Alex Kesselman, MPI Internet Algorithms: Design and Analysis MiniCourse, Oct. 2004.
Fast Distributed Algorithm for Convergecast in Ad Hoc Geometric Radio Networks Alex Kesselman, Darek...
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Transcript of Fast Distributed Algorithm for Convergecast in Ad Hoc Geometric Radio Networks Alex Kesselman, Darek...
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Fast Distributed Algorithm for Convergecast
in Ad Hoc Geometric Radio Networks
Alex Kesselman, Darek KowalskiMPI 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, Environment
al, 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 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
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Interference
Collision
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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.
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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 < 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.
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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 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
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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 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
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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/n=O(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 OPT’s 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
vi
v1 v2
v3
v4
r
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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)*n*log 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 n*d2 and has linear latency.
Theorem 4: Any convergecast algorithm that has latency O(log n) requires energy (n2*d2).
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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 isn(2id)2 = (n2*d2)
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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
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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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