XTC: A Practical Topology Control Algorithm for Ad-Hoc Networks Roger Wattenhofer Aaron Zollinger

Overview What is Topology Control? Context – related work XTC algorithm XTC analysis Worst case Average case Conclusions WMAN 2004

Topology Control Sometimes also clustering, Dominating Set construction Not in this presentation Drop long-range neighbors: Reduces interference and energy! But still stay connected (or even spanner) WMAN 2004

Overview What is Topology Control? Context – related work XTC algorithm XTC analysis Worst case Average case Conclusions WMAN 2004

Context – Previous Work Mid-Eighties: randomly distributed nodes [Takagi & Kleinrock 1984, Hou & Li 1986] Second Wave: constructions from computational geometry, Delaunay Triangulation [Hu 1993], Minimum Spanning Tree [Ramanathan & Rosales-Hain INFOCOM 2000], Gabriel Graph [Rodoplu & Meng J.Sel.Ar.Com 1999] Cone-Based Topology Control [Wattenhofer et al. INFOCOM 2000]; explicitly prove several properties (energy spanner, sparse graph) Collecting more and more properties [Li et al. PODC 2001, Jia et al. SPAA 2003, Li et al. INFOCOM 2002] (e.g. local, planar, distance and energy spanner, constant node degree [Wang & Li DIALM-POMC 2003]) Only neighbor direction and relative distance But: exact node positions known WMAN 2004

K-Neigh (Blough, Leoncini, Resta, Santi @ MobiHoc 2003) “Connect to k closest neighbors!” Very simple algorithm. On average as good as others… Tough question: What should k be? [Thanks to P. Santi] WMAN 2004

Percolation Node density such that the graph is just about to become connected (about 5 nodes per unit disk). What’s the value for k at percolation?!? (Tough question?) too sparse critical density too dense WMAN 2004

K-Neigh and the Worst Case? What if the network looks like this: Does a typical/average network (or parts of an average network) really look like this? Probably not… but… Still, cool simulation and analysis results by Blough et al. For example: energy to compute K-Neigh topology is much smaller than CBTC topology (figure right) k+1 nodes k+1 nodes WMAN 2004

Overview What is Topology Control? Context – related work XTC algorithm XTC analysis Worst case Average case Conclusions WMAN 2004

XTC Algorithm D C G B A E F Each node produces “ranking” of neighbors. Examples Distance (closest) Energy (lowest) Link quality (best) Not necessarily depending on explicit positions Nodes exchange rankings with neighbors A 1. C 2. E 3. B 4. F 5. D 6. G E F WMAN 2004

XTC Algorithm (Part 2) D C G B A E F Each node locally goes through all neighbors in order of their ranking If the candidate (current neighbor) ranks any of your already processed neighbors higher than yourself, then you do not need to connect to the candidate. A 1. C 2. E 3. B 4. F 5. D 6. G E F 3. E 7. A WMAN 2004

Overview What is Topology Control? Context – related work XTC algorithm XTC analysis Worst case Average case Conclusions WMAN 2004

XTC Analysis (Part 1) Symmetry: A node u wants a node v as a neighbor if and only if v wants u. Proof: Assume 1) u  v and 2) u  v Assumption 2)  9w: (i) w Áv u and (ii) w Áu v Contradicts Assumption 1) WMAN 2004

XTC Analysis (Part 1) Symmetry: A node u wants a node v as a neighbor if and only if v wants u. Connectivity: If two nodes are connected originally, they will stay so (provided that rankings are based on symmetric link-weights). If the ranking is energy or link quality based, then XTC will choose a topology that routes around walls and obstacles. WMAN 2004

XTC Analysis (Part 2) If the given graph is a Unit Disk Graph (no obstacles, nodes homogeneous, but not necessarily uniformly distributed), then … The degree of each node is at most 6. The topology is planar. The graph is a subgraph of the RNG. Relative Neighborhood Graph RNG(V): An edge e = (u,v) is in the RNG(V) iff there is no node w with (u,w) < (u,v) and (v,w) < (u,v). u v WMAN 2004

XTC Average-Case Unit Disk Graph XTC WMAN 2004

XTC Average-Case (Degrees) u UDG max UDG avg GG max GG avg XTC max XTC avg WMAN 2004

XTC Average-Case (Stretch Factor) XTC vs. UDG – Euclidean GG vs. UDG – Euclidean XTC vs. UDG – Energy GG vs. UDG – Energy WMAN 2004

XTC Average-Case (Geometric Routing) connectivity rate worse GFG/GPSR on GG GOAFR+ on GXTC better GOAFR+ on GG WMAN 2004

No complex assumptions Conclusion Even with minimal assumptions, only neighbor ranking, it is possible to construct a topology with provable properties: Symmetry Connectivity Bounded degree Planarity Simple algorithm + XTC lends itself to practical implementation No complex assumptions WMAN 2004

Really?!? Topology Control Drop long-range neighbors: Reduces interference and energy! But still stay connected (or even spanner) Really?!? WMAN 2004

What Is Interference? Model We want to minimize maximum interference! Transmitting edge e = (u,v) disturbs all nodes in vicinity Interference of edge e = # Nodes covered by union of the two circles with center u and v, respectively, and radius |e| We want to minimize maximum interference! At the same time topology must be connected or a spanner etc. 8 WMAN 2004

Low Node Degree Topology Control? Most researchers argue that low node degree is sufficient for low interference! This is not true since you can construct very bad topologies with minimum node degree but huge interference! WMAN 2004

Let’s Study the Following Topology! …from a worst-case perspective. WMAN 2004

Topology Control Algorithms Produce… All known topology control algorithms (XTC too!) include the nearest neighbor forest as a subgraph, and produce something like this: The interference of this graph is (n)! WMAN 2004

But Interference… Interference does not need to be high… This topology has interference O(1)!! WMAN 2004

Interesting Research Question We have some preliminary results: There is no local algorithm that finds a good interference topology The optimal topology will not be planar, etc. LISE, LLISE algorithms [Burkhart, von Rickenbach, Wattenhofer, Z. @ MobiHoc 2004] WMAN 2004

Simulation UDG, I = 50 XTC, I = 25 LLISE2, I = 23 LLISE10, I = 12 20 Knoten pro Unit Disk 10 Units Seitenlänge RNG kein Spanner! LLISE2, I = 23 LLISE10, I = 12 WMAN 2004

No complex assumptions Conclusion Even with minimal assumptions, only neighbor ranking, it is possible to construct a topology with provable properties: Symmetry Connectivity Bounded degree Planarity Simple algorithm No complex assumptions XTC lends itself to practical implementation + But does Topology Control really reduce interference? WMAN 2004

Questions? Comments? Distributed Computing Group