Wireless Networking Graph Theory Unplugged WG 2004

Wireless Networking Graph Theory Unplugged WG 2004
Distributed Computing Group Roger Wattenhofer

Roger Wattenhofer, ETH Zurich @ WG 2004
Overview Introduction Ad-Hoc and Sensor Networks Routing / Broadcasting Clustering Topology Control Conclusions Roger Wattenhofer, ETH WG 2004

Wireless ad-hoc nodes (“terminodes”) are distributed
Radio Power Wireless ad-hoc nodes (“terminodes”) are distributed Processor Sensors Picture is from the Berkeley Smart Dust project: The RF motes' final design was finished in the early part of 1999. Since then, various projects have been undertaken to test communication protocols for distributed sensor networks with the RF motes. Memory

What are Ad-Hoc/Sensor Networks?
Roger Wattenhofer, ETH WG 2004

Open Problem #1: Positioning and Virtual Coordinates
Unit Disk Graph: Link if and only if Euclidean distance at most 1. Positioning: Some nodes know their position (“anchor nodes”). Virtual Coordinates: Unit Disk Graph Embedding Graph Drawing? (Edge crossings  I couldn’t care less!!!) Known to be NP-hard [Breu & Kirkpatrick 1998] There is no PTAS [Kuhn et al, 2004] Approximation algorithms? Minimize ratio of longest edge over shortest non-edge. Mobile/dynamic nodes  Local updates, stability Roger Wattenhofer, ETH WG 2004

Routing in Ad-Hoc Networks
Multi-Hop Routing Moving information through a network from a source to a destination if source and destination are not within mutual transmission range Reliability Nodes in an ad-hoc network are not 100% reliable Algorithms need to find alternate routes when nodes are failing Mobile Ad-Hoc Network (MANET) It is often assumed that the nodes are mobile (“Moteran”) Roger Wattenhofer, ETH WG 2004

Simple Classification of Ad-hoc Routing Algorithms
Proactive Routing Small topology changes trigger a lot of updates, even when there is no communication  does not scale Reactive Routing Flooding the whole network does not scale Distance Vector Routing: as in a fixnet nodes maintain routing tables using update messages Flooding: when node received message the first time, forward it to all neighbors DSDV 10 tricks  2^10 algorithms no mobility critical mobility mobility very high Source Routing (DSR, AODV): flooding, but re-use old routes Roger Wattenhofer, ETH WG 2004

Discussion Lecture “Mobile Computing”: 10 Tricks  210 routing algorithms In reality there are almost that many! Q: How good are these routing algorithms?!? Any hard results? A: Almost none! Method-of-choice is simulation… Perkins: “if you simulate three times, you get three different results” Flooding is key component of (many) proposed algorithms At least flooding should be efficient Wattenhofer: “even if you don’t implement provably good algorithms (because they are too complicated, or too simple), they still help you to understand the problem” Roger Wattenhofer, ETH WG 2004

Overview Introduction Clustering Flooding vs. Dominating Sets Algorithm Overview Phase A Phase B Lower Bounds Topology Control Conclusions Roger Wattenhofer, ETH WG 2004

Finding a Destination by Flooding
One active node > weight 1 Roger Wattenhofer, ETH WG 2004

Finding a Destination Efficiently
One active node > weight 1 Roger Wattenhofer, ETH WG 2004

(Connected) Dominating Set
A Dominating Set DS is a subset of nodes such that each node is either in DS or has a neighbor in DS. A Connected Dominating Set CDS is a connected DS, that is, there is a path between any two nodes in CDS that does not use nodes that are not in CDS. It might be favorable to have few nodes in the (C)DS. This is known as the Minimum (C)DS problem. Dominating set problem: how many queens do you need to attack/own all 64 squares of the chess board. How many for a CDS? Popular in 1850 among chess players (also famous: maximum independent set) Answer 5. rows 2,3,4,5,8 in column D, is also CDS. Show it! Ask them what they think. Dominating set is an important problem. Domination number \gamma of a graph is the size of the smallest dominating set Roger Wattenhofer, ETH WG 2004

Formal Problem Definition: M(C)DS
Input: We are given an (arbitrary) undirected graph. Output: Find a Minimum (Connected) Dominating Set, that is, a (C)DS with a minimum number of nodes. Problems M(C)DS is NP-hard Find a (C)DS that is “close” to minimum (approximation) The solution must be local (global solutions are impractical for mobile ad-hoc network) – topology of graph “far away” should not influence decision who belongs to (C)DS Roger Wattenhofer, ETH WG 2004

Algorithm Overview Input: Local Graph Fractional Dominating Set Dominating Set Connected Dominating Set 0.2 0.2 0.5 0.3 0.3 0.8 0.5 0.2 0.1 Phase A: Distributed linear program rel. high degree gives high value Phase B: Probabilistic algorithm Phase C: Connect DS by “tree” of “bridges” Roger Wattenhofer, ETH WG 2004

Phase A is a Distributed Linear Program
Nodes 1, …, n: Each node u has variable xu with xu ¸ 0 Sum of x-values in each neighborhood at least 1 (local) Minimize sum of all x-values (global) = 1.5 ¸ 1 Linear Programs can be solved optimally in polynomial time But not in a distributed fashion! That’s what we do here… Linear Program 0.2 0.2 0.5 0.3 0.3 0.8 0.5 0.2 0.1 Adjacency matrix with 1’s in diagonal Roger Wattenhofer, ETH WG 2004

Phase A Algorithm A constant series of steps. In each step each node talks to all the neighbors, and then does local computation… Roger Wattenhofer, ETH WG 2004

Result after Phase A Distributed Approximation for Linear Program Instead of the optimal values xi* at nodes, nodes have xi(), with The value of  depends on the number of rounds k (the locality) For some special kind of graphs alpha with only 1 round (regular graphs) Roger Wattenhofer, ETH WG 2004

Dominating Set as Integer Program
What we have after phase A What we want after phase B Roger Wattenhofer, ETH WG 2004

Phase B Algorithm Each node applies the following algorithm: Calculate (= maximum degree of neighbors in distance 2) Become a dominator (i.e. go to the dominating set) with probability Send status (dominator or not) to all neighbors If no neighbor is a dominator, become a dominator yourself From phase A Highest degree in distance 2 1 takes two rounds… Roger Wattenhofer, ETH WG 2004

Related Work on (Connected) Dominating Sets
Global algorithms Johnson (1974), Lovasz (1975), Slavik (1996): Greedy is optimal Guha, Kuller (1996): An optimal algorithm for CDS Feige (1998): ln  lower bound unless NP 2 nO(log log n) Local (distributed) algorithms “Handbook of Wireless Networks and Mobile Computing”: All algorithms presented have no guarantees Gao, Guibas, Hershberger, Zhang, Zhu (2001): “Discrete Mobile Centers” O(loglog n) time, but nodes know coordinates MIS-based algorithms (e.g. Alzoubi, Wan, Frieder, 2002) that only work on unit disk graphs. Kuhn, Wattenhofer (2003): Tradeoff time vs. approximation Roger Wattenhofer, ETH WG 2004

Recent Improvements Improved algorithms (Kuhn, Wattenhofer, 2004): O(log2 / 4) time for a (1+)-approximation of phase A with logarithmic sized messages. If messages can be of unbounded size there is a constant approximation of phase A in O(log n) time, using the graph decomposition by Linial and Saks. An improved and generalized distributed randomized rounding technique for phase B. Works for quite general linear programs. Is it any good…? Roger Wattenhofer, ETH WG 2004

Lower Bound for Dominating Sets: Intuition…
Two graphs (m << n). Optimal dominating sets are marked red. complete n n n … n-1 m m m n |DSOPT| = 2. |DSOPT| = m+1. Roger Wattenhofer, ETH WG 2004

In local algorithms, nodes must decide only using local knowledge. In the example green nodes see exactly the same neighborhood. So these green nodes must decide the same way! … n-1 m m n Roger Wattenhofer, ETH WG 2004

The Lower Bound Lower bounds (Kuhn, Moscibroda, Wattenhofer, 2004): Model: In a network/graph G (nodes = processors), each node can exchange a message with all its neighbors for k rounds. After k rounds, node needs to decide. We construct the graph such that there are nodes that see the same neighborhood up to distance k. We show that node ID’s do not help, and using Yao’s principle also randomization does not. Results: Many problems (vertex cover, dominating set, matching, etc.) can only be approximated (nc/k2 / k) and/or (1/k / k). It follows that a polylogarithmic dominating set approximation (or maximal independent set, etc.) needs at least (log  / loglog ) and/or ((log n / loglog n)1/2) time. Roger Wattenhofer, ETH WG 2004

Graph Used in Dominating Set Lower Bound
The example is for k = 3. All edges are in fact special bipartite graphs with large enough girth. Roger Wattenhofer, ETH WG 2004

Clustering for Unstructured Radio Networks
“Big Bang” (deployment) of a sensor and/or ad-hoc network: Nodes wake up asynchronously (very late, maybe) Neighbors unknown Hidden terminal problem No global clock No established MAC protocol No reliable collision detection Limited knowledge of the number of nodes or degree of network. We have randomized algorithms that compute DS (or MIS) in polylog(n) time even under these harsh circumstances, where n is an upper bound on the number of nodes in the system. [Kuhn, Moscibroda, Wattenhofer, 2004] Roger Wattenhofer, ETH WG 2004

Overview Introduction Clustering Topology Control What is it? What is it good for? Does Topology Control Reduce Interference? Cellular Networks, Sensor Networks, etc. Conclusions Roger Wattenhofer, ETH WG 2004

Topology Control Drop long-range neighbors: Reduces interference and energy! But still stay connected (or even spanner) Roger Wattenhofer, ETH WG 2004

Topology Control as a Trade-Off
Sometimes also clustering (first part of the talk) is called topology control Topology Control Network Connectivity Spanner Property Conserve Energy Reduce Interference d(u,v) ¢ t ¸ dTC(u,v) Roger Wattenhofer, ETH WG 2004

Classic Solution: Gabriel Graph
Let disk(u,v) be a disk with diameter (u,v) that is determined by the two points u,v. The Gabriel Graph GG(V) is defined as an undirected graph (with E being a set of undirected edges). There is an edge between two nodes u,v iff the disk(u,v) including boundary contains no other points. Gabriel Graph is planar Gabriel Graph is energy optimal [energy of link is at least distance squared] disk(u,v) v u Extra pages Explain RNG Roger Wattenhofer, ETH WG 2004

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

Related 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), locality. 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]) Explicit interference [Meyer auf der Heide et al. SPAA 2002]. Interference between edges, time-step routing model, congestion; trade-offs; however, interference model based on network traffic Roger Wattenhofer, ETH WG 2004

What Is Interference? Model Problem statement
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| Problem statement We want to minimize maximum interference! At the same time topology must be connected or a spanner etc. 8 Exact size of interference range does not change the results Roger Wattenhofer, ETH WG 2004

Low Node Degree Topology Control?
Low node degree does not necessarily imply low interference: Very low node degree but huge interference Roger Wattenhofer, ETH WG 2004

Let’s Study the Following Topology!
…from a worst-case perspective Roger Wattenhofer, ETH WG 2004

Topology Control Algorithms Produce…
All known topology control algorithms (with symmetric edges) include the nearest neighbor forest as a subgraph and produce something like this: The interference of this graph is (n)! Roger Wattenhofer, ETH WG 2004

But Interference… Interference does not need to be high… This topology has interference O(1)!! Roger Wattenhofer, ETH WG 2004

Algorithms and Lower Bounds
[Burkhart, von Rickenbach, Wattenhofer, Zollinger, 2004] Interference-optimal connectivity-preserving topology Local interference-optimal spanner topology Algorithms also work if interference radius >> transmission radius No local algorithm can find a good topology Optimal topology is not planar UDG, I = 50 RNG, I = 25 LLISE10, I = 12 Roger Wattenhofer, ETH WG 2004

New Results… Interference-driven topology control is exciting new paradigm… We have a few other upcoming results: For cellular networks: minimize number of base stations a mobile station overhears by reducing the transmission power of the base stations  “minimum membership set cover” problem For sensor networks: data gathering without listening to lots of unwanted traffic… Roger Wattenhofer, ETH WG 2004

Open Problem #2: In-Interference
Given ad-hoc network represented by nodes in a plane. Connect nodes by spanning tree. Circle of each node centered at node with the radius being the length of longest adjacent edge in spanning tree. Coverage of node is the number of circles node falls into. Minimize the maximum (or average) coverage. 4 4 2 3 3 3 Roger Wattenhofer, ETH WG 2004

Overview Introduction Clustering Topology Control Conclusions Clustering vs. Topology Control More realism, more realism, more realism, … … Practice! Roger Wattenhofer, ETH WG 2004

Clustering vs. Topology Control
(Connected) Dominating Set (Connected) Domatic Partition Both approaches sparsen the graph in order to reduce energy… … by turning off fraction of the nodes, and thus interference. Two sides of the same medal? Topology Control Interference-Driven T.C. … by turning off long-range links, and thus interference. Roger Wattenhofer, ETH WG 2004

What does a typical ad-hoc network look like?

Like this? Roger Wattenhofer, ETH WG 2004

Like this? One active node > weight 1 Roger Wattenhofer, ETH WG 2004

Or rather like this? One active node > weight 1 Roger Wattenhofer, ETH WG 2004

Or even like this? Roger Wattenhofer, ETH WG 2004

What about typical mobility?
Brownian Motion? Random Way-Point? Statistical Data Model? Roger Wattenhofer, ETH WG 2004

Combine Theory with Practice
Practical experiments… btnodes of NCCR/MICS Scatterweb Shockfish Roger Wattenhofer, ETH WG 2004

Some credit… Problem (Algorithm) Reference Dominating Set Kuhn, PODC 2003 Kuhn, Moscibroda, PODC 2004 Topology Control and Interference (LISE, XTC, MMSC, Sensor Networks) Burkhart et MobiHoc 2004 W., WMAN 2004 von Rickenbach et submitted Zollinger et submitted Geo-Routing (GOAFR) Kuhn, W., MobiHoc 2003 Kuhn, W., Zhang, PODC 2003 Positioning (GHoST) Bischoff, PerCom 2004 Kuhn et submitted, Bischoff et subm. Data gathering Cristescu et submitted von Rickenbach, submitted Models: Quasi-UDG Kuhn, W., DIALM 2003 “Big Bang” problem Moscibroda, Kuhn, ESA & MobiCom 2004 Roger Wattenhofer, ETH WG 2004

Questions? Comments? Distributed Computing Group Roger Wattenhofer
Thanks to my students Fabian Kuhn, Aaron Zollinger, Regina Bischoff, Thomas Moscibroda, Pascal von Rickenbach, Martin Burkhart, etc.

BACKUP

Distributed Computing and Networking Graph Theory
Mobile Computing Distributed Computing and Networking Graph Theory Computational Geometry

TODO Invitation back: graph theory is needed urgently…! Only few groups study provable results. Graphs, UDG, QUDG, etc. New Dominating set, lower bound GOAFR Open Problems

Open Problems UDG Embedding Edge crossings  I couldn’t care less!!!
Applications in ad-hoc/sensor networks (GPS) Known to be NP-hard [Breu & Kirkpatrick 1998] There is no PTAS [Kuhn et al, 2004] Approximation algorithms? Minimize ratio of longest edge over shortest non-edge. Mobile/dynamic nodes  Local updates, stability

Interference Tree In-Interference Highway-Model 3. The routing issue
(few bits only, for example)

Overview Introduction Routing in Ad-Hoc Networks Dominating Set Based Routing Geometric Routing What is geometric routing? Correct geometric routing Worst-case efficient geometric routing Average-case efficient geometric routing Topology Control Conclusions Roger Wattenhofer, ETH WG 2004

Geometric Routing ??? t s s Roger Wattenhofer, ETH WG 2004

Greedy Routing Each node forwards message to “best” neighbor t s Roger Wattenhofer, ETH WG 2004

Greedy Routing Each node forwards message to “best” neighbor But greedy routing may fail: message may get stuck in a “dead end” Needed: Correct geometric routing algorithm t ? s Roger Wattenhofer, ETH WG 2004

What is Geometric Routing?
A.k.a. location-based, position-based, geographic, etc. Chapter 18 (“Routing […] in Geometric and Wireless Networks”) in “Handbook of Wireless Networking and Mobile Computing” Each node knows its own position and position of neighbors Source knows the position of the destination No routing tables stored in nodes! Geometric routing makes sense Own position: GPS/Galileo, local positioning algorithm Destination: overlay P2P net, geocasting, source routing++ Learn about ad-hoc routing in general Roger Wattenhofer, ETH WG 2004

Face Routing Based on ideas by [Kranakis, Singh, Urrutia CCCG 1999] Here simplified (and actually improved) Roger Wattenhofer, ETH WG 2004

Face Routing Remark: Planar graph can easily (and locally!) be computed with the Gabriel Graph, for example. Roger Wattenhofer, ETH WG 2004

Face Routing s t Roger Wattenhofer, ETH WG 2004

Face Routing Properties
All necessary information is stored in the message Source and destination positions Point of transition to next face Completely local: Knowledge about direct neighbors‘ positions sufficient Faces are implicit Planarity of graph is computed locally (not an assumption) “Right Hand Rule” Roger Wattenhofer, ETH WG 2004

Face Routing Works on Any Graph

Face Routing Theorem: Face Routing reaches destination in O(n) steps But: Can be very bad compared to the optimal route Roger Wattenhofer, ETH WG 2004

Bounding Searchable Area
t Roger Wattenhofer, ETH WG 2004

Adaptively Bound Searchable Area
What is the correct size of the bounding area? Start with a small searchable area Grow area each time you cannot reach the destination In other words, adapt area size whenever it is too small Theorem: Algorithm finds destination after O(c2) steps, where c is the cost of the optimal path from source to destination. Proof: Not in this presentation. Roger Wattenhofer, ETH WG 2004

Algorithm is worst-case optimal
Can we do any better? No, with “wheel” example Destination is central node Source is any node on ring Any spoke can go to middle Geometric routing: no routing tables  test many spines Best path of size O(c)+O(c) Test (c) spokes of length (c) Cost (c2) instead of O(c) Theorem: Algorithm is asymptotically worst-case optimal. Roger Wattenhofer, ETH WG 2004

GOAFR – Greedy Other Adaptive Face Routing
Algorithm is not very efficient (especially in dense graphs) Combine Greedy and (Other Adaptive) Face Routing Route greedily as long as possible Circumvent “dead ends” by use of face routing Then route greedily again Theorem: GOAFR is still asymptotically worst-case optimal… …and it is efficient in practice, in the average-case. What does “practice” mean? Usually nodes placed uniformly at random Roger Wattenhofer, ETH WG 2004

Average Case Not interesting when graph not dense enough Not interesting when graph is too dense Critical density range (“percolation”) Shortest path is significantly longer than Euclidean distance too sparse critical density too dense Roger Wattenhofer, ETH WG 2004

Critical Density: Shortest Path vs. Euclidean Distance
Shortest path is significantly longer than Euclidean distance Critical density range mandatory for the simulation of any routing algorithm (not only geometric) Roger Wattenhofer, ETH WG 2004

Simulation on Randomly Generated Graphs
9 1 GPSR worse 0.9 8 Connectivity 0.8 7 0.7 6 Greedy success 0.6 Performance 5 0.5 Frequency 0.4 4 GOAFR+ 0.3 3 better 0.2 2 0.1 critical 1 2 4 6 8 10 12 Network Density [nodes per unit disk] Roger Wattenhofer, ETH WG 2004

Discussion Previously known Non-competitive algorithms (Face Routing, GPSR, …) Three papers [DIALM 2002, MOBIHOC 2003, PODC 2003]: The first worst-case optimal algorithm The first worst-case optimal and average-case efficient algorithm Percolation theory to evaluate routing algorithms Various results for different cost metrics (“super-linear” not competitive) Various related results (“bounded degree graphs”) Algorithm is simple: Can be implemented in network processor Quite a few implementations available Algorithm can be married with other approaches Geometric routing as a more stable form of source routing First tight result in ad-hoc routing (to our knowledge) Roger Wattenhofer, ETH WG 2004

Topology Control Drop long-range neighbors: Reduces interference, and energy! But still stay connected (or even spanner) Roger Wattenhofer, ETH WG 2004

Overview Introduction Routing in Ad-Hoc Networks Dominating Set Based Routing Geometric Routing Topology Control What’s that? XTC Interference Conclusions Roger Wattenhofer, ETH WG 2004

XTC Algorithm (W., Zollinger)
D C G B Each node produces “ranking” of neighbors. Examples Distance (closest) Energy (lowest) Link Quality (best) Nodes exchange rankings with neighbors A 1. C 2. E 3. B 4. F 5. D 6. G E F Roger Wattenhofer, ETH WG 2004

XTC Algorithm (Part 2) 3. B 4. A 6. G 8. D 2. C 4. G 5. A 4. B 6. A 7. C D C G 7. A 8. C 9. E B 1. F 3. A 6. D 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 don’t 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 Roger Wattenhofer, ETH WG 2004

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

XTC Analysis (Part 2) If the graph was a Euclidean graph (no obstacles, nodes homogeneous, but not necessarily uniformly distributed), then … The degree of each node is at most 6. The topology will be planar. The graph will be a sub-graph 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 Roger Wattenhofer, ETH WG 2004

XTC Average-Case Remark: We are currently implementing XTC… Unit Disk Graph XTC Roger Wattenhofer, ETH WG 2004

XTC Average-Case (Degrees)

XTC Average-Case (Span)

K-Neigh (Blough, Leoncini, Resta, Santi @ MobiHoc 2003)
“Connect to k closest neighbors!” Tough Question: What should k be? Algorithm is dead simple. On average as good as others… [Thanks to P. Santi] W. et al. Infocom 2001 Roger Wattenhofer, ETH WG 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 Roger Wattenhofer, ETH WG 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 Roger Wattenhofer, ETH WG 2004

Lessons to be learned? Average-case µ Worst-case (A worst-case algorithm also works in the average-case, but not vice versa) Average-case as great source of inspiration Algorithm should at least be correct (whatever that means)! It seems to be easier to tune a worst-case algorithm for the average case than vice versa (e.g. AFR  GOAFR+) Roger Wattenhofer, ETH WG 2004

Expected number of dominators in step 2
By definition By step 2: Using Phase A: xi* is the optimum Roger Wattenhofer, ETH WG 2004

Expected number of additional dominators in step 4
Pr[node i not dominated] No neighbor dominator after step 2 Roger Wattenhofer, ETH WG 2004

Result after Phase B With Uncovered nodes become dominators Previous slide Solution of Dual LP, and Dual · Primal Theorem: E[|DS|] · O( ln  ¢ |DSOPT|) Roger Wattenhofer, ETH WG 2004

Algorithms – Requirement: Retain Graph Connectivity
LIFE (Low Interference Forest Establisher) Attribute interference values as weights to edges Compute minimum spanning tree/forest (Kruskal’s algorithm) Theorem: LIFE constructs a Minimum Interference Forest Proof: Algorithm computes forest MST also minimizes maximum interference value Roger Wattenhofer, ETH WG 2004

Algorithms – Requirement: Construct Spanner
LISE (Low Interference Spanner Establisher) Add edges with increasing interference until spanner property fulfilled Theorem: LISE constructs a Minimum Interference t-Spanner Proof: Algorithm computes t-spanner Algorithm inserts edges with increasing coverage only “as long as necessary” Roger Wattenhofer, ETH WG 2004

Algorithms – Requirement: Construct Spanner Locally
LLISE Local algorithm: scalable Nodes collect (t/2)-neighborhood Locally compute interference-minimal paths guaranteeing spanner property Only request that path to stay in the resulting topology Theorem: LLISE constructs a Minimum Interference t-Spanner Roger Wattenhofer, ETH WG 2004

Average-Case Interference: Preserve Connectivity
UDG GG RNG LIFE Roger Wattenhofer, ETH WG 2004

Interference-Optimal Topology
There is no local algorithm that can find a good interference topology The optimal topology will not be planar Roger Wattenhofer, ETH WG 2004

Average-Case Interference: Spanners
RNG LLISE, t=2 t=4 t=6 t=8 t=10 LIFE Roger Wattenhofer, ETH WG 2004

Algorithms and Lower Bounds
[Burkhart, von Rickenbach, Wattenhofer, Zollinger, 2004] Interference-optimal connectivity-preserving topology Local interference-optimal spanner topology Algorithms also work if interference radius >> transmission radius No local algorithm can find a good topology Optimal topology is not planar Roger Wattenhofer, ETH WG 2004

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

Wireless Networking Graph Theory Unplugged WG 2004

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