, v) stand for p(  H min (u, v)), the power stretch factor of H with respect to G is defined as

10. Related Works  Our major work is to develop a localized topology control algorithm where each node makes a decision about its transmission power based on only its local information  The two widely used energy conservation approaches in literature are to - Reduce the total transmission power - Reduce the power stretch factor  However, these two approaches may offset each other

11.  The problem of finding a complete f whose total transmission power is the minimum among all of the complete assignments is called the min-total assignment problem  The min-total assignment problem is NP-hard when the nodes are deployed in a d-dimensional space, d  2  The general structure of the minimum-power topology for rp  0 is still unknown Related Works (cont.)

12. Basic Ideas  The proposed algorithm is based on the following ideas - First construct a connected subgraph H = (V, E’) - Assure that the power stretch factor of H is bounded - The total transmission power is then minimized as much as possible  We use the local information of each node to excise some links of G while still keeps the power stretch factor being bounded by a predetermined value cb

13. Our Localized Topology Control Algorithm  The proposed algorithm consists of two phases - Phase I: Local shortest tree construction - Phase II: Path search replacement  A simple illustrative example is shown below p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7

14. Phase I of the Proposed Algorithm  Definition 1 (Local Topology View) The local topology view of node u, LTV (u, k) = (V’, E’)), is a subgraph of G such that (1) v i  V’ if the hop distance between v i and u is no more than k (2) (v i, v j )  E’ if both v i and v j belong to V’  Suppose that a subgraph of G is associated with a transmission power assignment f - For each node u, if link (u, v) satisfies w (u, v) = f (u) then (u, v) is called a critical link of u

15. Phase I of the Proposed Algorithm (cont.)  Each node u individually applies Dijkstra’s algorithm to get the shortest-paths from the source u to the other nodes in LTV (u, 1)  The local shortest path tree of node u (denoted by LSPT (u)) can thus be obtained  DC (u) = {v  V’ | h (LSPT (u), v) = 1}, where h (LSPT (u), v) is the height of a child node v in LSPT (u)  Node u then deletes the edges {(u, w) | w  DC (u)}  The topology generated is denoted as G I

16. Phase II of the Proposed Algorithm  For each node u, its transmission power could be further reduced by trying to eliminate the critical links that are replaceable with alternative paths  For each critical link (u, v), node u tries to search another path that reaches node v based on LTV (u, k) - We call such path the replacing path of (u, v) - The entire replacing paths of node u is denoted as RP (u)  The searching procedure is applying Dijkstra’s algorithm again on LTV (u, k)

17. Phase II of the Proposed Algorithm (cont.)  If no such path exists or no replacing path has transmission power  cb  w (u, v) - The search process is ended - RP (u) is set as an empty list - ps (u) is set to 0  The priority of node u is a pair pri(u) = - pri(v 1 ) = (ps(v 1 ), ID 1 ), pr (v 2 ) = (ps(v 2 ), ID 2 ) - pri(v 1 ) > pri(v 2 ) if ps(v 1 ) > ps(v 2 )  (ps(v 1 ) = ps(v 2 )  ID 1 < ID 2 )  The above procedure for deciding RP (u) starts if node u has the highest priority in its k-hop neighborhood

18. Phase II of the Proposed Algorithm (cont.)  After Phase I, each node deletes its uni-directional links and the resulting topology is denoted as G II  The constructed topology after Phase II is denoted as G III  A simple heuristic for further decreasing the total transmission power is also proposed

19. Important Properties  The minimum-energy path between any two nodes in G is preserved in G I  The minimum-energy path between the two end nodes of each deleted link in G I is preserved in G II  G II preserves the network connectivity of G  is bounded by cb  G III preserves the network connectivity of G and has a bounded power stretch factor cb

20. Dealing with Mobility  We also consider the case of modest movement of the nodes  It would be extremely difficult for a topology control algorithm to even effectively guarantee network connectivity if network topology changes too fast  As mentioned in previous works, node movement can be viewed as two events, namely node addition and node deletion  In our case, however, at the beginning of each beacon interval each node u should check if there is a change in transmission radius after deciding the new logical links

21. Performance Comparisons  We compared the performance via extensive simulations  We observe the following metrics of each constructed topology H - Total transmission power (denoted by tpc) - Power stretch mean (denoted by psm) - The maximum power stretch factor (denoted by max psf) - The variance of transmission power (denoted by var tp) - Average node degree (denoted by avg nd) - The maximum node degree (denoted by max nd)

22. Simulation Results The Performance Measurements with s = 1 The Performance Measurements with s = 3 The Performance Measurements with s = 5 The Performance Measurements with s = 7

23. Network topologies constructed by various algorithms I UDG SMEN GGAMST

24. Network topologies constructed by various algorithms II LMST ESPT 2, cb = 1.5 ESPT 1 ESPT 2, cb = 2.0

25. Conclusions  In this paper we develop a localized algorithm that requires only local information for constructing a logical topology on a given unit disk graph  The topology constructed by our algorithm has several desired features such as bounded power stretch factor, low total power consumption, and small variance of transmission power  The simulation results show that our algorithm outperforms others in terms of various important metrics

26. Future Research Power-aware topology control Topology control of ad-hoc networks in three- dimensional space Secure topology control algorithm Applications in overlay control for P2P communications

27. Thanks for Your Attention