CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch.
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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch
Discrete Algs for Mobile Wireless Sys2 Lecture 21 Topic: More on Topology Control Sources: Wang & Li von Rickenback et al. MIT 6.885 Fall 2008 slides
Discrete Algs for Mobile Wireless Sys3 Planar Subgraphs We've already seen uses for the Relative Neighborhood Graph and Gabriel Graph both are planar subgraphs of a geometric graph used to support face routing / perimeter routing What about other planar subgraphs? what would be additional good properties for a subgraph to have?
Discrete Algs for Mobile Wireless Sys4 Connectivity Subgraph should be connected Supports communication
Discrete Algs for Mobile Wireless Sys5 Sparseness Subgraph should have O(n) edges, where n is the number of nodes Ensures that algorithms running on it, such as routing, are time and power efficient
Discrete Algs for Mobile Wireless Sys6 Spanner There should exist some constant t such that for all nodes u and v, dist(u,v) in subgraph is at most t times dist(u,v) in original graph t is called stretch factor Makes routing be efficient only constant factor more expensive
Discrete Algs for Mobile Wireless Sys7 Bounded Degree Maximum number of neighbors of any node in the subgraph should be constant Small node degree reduces MAC-level contention and interference Improves overall throughput
Discrete Algs for Mobile Wireless Sys8 Computation of Subgraph Distributed "localized": each node can decide which of its incident edges are in the subgraph based only on info about nodes a constant number of hops away few, small, messages quick local computation
Discrete Algs for Mobile Wireless Sys9 RNGGG Yao ( ) Yao + Sink LDelRDGBoseLi & Wang Planar?Yes No Yes Stretchn-1 (n-1) O(1) ~10 f( ) Degree (n) O(1) (n) 27 19 + g(1/ ) Comparison of Connected Subgraph Structures Li & Wang centralized alg: is a parameter, trade off stretch and degree; can achieve (separately) stretch 5 and degree 25 Wang & Li paper describes a distributed localized algorithm to achieve essentially same results as in Li & Wang
Discrete Algs for Mobile Wireless Sys10 Some Ideas in the Centralized Algorithm compute Delaunay triangulation of the nodes remove edges longer than transmission radius; call result UDel order the nodes: G := UDel for i = 1 to n do let u be node with smallest degree in G rank of u in the ordering is n - i + 1 remove u from G Since UDel is planar, each node has at most 5 neighbors that precede it in the ordering Next compute the edges of the spanner…
Discrete Algs for Mobile Wireless Sys11 Some Ideas in the Centralized Algorithm E' := Ø; all nodes in V marked unprocessed repeat until all nodes are processed: let u be unprocessed node with smallest rank in the ordering Let v 1, v 2, …, v k be the processed neighbors of u in UDel (k ≤ 5) divide each sector formed by rays from u to the v i 's into open cones of degree ≤ for each cone, let s 1,…,s m be the geometrically ordered (?) neighbors (in UDel) of u add to E' the shortest edge (in UDel) from u to some s i add to E' all edges (in original graph?) from s j to s j+1 (1 ≤ j < m) mark u as processed
Discrete Algs for Mobile Wireless Sys12 Example u v3v3 v2v2 v1v1 v5v5 v4v4 s3s3 s1s1 s2s2
Discrete Algs for Mobile Wireless Sys13 How to Achieve this in a Distributed Localized Fashion? First, find a particular planar spanner (called 2-localized Delaunay graph) use algorithm by Calinescu et al. to efficiently have every node collect its set of 2-hop neighbors (based on a connected dominating set) do some local computation and communication with neighbors to decide which incident edges form part of this spanner But this spanner can have degree as large as O(n)…
Discrete Algs for Mobile Wireless Sys14 Reducing the Degree Nodes exchange info to determine their 2-hop neighbors in the spanner just constructed Each node constructs a local ordering on the nodes in its 2-hop neighborhood A similar approach is used as in the centralized algorithm to choose the edges to put into the new spanner (each node divides its range into sectors, then each sector is divided into cones, order unprocessed neighbors, etc.) but achieved by nodes exchanging messages
Discrete Algs for Mobile Wireless Sys15 Dynamic Updates What happens if topology changes? Claim is that since algorithm is a localized one, update only needs to occur within 2- hop neighborhood of the change. Is this complete / convincing?
Discrete Algs for Mobile Wireless Sys16 Reducing Interference Through Topology Control Previous paper, as well as others, claim that interference is reduced by having a sparse subgraph (few edges) with low degree (few neighbors). However, the relationship between these graph aspects and the interference needs to be quantified In fact, what is an appropriate way to measure interference?
Discrete Algs for Mobile Wireless Sys17 Robust Interference Model [RSWZ] Proposes a measure of interference that is receiver-centric where interference actually occurs Definition is robust to addition/removal of a single node a small change to the topology should not have a big impact on the value of the measure W.r.t. the new measure, existing topology control algorithms do a poor job of reducing interference any structure that contains edges from each node to its nearest neighbor can have large interference For all 1-D networks, present an algorithm that constructs a topology with O( (n)) interference, and show this is optimal
Discrete Algs for Mobile Wireless Sys18 Low Degree Spanner = Low Interference? There is a problem with equating low degree in a spanner (or any subgraph) with low interference. The spanner indicates point-to-point communication. But in a wireless network, messages are broadcast in all directions, and cause interference even at nodes for whom the message is not intended Next definition addresses this point
Discrete Algs for Mobile Wireless Sys19 Definition of Interference Measure Let G' be any subgraph of the unit disk graph on a set of nodes Let r u be the maximum distance from u to any of its neighbors in G' Interference of node v is the number of nodes u (other than v) such that distance between v and u is at most r u Interference of G' is maximum interference of any node in G' Essentially augment G' with the "missing" edges due to the broadcast nature of communication and measure the degree of the resulting graph
Discrete Algs for Mobile Wireless Sys20 Nearest Neighbor Forest Most known topology control algorithms (e.g., spanner constructions) include the edge between each node and its nearest neighbor Theorem: If the subgraph includes these nearest- neighbor edges, then the interference can be (n) times larger than the interference of the optimum connected topology. Proof: Consider graph in Fig. 3 consisting of two exponential chains: nearest neighbor tree has (n) interference there is another tree with O(1) interference
Discrete Algs for Mobile Wireless Sys21 One-Dimensional Case First an algorithm is given to construct a topology for the exponential chain network with O( (n)) interference. Then an algorithm that works for an arbitrary 1-D network is given, again providing O( (n)) interference A lower bound is proved for the exponential chain network to show that no algorithm on this graph can do better than ( (n)) interference Since the general algorithm actually behaves poorly on evenly distributed chains (O( (n)) vs. O(1)), they describe an algorithm that results in a topology whose interference is always at most O( 4 (n)) of the optimal
Discrete Algs for Mobile Wireless Sys22 Discussion This measure of interference seems to be missing an important aspect: the amount of time that a node uses the broadcast channel and thus interferes with other nodes For instance, in the graph of Fig 8, when two non- hub nodes want to communicate, they must do so through a hub node. The hub node has to relay messages, but these relayed messages interfere with all nodes to the left of the hub (Thanks to Mike George for pointing this out)