Chapter 7 TOPOLOGY CONTROL

Transcription

1 Chapter TOPOLOGY CONTROL Oeriew Topology Control Gabriel Graph et al. XTC Interference SINR & Schedling Complexity Distribted Compting Grop Mobile Compting Winter 00 / 00 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Topology Control Topology Control as a Trade-Off Sometimes also clstering, Dominating Set constrction (See later) Topology Control Drop long-range neighbors: Redces interference and energy! Bt still stay connected (or een spanner) Network Connectiity Spanner Property d(,) t d TC (,) Consere Energy Redce Interference Sparse Graph, Low Degree Planarity Symmetric Links Less Dynamics Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /

2 Gabriel Graph Delanay Trianglation Let disk(,) be a disk with diameter (,) that is determined by the two points,. The Gabriel Graph GG(V) is defined as an ndirected graph (with E being a set of ndirected edges). There is an edge between two nodes, iff the disk(,) inclding bondary contains no other points. disk(,) Let disk(,,w) be a disk defined by the three points,,w. The Delanay Trianglation (Graph) DT(V) is defined as an ndirected graph (with E being a set of ndirected edges). There is a triangle of edges between three nodes,,w iff the disk(,,w) contains no other points. disk(,,w) w As we will see the Gabriel Graph has interesting properties. The Delanay Trianglation is the dal of the Voronoi diagram, and widely sed in arios CS areas; the DT is planar; the distance of a path (s,,t) on the DT is within a constant factor of the s-t distance. Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Other planar graphs Properties of planar graphs Relatie Neighborhood Graph RNG(V) Theorem 1: MST( V) RNG( V) GG( V) DT( V) An edge e = (,) is in the RNG(V) iff there is no node w with (,w) < (,) and (,w) < (,). Corollary: Since the MST(V) is connected and the DT(V) is planar, all the planar graphs in Theorem 1 are connected and planar. Minimm Spanning Tree MST(V) A sbset of E of G of minimm weight which forms a tree on V. Theorem : The Gabriel Graph contains the Minimm Energy Path (for any path loss exponent α ) Corollary: GG(V) UDG(V) contains the Minimm Energy Path in UDG(V) Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /

3 More examples XTC: Lightweight Topology Control β-skeleton Generalizing Gabriel (β = 1) and Relatie Neighborhood (β = ) Graph Yao-Graph Each node partitions directions in k cones and then connects to the closest node in each cone Cone-Based Graph Dynamic ersion of the Yao Graph. Neighbors are isited in order of their distance, and sed only if they coer not yet coered angle Topology Control commonly assmes that the node positions are known. What if we do not hae access to position information? XTC algorithm XTC analysis Worst case Aerage case Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / XTC: lightweight topology control withot geometry XTC Algorithm (Part ) D F E A 1. C. E. B. F. D. G C B G Each node prodces ranking of neighbors. Examples Distance (closest) Energy (lowest) Link qality (best) Not necessarily depending on explicit positions Nodes exchange rankings with neighbors. A. C. E D 1. F. A. D F E. E. A. B. A. G. D A 1. C. E. B. F. D. G C B. C. G. A G. B. A. C Each node locally goes throgh all neighbors in order of their ranking If the candidate (crrent neighbor) ranks any of yor already processed neighbors higher than yorself, then yo do not need to connect to the candidate. Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1

4 XTC Analysis (Part 1) XTC Analysis (Part 1) Symmetry: A node wants a node as a neighbor if and only if wants. Symmetry: A node wants a node as a neighbor if and only if wants. Proof: Assme 1) and ) Assmption ) w: (i) w and (ii) w Contradicts Assmption 1) Connectiity: If two nodes are connected originally, they will stay so (proided that rankings are based on symmetric link-weights). If the ranking is energy or link qality based, then XTC will choose a topology that rotes arond walls and obstacles. Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 XTC Analysis (Part ) XTC Aerage-Case If the gien graph is a Unit Disk Graph (no obstacles, nodes homogeneos, bt not necessarily niformly distribted), then The degree of each node is at most. The topology is planar. The graph is a sbgraph of the RNG. Relatie Neighborhood Graph RNG(V): An edge e = (,) is in the RNG(V) iff there is no node w with (,w) < (,) and (,w) < (,). Unit Disk Graph XTC Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1

5 XTC Aerage-Case (Degrees) XTC Aerage-Case (Stretch Factor) 1. UDG max GG max XTC max Node Degree Network Density [nodes per nit disk] UDG ag GG ag XTC ag Stretch Factor Network Density [nodes per nit disk] XTC s. UDG Eclidean GG s. UDG Eclidean XTC s. UDG Energy GG s. UDG Energy Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 XTC Aerage-Case (Geometric Roting) k-xtc: More connectiity better Performance worse 1 0 connectiity rate GFG/GPSR on GG GOAFR+ on G XTC GOAFR+ on GG 0 1 Network Density [nodes per nit disk] Freqency A graph is k-(node)-connected, if k-1 arbitrary nodes can be remoed, and the graph is still connected. In k-xtc, an edge (,) is only remoed if there exist k nodes w 1,, w k sch that the k edges (w 1, ),, (w k, ), (w 1,),, (w k,) are all better than the original edge (,). Theorem: If the original graph is k-connected, then the prned graph prodced by k-xtc is as well. Proof: Let (,) be the best edge that was remoed by k-xtc. Using the constrction of k-xtc, there is at least one common neighbor w that sries the slaghter of k-1 nodes. By indction assme that this is tre for the j best edges. By the same argment as for the best edge, also the j+1 st edge (, ), since at least one neighbor sries w sries and the edges (,w ) and (,w ) are better. Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /0

6 Implementing XTC, e.g. BTnodes Implementing XTC, e.g. on mica motes Idea: XTC chooses the reliable links The qality measre is a moing aerage of the receied packet ratio Sorce roting: rote discoery (flooding) oer these reliable links only Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Topology Control as a Trade-Off What is Interference? Exact size of interference range does not change the reslts Link-based Interference Model Node-based Interference Model Topology Control Interference Interference Network Connectiity Spanner Property Really?!? Consere Energy Redce Interference Sparse Graph, Low Degree Planarity Symmetric Links Less Dynamics How many nodes are affected by commnication oer a gien link? By how many other nodes can a gien network node be distrbed? Problem statement We want to minimize maximm interference At the same time topology mst be connected or a spanner etc. Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /

7 Low Node Degree Topology Control? Let s Stdy the Following Topology! Low node degree does not necessarily imply low interference: from a worst-case perspectie Very low node degree bt hge interference Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Topology Control Algorithms Prodce Bt Interference All known topology control algorithms (with symmetric edges) inclde the nearest neighbor forest as a sbgraph and prodce something like this: Interference does not need to be high The interference of this graph is Ω(n)! This topology has interference O(1)!! Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /

8 Link-based Interference Model Interference-optimal topologies: Link-based Interference Model LIFE (Low Interference Forest Establisher) There is no local algorithm that can find a good interference topology The optimal topology will not be planar Preseres Graph Connectiity LIFE Attribte interference ales as weights to edges Compte minimm spanning tree/forest (Krskal s algorithm) LIFE constrcts a minimminterference forest Interference Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /0 Link-based Interference Model Link-based Interference Model LISE (Low Interference Spanner Establisher) LocaLISE Scalability Constrcts a spanning sbgraph Constrcts a spanner locally LISE Add edges with increasing interference ntil spanner property flfilled LISE constrcts a minimminterference t-spanner -hop spanner with Interference LocaLISE Nodes collect (t/)-neighborhood Locally compte interferenceminimal paths garanteeing spanner property Only reqest that path to stay in the reslting topology LocaLISE constrcts a minimm-interference t-spanner Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /

9 Link-based Interference Model Aerage-Case Interference: Presere Connectiity LocaLISE (Low Interference Spanner Establisher) Constrcts a spanner locally LocaLISE Nodes collect (t/)-neighborhood Locally compte interferenceminimal paths garanteeing spanner property Only reqest that path to stay in the reslting topology LocaLISE constrcts a minimm-interference t-spanner Interference 0 0 UDG 0 GG 0 0 RNG LIFE Network Density [nodes per nit disk] Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Aerage-Case Interference: Spanners Link-based Interference Model RNG LLISE, t= Interference 0 1 t= t= t= t= UDG, I = 0 RNG, I = LIFE Network Density [nodes per nit disk] LocaLISE, I = LocaLISE, I = 1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /

10 Node-based Interference Model Node-based Interference Model Already 1-dimensional node distribtions seem to yield inherently high interference... Connecting linearly reslts in interference O(n) 1 Already 1-dimensional node distribtions seem to yield inherently high interference... Connecting linearly reslts in interference O(n)...bt the exponential node chain can be connected in a better way...bt the exponential node chain can be connected in a better way Interference Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Matches an existing lower bond Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Node-based Interference Model Arbitrary distribted nodes in one dimension Approximation algorithm with approximation ratio in O( ) Towards a More Realistic Interference Model... Signal-to-interference and noise ratio (SINR) Power leel of node Path-loss exponent Two-dimensional node distribtions Randomized algorithm reslting in interference O( ) No deterministic algorithm so far... Noise Problem statement Distance between two nodes Minimm signal-tointerference ratio Determine a power assignment and a schedle for each node sch that all message transmissions are sccessfl SINR is always assred Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /0

11 Qiz: Can these two links transmit simltaneosly? Let s try harder A C D B A C D B 0m 1m 0m 1m Graph-theoretical models: No! Neither in- nor ot-interference SINR model: constant power: No! Node B will receie the transmission of node C SINR model: power according to distance-sqared: No! Node D will receie the transmission of node A Let s forget abot noise first Then the SINR at B is: P A /0 /(P C /0 ) = ρ/, with ρ = P A /P C. And the SINR at D is: P C /1 /(P A /0 ) = 00/ρ. Making both SINR eqal, ρ = 000 ρ = 0. Hge! Let s try with noise N = 1, P A = 0 000, P C = 1 000: Then SINR at B = P A /0 /(P C /0 +1) > And SINR at D = P C /1 /(P A /0 +1) > (Inclde noise directly, and yo get both SINR s aboe.) Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /1 Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer / A Simple Problem Each node in the network wants to send a message to an arbitrary other node Commonly assmed power assignment schemes Uniform Linear Proportional to (receier distance) α Constant power leel Both lead to a schedle of length A cleer power assignment reslts in a schedle of length This has strong implications to MAC layer protocols Distribted Compting Grop MOBILE COMPUTING R. Wattenhofer /