Fast Distributed Algorithm for Convergecast

in Ad Hoc Geometric Radio Networks

Alex Kesselman, Darek KowalskiMPI Informatik

Presentation Flow

Introduction Problem Description System Model Algorithm Analysis Conclusions and Future Work

Applications of Sensor Networks

Military, Environment

al, Rescue ...

Wireless ad-hoc networks

System characteristics: Large number of wireless nodes Each node has a limited battery

power Adjustable transmission ranges

Several challenging problems: Fast communication Low energy operation

Main Communication Tasks

Collecting data – Convergecast Distributing data – Broadcast

Motivation

We study the convergecast problem

Prior work concentrated on energy efficiency alone

Many new applications have stringent latency requirements

We have dual objective – Low-Latency and Energy-Efficiency

Problem Description

There are n nodes in the network Data from all the nodes to be

collected at a central node Metrics

Time complexity Energy consumption

System Model

Energy consumed for communication at distance d is dα (α between 2 and 4)

Nodes are static and clocks are synchronized Each node can learn the distance to the

closest active neighbor (using GPS) A node can either transmit or receive at a

time Collision Detection (CD): each node can

detect a collision within its transmission range Intermediate nodes merge the data into one

message

Interference

Collision

Distributed Convergecast Algorithm

1. Set the transmission range of each node to the distance to the closest active node.

2. Transmit MSG(data, u) with a constant probability p.

3. If a message MSG(data,u) has been transmitted and there is no collision

enter the inactive mode, otherwise, merge the received data (if

any) with u’s own data.

DC Algorithm Example

Convergence UB

Observation 1: The data is passed to nodes that remain active.

Theorem 1: The expected running time of the DC algorithm is O(log n) and the algorithm terminates properly.

Convergence UB Cont.

Let G be the communication graph.Claim 1: The in-degree of any node in

G is at most 6.

Convergence UB Cont.

Lemma 1: There is a constant 0 < c < 1 such that with probability at least c, the fraction of active nodes that perform successful transmission in round t is at least c.

Proof of Lemma 1

Claim 1 implies that the average out-degree among the nodes in G is bounded by 6

At least half of the nodes in G have out-degree of at most 12

Proof of Lemma 1 Cont.

The probability of u’s successful transmission: all its out-neighbors and the in-neighbors of

its out-neighbors remain silent Each of u's out-neighbors may have at

most 6 in-neighbors The probability of successful

transmission is at least ps=p(1-p)72

Proof of Lemma 1 Cont.

The expected number of nodes that do not transmit successfully during a round is at most n(1-ps)

Let c=ps/2 Using Markov inequality, “the number

of nodes which transmit successfully during a round is at least n*c” holds with probability at least c

Proof of Theorem 1

We say that a round is progressive if a fraction c of active nodes become inactive

The algorithm terminates after log1-

c1/n progressive rounds By Lemma 1, the expected running

time is (1/c)*log1-c1/n=O(log n)

Convergence LB

Theorem 2: The expected running time of any (centralized) convergecast algorithm in an arbitrary network is at least (log n).

Proof of Theorem 2

Each node must successfully transmit once

When a node transmits, the receiving node is busy and cannot transmit itself

The number of nodes that have not transmitted yet is decreased by at most a factor of two during a time step

Energy UB

Observation 2: The MST algorithm achieves the optimum energy.

Lemma 2: The energy spent by the DC algorithm during any round is at most (2/6)*n times the optimum energy.

Proof of Lemma 2

Consider a round t and let m be the number of active nodes

Enumerate the nodes in the order of non-increasing transmission range: R1… Rm

Let Z be the sum of the transmission ranges of the nodes under OPT (during OPT’s whole execution)

Proof of Lemma 2 Cont.

Claim 2: We have that Ri Z/i. Consider the set S of the first i active

nodes The distance between any two nodes in

S is at least Ri

Otherwise, at least one node has its itransmission range larger than the distance to the closest active node

The claim follows since OPT must connect all nodes in S to the root

Proof of Lemma 2 Cont.

Each distance is at least Ri Z iRi

vi

v1 v2

v3

v4

r

Proof of Lemma 2 Cont.

The energy consumption of the DC algorithm during round t is at most

(Z/i)2 = Z2 (1/i)2 (2/6)*Z2

On the other hand, the optimum energy is at least n*(Z/n)2

Energy UB Cont.

Theorem 3: The total energy consumption of the DC algorithm at most O((2/6)*n*log n) times the optimum energy.

Energy LB

Consider a line topology and let d be the distance between two consecutive nodes.

Claim 3: OPT requires energy n*d2 and has linear latency.

Theorem 4: Any convergecast algorithm that has latency O(log n) requires energy (n2*d2).

Line Example: OPT

Proof of Theorem 4

In each round a constant fraction of active nodes pass their data to adjacent active neighbors and become inactive

In this case the transmission ranges of active nodes grow exponentially

The total energy consumption isn(2id)2 = (n2*d2)

Conclusion

First sub-linear convergecast algorithm (assuming variable transmission ranges)

Asymptotically optimal running time Can be used for fast gossiping

(convergecast+broadcast) Analysis of energy/latency tradeoff

Open Problems

Relax the collision detection and GPS assumptions

Design deterministic algorithms Analyze the energy/latency

tradeoff for the whole range of latency bounds