Algorithms for Ad Hoc Networks : Algorithms for Ad Hoc Networks Roger Wattenhofer
MedHocNet 2005
Distributed Algorithms vs. Ad Hoc Networking : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 2 Distributed Algorithms vs. Ad Hoc Networking Small community
O(…), (…), (…)
Everybody knows best paper
New algorithm: Compare it with the best previous
Sometimes study the wrong problem; propose protocols that are way too complicated Big community
Milliseconds
Everybody knows first* paper
New protocol: Compare it with the first that was proposed
Reinvent the wheel; many papers do not offer any progress
Algorithmic Research in Ad Hoc and Sensor Networking : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 3 Algorithmic Research in Ad Hoc and Sensor Networking Link Layer
Network Layer
Services
Theory/Models Clustering (Dominating Sets, etc.)
MAC Layer and Coloring
Topology and Power Control
Interference and Signal-to-Noise-Ratio
Deployment (Unstructured Radio Networks)
New Routing Paradigms (e.g. Link Reversal)
Geo-Routing
Broadcast and Multicast
Data Gathering
Location Services and Positioning
Time Synchronization
Modeling and Mobility
Lower Bounds for Message Passing
Selfish Agents, Economic Aspects, Security
Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 4 Overview Introduction
Ad Hoc and Sensor Networks
Routing / Broadcasting
Clustering
Conclusions
Routing in Ad Hoc Networks : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 5 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
Simple Classification of Ad hoc Routing Algorithms : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 6 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 Flooding:
when node received
message the first time,
forward it to all neighbors Distance Vector Routing:
as in a fixnet nodes
maintain routing tables
using update messages Reactive Routing
Flooding the whole network does not scale no mobility mobility very high critical mobility Source Routing (DSR, AODV):
flooding, but re-use old routes
Discussion : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 7 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
Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 8 Overview Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Conclusions
Finding a Destination by Flooding : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 9 Finding a Destination by Flooding
Finding a Destination Efficiently : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 10 Finding a Destination Efficiently
(Connected) Dominating Set : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 11 (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 tohave few nodes in the (C)DS. This is known as theMinimum (C)DS problem.
Formal Problem Definition: M(C)DS : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 12 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
Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 13 Overview Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Algorithm Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 14 Algorithm Overview 0.2 0.5 0.2 0.8 0 0.2 0.3 0.1 0.3 0 Input:
Local Graph Fractional
Dominating Set Dominating
Set Connected
Dominating Set 0.5 Phase C:
Connect DS
by “tree” of
“bridges” Phase B:
Probabilistic
algorithm Phase A:
Distributed
linear program
rel. high degree
gives high value
Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 15 Overview Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Phase A is a Distributed Linear Program : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 16 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)
0.5+0.3+0.3+0.2+0.2+0 = 1.5 ¸ 1
Linear Programs can be solved optimally in polynomial time
But not in a distributed fashion! That’s what we do here… 0.2 0.5 0.2 0.8 0 0.2 0.3 0.1 0.3 0 0.5 Linear Program Adjacency matrix
with 1’s in diagonal
Phase A Algorithm : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 17 Phase A Algorithm
Result after Phase A : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 18 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)
Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 19 Overview Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Dominating Set as Integer Program : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 20 Dominating Set as Integer Program What we have after phase A
What we want after phase B
Phase B Algorithm : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 21 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
Result after Phase B : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 22 Result after Phase B Randomized rounding technique
Expected number of nodes joining the dominating set in step 2 is bounded by log(+1) ¢ |DSOPT|.
Expected number of nodes joining the dominating set in step 4 is bounded by |DSOPT|. Theorem: E[|DS|] · O( ln ¢ |DSOPT|)
Related Work on (Connected) Dominating Sets : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 23 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
Recent Improvements : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 24 Recent Improvements Improved algorithms (in submission):
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…?
Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 25 Overview Introduction
Clustering
Flooding vs. Dominating Sets
Algorithm Overview
Phase A
Phase B
Lower Bounds
Topology Control
Conclusions
Lower Bound for Dominating Sets: Intuition… : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 26 Lower Bound for Dominating Sets: Intuition… m n-1 complete n m m … n n n Two graphs (m << n). Optimal dominating sets are marked red. |DSOPT| = 2. |DSOPT| = m+1.
Lower Bound for Dominating Sets: Intuition… : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 27 Lower Bound for Dominating Sets: Intuition… 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! m n-1 n m …
Lower Bound for Dominating Sets: Intuition… : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 28 Lower Bound for Dominating Sets: Intuition… m n-1 complete n m m … n n n But however they decide, one way will be devastating (with n = m2)! |DSOPT| = 2.
|DSOPT without green| ¸ m. |DSOPT| = m+1.
|DSOPT with green| > n
The Lower Bound : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 29 The Lower Bound Lower bounds (Kuhn, Moscibroda, Wattenhofer @ PODC 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.
Graph Used in Dominating Set Lower Bound : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 30 Graph Used in Dominating Set Lower Bound The example is for k = 3.
All edges are in fact special bipartite graphswith large enough girth.
A Theory of “Locality”? : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 31 A Theory of “Locality”? Ad hoc and sensor networks
The largest network in the world?!?
Managing organizations? Society?!?
Matrix multiplication, etc.
A better and faster algorithm : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 32 A better and faster algorithm Assume that nodes know their position (GPS)
Assume that nodes are in the plane; two nodes are within their transmission radius if and only if their Euclidean distance is at most 1 (UDG, unit disk graph)
Then… : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 33 Then… half of tx radius
Algorithm : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 34 Algorithm Beacon your position
If, in your virtual grid cell, you are the node closest to the center of the cell, then join the CDS, else do not join.
That’s it.
1 transmission per node, O(1) approximation, even for CDS
If you have mobility, then simply “loop” through algorithm, as fast as your application/mobility wants you to.
Comparison : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 35 Comparison First algorithm (distributed linear program)
Algorithm computes CDS
k2+O(1) transmissions/node
O(O(1)/k log ) approximation
General graph
No position information Second algorithm (virtual grid)
Algorithm computes CDS
1 transmission/node
O(1) approximation
Unit disk graph (UDG)
Position information (GPS)
Let’s talk about models… : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 36 Let’s talk about models… General Graph
Captures obstacles
Captures directional radios
Often too pessimistic UDG & GPS
UDG is not realistic
GPS not always available
Indoors
2D 3D?
Often too optimistic too pessimistic too optimistic Are there any models in
between these extremes?
Models : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 37 Models too pessimistic too optimistic General
Graph UDG
GPS UDG
No GPS Quasi
UDG d 1 Bounded
Growth Unit Ball
Graph In a doubling
metric: Number of
independent
neighbors
is bounded
(UDG: 5)
Another Algorithm 1: MIS : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 38 Another Algorithm 1: MIS Build maximal independent set (MIS), then connect MIS for CDS
Proposed by many, patented(!) by Alzoubi et al.
A MIS is by definition also a DS
Connecting with independent 1- and 2-hop bridges
Slow! Works well only on UDGs; robust for general graphs
Another Algorithm 2: Election : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 39 Another Algorithm 2: Election Every node elects a leader; every elected node goes into DS
First analyzed by Jie Gao et al.
1 round of communication for DS only; lots of practical appeal
In the worst case very bad, even for UDGs only a √n approximation 5 6 1 9 4 7 2 3 8
Another Algorithm 3: Non-neighboring neighbors : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 40 Another Algorithm 3: Non-neighboring neighbors If a node has neighbors who are not neighbors, join CDS
Proposed by Jie Wu et al.
Renders a CDS directly
Almost as bad as choosing all nodes, even for random UDGs
Only DS algorithm reviewed in several books
Lots of improvements, also proposed by Jie Wu et al. ?
Another Algorithm 4: Covering connected neighbors : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 41 Another Algorithm 4: Covering connected neighbors If higher priority neighbors are connected and cover all other neighbors, then don’t join CDS, else join CDS
This talk, inspired by an improvement of Jie Wu
2 rounds of communication for CDS only; lots of practical appeal
In the worst case very bad, even for UDGs only a √n approximation
However, on random UDGs, this gives a O(1) approximation 5 6 1 9 4 7 2 3 8
Result Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 42 Result Overview tx / node quality O(1) log √n 1 2 O(log*) O(log) General Graph2 UDG67 UDG4 UDG5 UDG/GPS1 GBG8 UDG = Unit Disk Graph
UBG = Unit Ball Graph
GBG = Growth Bounded G.
/GPS = With Position Info
/D = With Distance Info Lower Bound for General Graphs9 better better UBG/D3 loglog ?
References : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 43 References Folk theorem, e.g. Kuhn, Wattenhofer, Zhang, Zollinger, PODC 2003
Kuhn, Wattenhofer, PODC 2003; improvement submitted
CDS improvement by Dubhashi et al, SODA 2003
Kuhn, Moscibroda, Wattenhofer, PODC 2005
Alzoubi, Wan, Frieder, MobiHoc 2002
Wu and Li, DIALM 1999
Gao, Guibas, Hershberger, Zhang, Zhu, SCG 2001
This Talk, improving on Wu and Li
Kuhn, Moscibroda,Nieberg, Wattenhofer, submitted
Kuhn, Moscibroda, Wattenhofer, PODC 2004
More Models : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 44 More Models Random Distribution
for all geometric models
“Infocom vs. PODC”
Related Problems
e.g. (Connected) Domatic Partition Moscibroda et al., WMAN 2005
Facility Location Moscibroda et al., PODC 2005
Weighted Graph Models
Signal-to-Interference-and-Noise-Ratio (SINR)
Communication Models
Message Size
Unstructured Radio Network (no established MAC layer)
Clustering for Unstructured Radio Networks : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 45 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 @ MobiCom 2004]
[Moscibroda, Wattenhofer @ PODC 2005]
Overview : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 46 Overview Introduction
Clustering
Conclusions
Big Research Opportunities : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 47 Big Research Opportunities Link Layer
Network Layer
Services
Theory/Models Clustering (Dominating Sets, etc.)
MAC Layer and Coloring
Topology and Power Control
Interference and Signal-to-Noise-Ratio
Deployment (Unstructured Radio Networks)
New Routing Paradigms (e.g. Link Reversal)
Geo-Routing
Broadcast and Multicast
Data Gathering
Location Services and Positioning
Time Synchronization
Modeling and Mobility
Lower Bounds for Message Passing
Selfish Agents, Economic Aspects, Security
Check yourself: www.dcg.ethz.ch Reading List : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 48 Check yourself: www.dcg.ethz.ch Reading List …
Conclusions & Open Problems : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 49 Conclusions & Open Problems You don’t have to do algorithms and proofs…
… but it would be good to be aware of them.
Open Problems and Research Directions
Fast good algorithm (for standard UDG) or new lower bound
Study problems for models in-between UDG and general graph
Mobility and dynamics
Study new models: e.g. SINR
Real implementations
Questions?Comments? : Roger Wattenhofer, ETH Zurich @ MedHocNet 2005 Questions?Comments? Thank you for your attention