Energy-Efficient Distributed Algorithms for Ad hoc Wireless Networks

Gopal Pandurangan

Department of Computer SciencePurdue University

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Energy-Efficient Distributed Algorithms

Ad hoc wireless sensor networks operate under severe

energy constraints. Energy-Efficient distributed algorithms are critical. Low energy algorithms even possibly at the cost of

reduced quality of solution : Distributed approximation algorithms.

Algorithms use only “local” knowledge: Localized algorithms

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Distributed Algorithms

Traditionally complexity measures: messages, time.

Much of theory assumes point-to-point network communication model.

Wireless needs new models for designing distributed algorithms.

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Traffic Monitoring with Sensors

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Data Aggregation - Low Cost Tree

Data aggregation

Aggregate data on a tree Use a low cost tree

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Desirable Features

Simple and local

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Desirable Features

Simple and local

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Desirable Features

Simple and local Dynamic- handle node failures

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Desirable Features

Simple and local Dynamic- handle node failures Distributed Low energy Low synchronization Small number of messages Low degree

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Problem

Network Model: Weighted unit disk graph (UDG)

Find a Minimum Spanning Tree (MST) rooted at a given

node

MST is a difficult problem

Can we construct an approximately good spanning tree?

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Nearest Neighbor Tree (NNT) SchemeKhan and Pandurangan. DISC, 2006, Best Student Paper

Award.

Given: A (connected) undirected weighted graph G.

Each node chooses a unique rank.

Each node connects to its nearest node (via a shortest path) of higher rank.

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NNT Construction

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3

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6

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Output is a spanning tree called NNT.

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NNT Theorem

(Khan, Pandurangan, and Kumar. Theoretical Computer Science, 2007

Theorem 1:

On any graph G, NNT scheme

produces a spanning tree that has a

cost of at most O(log n) times the

(optimal) MST.

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Distributed NNT AlgorithmEach node executes the same algorithm

simultaneously:

Rank selection.

Finding the nearest node of higher rank.

Connecting to the nearest node of higher rank.

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u s

Rank Selection

Root s selects a number p(s) from [b-1, b] s sends ID(s) and p(s) to all of its neighbor in one time step. Any other node u after receiving the first message with ID(v)

and p(v) from a neighbor v: Selects a number p(u) from [p(v)-1, p(v)) Sends ID(u) and p(u) to all of its neighbors

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Defining Rank

For any u and v, r(u) < r(v) iff p(u) < p(v) or p(u) = p(v) and ID(u) < ID(v)

A node with lower random number p() has lower rank. Ties are broken using ID()

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Tree construction

Each node knows the rank of all of its neighbors. The leader s has the highest rank among all nodes

in the graph. For every node (except s), there is a neighbor with

higher rank. It connects to that node.

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NNT algorithm

Very localized. O(|E|) messages. O(Diameter) time. Low energy complexity.

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Energy complexity of a distributed algorithm Energy complexity is a measure of the

energy needed by the distributed algorithm. Various factors affect energy complexity

Time needed. Number of messages exchanged. Radiation energy needed to transmit a message through a certain distance --- typically assumed proportional to

some power of the distance. Energy overheads of the hardware (startup energy,

receiver energy etc.) ….

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Energy Complexity

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M

ii

W r

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A Comparison of Algorithms

Algorithm Energy Complexity MST Quality

GHS (log^2 n) optimal

KPK (TPDS 08) O(log n) on average O(log n)approximation

CKKP (SPAA 08) O(log n) on average optimal

CKKP (SPAA 08) O(1) on average O(1)-approximation

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Questions

Good energy model of hardware? Distributed network computing model for

wireless ? How to design energy-efficient distributed

algorithms? Approximation algorithms? How do cross layer issues affect design? A new theory needed.