1 / 20

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems. Spring 2009 Prof. Jennifer Welch. Lecture 20. Topic: Part III: Building and Maintaining Network Structures Topology Control Sources: Li et al. Bahramgiri et al. MIT 6.885 Fall 2008 slides. Topology Control.

cutler
Download Presentation

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

  2. Lecture 20 • Topic: • Part III: Building and Maintaining Network Structures • Topology Control • Sources: • Li et al. • Bahramgiri et al. • MIT 6.885 Fall 2008 slides Discrete Algs for Mobile Wireless Sys

  3. Topology Control • Suppose you have a set of nodes V, where each node v has a Euclidean position and a maximum transmission power. • If all nodes transmit using maximum power, they induce a set of edges E representing the communication links • Starting with connected graph G = (V,E), reduce number of neighbors of each node by reducing transmission power at that node to produce a subset E' of E with G' = (V,E') still connected Discrete Algs for Mobile Wireless Sys

  4. How to Optimize Power? • Focus on reducing transmission power of each node • minimize the maximum transmission power per node Discrete Algs for Mobile Wireless Sys

  5. Cone-Based Topology Control [Li et al.] • Problem studied is how to reduce power while preserving connectivity • Assumes interference is implicitly handled by reducing node degree and edge set • Claims not to require position of each node and not to be coupled with any radio propagation model • Assumes a MAC layer that provides a reliable broadcast operation; synchronous rounds • Mainly theoretical with some simulations to validate theoretical results Discrete Algs for Mobile Wireless Sys

  6. Central Idea of Algorithm • Parameter to algorithm • Each node u starts broadcasting and collecting Acks with minimum power • Node u gradually increases its transmission power • stops when reaching maximum power or every cone of degree  contains at least one node (neighbor) Discrete Algs for Mobile Wireless Sys

  7. Sample Execution << see CBTCex.pdf >> Discrete Algs for Mobile Wireless Sys

  8. CBTC(), Code for Node u Nu := Ø // neighbors discovered so far Du := Ø // directions from which Acks have arrived pu := pmin while (pu < pmax and there exists gap(,Du)) pu := increase(pu) bcast(u,pu,("Hello",pu)) wait for Acks Nu := Nu U {v : v just sent an Ack} Du := Du U {diru(v) : v just sent an Ack} Discrete Algs for Mobile Wireless Sys

  9. Some Definitions • Let N(u) be final set of neighbors discovered by node u • Let E– = {(u,v) : v is in N(u) and u is in N(v)} • Let E+ = {(u,v) : v is in N(u) or u is in N(v)} Discrete Algs for Mobile Wireless Sys

  10. Optimizations • Shrink-back: for nodes which terminate with p = pmax, reduce power while keeping coverage. • Asymmetric edge removal: transform E+ into E–, as long as E– preserves connectivity • Pairwise edge removal: remove implicit 3-cycles; use node ids to get unique edge weights and remove longest edge Discrete Algs for Mobile Wireless Sys

  11. Proof of Correctness • INFOCOMM paper proves that when  = 2/3, then E+ is connected. • PODC paper proves that • when  = 2/3, then E– is connected and • when  = 5/6, then E+ is connected • Proofs are geometrical in nature, subtle and somewhat long • From MIT course notes, prove something stronger: • when  = 2/3, then E– is not only connected, but contains the minimum spanning tree Discrete Algs for Mobile Wireless Sys

  12. About Minimum Spanning Trees • Given a set of points in the plane and all straight-line edges between them, want to find MST of the resulting graph (cost of an edge is its length). • Yao's Lemma: If edge (u,v) is in MST, then v is u's closest neighbor in every /3 cone centered at u containing v. Discrete Algs for Mobile Wireless Sys

  13. Using Yao's Lemma • Suppose in contradiction the CBCT output graph G' is missing some edge (u,v) of the MST M of the maximum power graph G • Consider the cone of angle 2/3 centered at u with v on its bisector; each half of the cone has angle /3 • By construction, CBCT guarantees that there is some neighbor w of u inside this cone in G'. • Since v is not a neighbor of u, but w is, it must be that v is farther away from u than w is. • Contradicts Yao's Lemma! v w u Discrete Algs for Mobile Wireless Sys

  14. Extension to k-Connectivity [Bahramgiri] • A connected graph is k-connected if removing any set of less than k nodes results in a graph that is still connected. • How can we find minimum power needed by each node to ensure that resulting graph is k-connected? • Use the cone algorithm CBTC with  = 2/3k. Discrete Algs for Mobile Wireless Sys

  15. Proof of Correctness • Theorem: If G is k-connected, then G, result of running CBTC(/k) with  = 2/3 on G, is also k-connected. • Proof: Suppose in contradiction G/k is not k-connected: removing nodes v1, …, vk-1 disconnects G/k. Discrete Algs for Mobile Wireless Sys

  16. CBTC(/k) G/k remove v1,…,vk-1 remove v1,…,vk-1 G1 disconnected G' connected show G' is a subgraph of G1, and thus G1 is connected CBTC() G' connected by previous result Proof of Correctness  = 2/3 G k-connected Discrete Algs for Mobile Wireless Sys

  17. Proof of Correctness • Suppose in contradiction G' is not a subgraph of G1. • So there is some edge (u,v) in G' but not in G1. • (u,v) is also not in G/k, since no edges between non-removed nodes are removed in going from G/k to G1 • So (u,v) is not needed in G/k: there are some other nodes w1, w2,… that are closer to u than v is, s.t. max angle b/w wi and wi+1 is at most /k Discrete Algs for Mobile Wireless Sys

  18. CBTC(/k) G/k remove v1,…,vk-1 remove v1,…,vk-1 G1 disconnected G' connected show G' is a subgraph of G1, and thus G1 is connected CBTC() G' connected by previous result Proof of Correctness  = 2/3 G doesn't contain (u,v) k-connected doesn't contain (u,v) contains (u,v) Discrete Algs for Mobile Wireless Sys

  19. Proof of Correctness • In worst case, the k nodes removed are consecutive w-nodes. • Resulting empty cone has angle at most (/k)*k =  • So in G', there is no need to use enough power to reach v; it is sufficient to reach the remaining w-nodes • Thus (u,v) is not an angle of G', contradiction. Discrete Algs for Mobile Wireless Sys

  20. u u u u Proof of Correctness u Discrete Algs for Mobile Wireless Sys

More Related