Fast Distributed Algorithm for Convergecast in Ad Hoc Geometric Radio Networks Alex Kesselman, Darek Kowalski MPI Informatik

Presentation Flow Introduction Problem Description System Model Algorithm Analysis Conclusions and Future Work

Applications of Sensor Networks Military, Environmental, 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 d α (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 p s =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-p s ) Let c=p s /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 log 1-c 1/n progressive rounds By Lemma 1, the expected running time is (1/c)*log 1-c 1/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: R 1  …  R m 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 R i  Z/i. Consider the set S of the first i active nodes The distance between any two nodes in S is at least R i 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 R i  Z  iR i vivi v1v1 v2v2 v3v3 v4v4 r

Proof of Lemma 2 Cont. The energy consumption of the DC algorithm during round t is at most  (Z/i) 2 = Z 2  (1/i) 2  (  2 /6)*Z 2 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*d 2 and has linear latency. Theorem 4: Any convergecast algorithm that has latency O(log n) requires energy  (n 2 *d 2 ).

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 is n  (2 i d) 2 =  (n 2 *d 2 )

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