Fast%20Distributed%20Algorithm%20for%20Convergecast%20in%20Ad%20Hoc%20Geometric%20Radio%20Networks

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Fast Distributed Algorithm for Convergecastin Ad
Hoc Geometric Radio Networks

• Alex Kesselman, Darek Kowalski
• MPI Informatik

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Presentation Flow

• Introduction
• Problem Description
• System Model
• Algorithm
• Analysis
• Conclusions and Future Work

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Applications of Sensor Networks

• Military,
• Environmental,
• Rescue ...

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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

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Main Communication Tasks

• Collecting data Convergecast
• Distributing data Broadcast

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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

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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

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System Model

• Energy consumed for communication at distance d
  is da (a 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

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Interference
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Distributed Convergecast Algorithm

• Set the transmission range of each node to the
  distance to the closest active node.
• Transmit MSG(data, u) with a constant probability
  p.
• 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
  us own data.

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DC Algorithm Example
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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.

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Convergence UB Cont.

• Let G be the communication graph.
• Claim 1 The in-degree of any node in G is at
  most 6.

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Convergence UB Cont.

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

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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

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Proof of Lemma 1 Cont.

• The probability of us 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 psp(1-p)72

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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 cps/2
• Using Markov inequality, the number of nodes
  which transmit successfully during a round is at
  least nc holds with probability at least c

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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/nO(log n)

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Convergence LB

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

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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

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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.

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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 OPTs whole execution)

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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

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Proof of Lemma 2 Cont.

• Each distance is at least Ri ? Z ? iRi

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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

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Energy UB Cont.

• Theorem 3 The total energy consumption of the DC
  algorithm at most O((?2/6)nlog n) times the
  optimum energy.

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Energy LB

• Consider a line topology and let d be the
  distance between two consecutive nodes.
• Claim 3 OPT requires energy nd2 and has linear
  latency.
• Theorem 4 Any convergecast algorithm that has
  latency O(log n) requires energy ?(n2d2).

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Line Example OPT
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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 is
• n?(2id)2 ?(n2d2)

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Conclusion

• First sub-linear convergecast algorithm (assuming
  variable transmission ranges)
• Asymptotically optimal running time
• Can be used for fast gossiping (convergecastbroad
  cast)
• Analysis of energy/latency tradeoff

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Open Problems

• Relax the collision detection and GPS assumptions
• Design deterministic algorithms
• Analyze the energy/latency tradeoff for the whole
  range of latency bounds

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• Thank You!