Fast Distributed Algorithm for Convergecast
in Ad Hoc Geometric Radio Networks
Alex Kesselman, Darek Kowalski
MPI Informatik
Presentation Flow
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Introduction
Problem Description
System Model
Algorithm
Analysis
Conclusions and Future Work
Applications of Sensor Networks
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Military,
Environmental,
Rescue ...
Wireless ad-hoc networks
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System characteristics:
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Large number of wireless nodes
Each node has a limited battery power
Adjustable transmission ranges
Several challenging problems:
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Fast communication
Low energy operation
Main Communication Tasks
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Collecting data – Convergecast
Distributing data – Broadcast
Motivation
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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
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There are n nodes in the network
Data from all the nodes to be collected
at a central node
Metrics
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Time complexity
Energy consumption
System Model
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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
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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
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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
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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.
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The probability of u’s successful transmission:
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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.
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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
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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
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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
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Consider a round t and let m be the
number of active nodes
Enumerate the nodes in the order of nonincreasing 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.
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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
v1
r
vi
v2
v3
v4
Proof of Lemma 2 Cont.
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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
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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 = (n2*d2)
Conclusion
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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
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Relax the collision detection and GPS
assumptions
Design deterministic algorithms
Analyze the energy/latency tradeoff for
the whole range of latency bounds