## Fast distributed algorithms for computing separable functions

Venue: | IEEE Trans. Inform. Theory |

Citations: | 29 - 5 self |

### BibTeX

@ARTICLE{Mosk-aoyama_fastdistributed,

author = {Damon Mosk-aoyama and Devavrat Shah},

title = {Fast distributed algorithms for computing separable functions},

journal = {IEEE Trans. Inform. Theory},

year = {},

pages = {2997--3007}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—The problem of computing functions of values at the nodes in a network in a fully distributed manner, where nodes do not have unique identities and make decisions based only on local information, has applications in sensor, peer-to-peer, and adhoc networks. The task of computing separable functions, which can be written as linear combinations of functions of individual variables, is studied in this context. Known iterative algorithms for averaging can be used to compute the normalized values of such functions, but these algorithms do not extend in general to the computation of the actual values of separable functions. The main contribution of this paper is the design of a distributed randomized algorithm for computing separable functions. The running time of the algorithm is shown to depend on the running time of a minimum computation algorithm used as a subroutine. Using a randomized gossip mechanism for minimum computation as the subroutine yields a complete fully distributed algorithm for computing separable functions. For a class of graphs with small spectral gap, such as grid graphs, the time used by the algorithm to compute averages is of a smaller order than the time required by a known iterative averaging scheme. Index Terms—Data aggregation, distributed algorithms, gossip algorithms, randomized algorithms. I.

### Citations

715 |
Epidemic algorithms for replicated database maintenance
- Demers, Greene, et al.
- 1987
(Show Context)
Citation Context ...on spreading, gossip algorithms for disseminating a message to all nodes in a complete graph in which communication partners are chosen uniformly at random have been studied for some time [12], [21], =-=[9]-=-. Karp, Schindelhauer, Shenker, and Vöcking presented a push and pull gossip algorithm, in which communicating nodes both send and receive messages, that disseminates a message to all n nodes in a gra... |

668 |
Large Deviations Techniques and Applications
- Dembo, Zeitouni
- 1998
(Show Context)
Citation Context ...for any ε ∈ (0,1/2), �� ��� Pr Rk − 1 � � � � � ε λ� ≥ ≤ 2exp − λ ε2 � k . (2) 3 Proof: By definition, E[Rk] = 1 �k i=1 λ−1 = λ−1 . The inequality in (2) follows directly from Cramér’s k Theorem (see =-=[8]-=-, pp. 30, 35) and properties of exponential random variables. A direct implication of Lemma 1 is the following corollary, which bounds the probability that the absolute time Ck at which clock tick k o... |

646 |
Parallel and distributed computation – Numerical methods
- Bertsekas, Tsitsiklis
- 1989
(Show Context)
Citation Context ...r reaching consensus under appropriate conditions have been known since the classical work of Tsitsiklis [25] and Tsitsiklis, Bertsekas, and Athans [26] (see also the book by Bertsekas and Tsitsiklis =-=[4]-=-). Averaging algorithms compute the ratio of the sum of the input numbers to n, the number of nodes in the network, and not the exact value of the sum. Thus, such algorithms cannot be extended in gene... |

348 | Probabilistic counting algorithms for data base applications
- Flajolet, Martin
- 1985
(Show Context)
Citation Context ...for computing averages in a network. A. Related work In this section, we present a brief summary of related work. Algorithms for computing the number of distinct elements in a multiset or data stream =-=[11]-=-, [2] can be adapted to compute separable functions using information spreading [7]. We are not aware, however, of a previous analysis of the amount of time required for these algorithms to achieve a ... |

305 | Gossipbased computation of aggregate information
- KEMPE, DOBRA, et al.
- 2003
(Show Context)
Citation Context ...es in the graph (using one as the input at each node). Recently, Kempe, Dobra, and Gehrke showed the existence of a randomized iterative gossip algorithm for averaging with the optimal averaging time =-=[15]-=-. This result was restricted to complete graphs. The algorithm requires that the nodes begin the computation in an asymmetric initial state in order to compute separable functions, a requirement that ... |

249 | Approximate aggregation techniques for sensor databases
- Considine, Li, et al.
- 2004
(Show Context)
Citation Context ...rief summary of related work. Algorithms for computing the number of distinct elements in a multiset or data stream [11], [2] can be adapted to compute separable functions using information spreading =-=[7]-=-. We are not aware, however, of a previous analysis of the amount of time required for these algorithms to achieve a certain accuracy in the estimates of the function value when the computation is tot... |

219 | Synopsis diffusion for robust aggregation in sensor networks
- Nath, Gibbons, et al.
- 2004
(Show Context)
Citation Context ...Engineering and Computer Science, MIT. Emails: {damonma@cs.stanford.edu,devavrat@mit.edu} 1san algorithm for computing separable functions that relies on an order- and duplicate-insensitive statistic =-=[20]-=- of a set of numbers, the minimum. The algorithm is based on properties of exponential random variables, and reduces the problem of computing the value of a separable function to the problem of determ... |

198 |
Distributed asynchronous deterministic and stochastic gradient optimization algorithms
- Tsitsiklis, Bertsekas, et al.
- 1986
(Show Context)
Citation Context ...a distributed computation. Distributed algorithms for reaching consensus under appropriate conditions have been known since the classical work of Tsitsiklis [25] and Tsitsiklis, Bertsekas, and Athans =-=[26]-=- (see also the book by Bertsekas and Tsitsiklis [4]). Averaging algorithms compute the ratio of the sum of the input numbers to n, the number of nodes in the network, and not the exact value of the su... |

196 | Randomized rumor spreading
- Karp
- 2000
(Show Context)
Citation Context ...g presented a push and pull gossip algorithm, in which communicating nodes both send and receive messages, that disseminates a message to all n nodes in a graph in O(log n) time with high probability =-=[14]-=-. In this work, we have provided an analysis of the time required for a gossip algorithm to disseminate n messages to n nodes for the more general setting of arbitrary graphs and non-uniform random ch... |

179 |
Algorithms for Random Generation and Counting: A Markov Chain Approach
- Sinclair
- 1993
(Show Context)
Citation Context ...ochastic matrix P , the conductance of P , denoted Φ(P), is � i∈S,j /∈S Pij Φ(P) = min . S⊂V, 0<|S|≤n/2 |S| In general, the above definition of conductance is not the same as the classical definition =-=[24]-=-. However, we restrict our attention in this paper to doubly stochastic matrices P . When P is doubly stochastic, these two definitions are equivalent. Note that the definition of conductance implies ... |

162 | Gossip algorithms: design, analysis and applications
- Boyd, Ghosh, et al.
- 2005
(Show Context)
Citation Context ... for computing separable functions, we obtain an algorithm for computing separable functions whose performance on certain graphs compares favorably with that of known iterative distributed algorithms =-=[5]-=- for computing averages in a network. A. Related work In this section, we present a brief summary of related work. Algorithms for computing the number of distinct elements in a multiset or data stream... |

146 | Counting distinct elements in a data stream
- Bar-Yossef, Jayram, et al.
- 2002
(Show Context)
Citation Context ...mputing averages in a network. A. Related work In this section, we present a brief summary of related work. Algorithms for computing the number of distinct elements in a multiset or data stream [11], =-=[2]-=- can be adapted to compute separable functions using information spreading [7]. We are not aware, however, of a previous analysis of the amount of time required for these algorithms to achieve a certa... |

143 | Spatial gossip and resource location protocols
- Kempe, Kleinberg, et al.
- 2001
(Show Context)
Citation Context ...seminate n messages to n nodes for the more general setting of arbitrary graphs and non-uniform random choices of communication partners. For other related results, we refer the reader to [22], [16], =-=[17]-=-. We take note of the similar (independent) recent work of Ganesh, Massoulié, and Towsley [13], and Berger, Borgs, Chayes, and Saberi [3], on the spread of epidemics in a network. B. Organization The ... |

127 |
On Spreading Rumor
- Pittel
- 1987
(Show Context)
Citation Context ...ormation spreading, gossip algorithms for disseminating a message to all nodes in a complete graph in which communication partners are chosen uniformly at random have been studied for some time [12], =-=[21]-=-, [9]. Karp, Schindelhauer, Shenker, and Vöcking presented a push and pull gossip algorithm, in which communicating nodes both send and receive messages, that disseminates a message to all n nodes in ... |

127 | Entropy waves, the zig-zag graph product, and new constant-degree expanders
- Reingold, Vadhan, et al.
(Show Context)
Citation Context ...) and δ = 1/n, the computation time scales as O(log 2 n). B. Expander graph Expander graphs have been used for numerous applications, and explicit constructions are known for constantdegree expanders =-=[23]-=-. We consider here an undirected graph in which the maximum degree of any vertex, ∆, is a constant. Suppose that the edge expansion of the graph is |F(S,S min S⊂V, 0<|S|≤n/2 c )| = α, |S| where F(S,S ... |

126 |
Problems in decentralized decision making and computation
- Tsitsiklis
- 1984
(Show Context)
Citation Context ...greement or consensus among processors via a distributed computation. Distributed algorithms for reaching consensus under appropriate conditions have been known since the classical work of Tsitsiklis =-=[25]-=- and Tsitsiklis, Bertsekas, and Athans [26] (see also the book by Bertsekas and Tsitsiklis [4]). Averaging algorithms compute the ratio of the sum of the input numbers to n, the number of nodes in the... |

123 | The effect of network topology on the spread of epidemics
- Ganesh, Massoulie, et al.
- 2005
(Show Context)
Citation Context ...m random choices of communication partners. For other related results, we refer the reader to [22], [16], [17]. We take note of the similar (independent) recent work of Ganesh, Massoulié, and Towsley =-=[13]-=-, and Berger, Borgs, Chayes, and Saberi [3], on the spread of epidemics in a network. B. Organization The rest of the paper is organized as follows. Section II presents the distributed computation pro... |

118 | Size-estimation framework with applications to transitive closure and reachability
- Cohen
- 1997
(Show Context)
Citation Context ... randomized, and in particular uses exponential random variables. This usage of exponential random variables is analogous to that in an algorithm by Cohen1 for estimating the sizes of sets in a graph =-=[22]-=-. The basis for our algorithm is the following property of the exponential distribution. Property 1: Let W1, . . . , Wn be n independent random variables such that, for i = 1, . . . , n, the distribut... |

102 |
Maximizing throughput in wireless networks via gossiping
- Modiano, Shah, et al.
- 2006
(Show Context)
Citation Context ...e average of the sensed values at the nodes gives a good estimate of the ambient temperature. For more sophisticated applications of a distributed averaging algorithm, we refer the reader to [18] and =-=[19]-=-. Averaging is used for the distributed computation of the top k eigenvectors of a graph in [18], while in [19] averaging is used in a throughput-optimal distributed scheduling algorithm in a wireless... |

66 |
The shortest-path problem for graphs with random arc-lengths
- Frieze, Grimmett
- 1985
(Show Context)
Citation Context ...of information spreading, gossip algorithms for disseminating a message to all nodes in a complete graph in which communication partners are chosen uniformly at random have been studied for some time =-=[12]-=-, [21], [9]. Karp, Schindelhauer, Shenker, and Vöcking presented a push and pull gossip algorithm, in which communicating nodes both send and receive messages, that disseminates a message to all n nod... |

56 | Protocols and impossibility results for gossip-based communication mechanisms
- Kemepe, Kleinberg
(Show Context)
Citation Context ...to disseminate n messages to n nodes for the more general setting of arbitrary graphs and non-uniform random choices of communication partners. For other related results, we refer the reader to [22], =-=[16]-=-, [17]. We take note of the similar (independent) recent work of Ganesh, Massoulié, and Towsley [13], and Berger, Borgs, Chayes, and Saberi [3], on the spread of epidemics in a network. B. Organizatio... |

45 |
Rapid rumor ramification: Approximating the minimum broadcast time
- Ravi
- 1994
(Show Context)
Citation Context ...rithm to disseminate n messages to n nodes for the more general setting of arbitrary graphs and non-uniform random choices of communication partners. For other related results, we refer the reader to =-=[22]-=-, [16], [17]. We take note of the similar (independent) recent work of Ganesh, Massoulié, and Towsley [13], and Berger, Borgs, Chayes, and Saberi [3], on the spread of epidemics in a network. B. Organ... |

40 | On the spread of viruses on the internet
- Berger, Borgs, et al.
- 2005
(Show Context)
Citation Context ...For other related results, we refer the reader to [22], [16], [17]. We take note of the similar (independent) recent work of Ganesh, Massoulié, and Towsley [13], and Berger, Borgs, Chayes, and Saberi =-=[3]-=-, on the spread of epidemics in a network. B. Organization The rest of the paper is organized as follows. Section II presents the distributed computation problems we study and an overview of our resul... |

13 | Scale free aggregation in sensor networks - Enachescu, Goel, et al. - 2004 |

10 | The isoperimetric number of d-dimensional k-ary arrays
- Azizo˘glu, E˘gecio˘glu
- 1999
(Show Context)
Citation Context ...er of nodes in the graph is c d = (n 1/d ) d = n. For any two nodes a and b, there is an edge (a,b) in the graph if and only if, for some i ∈ {1,... ,d}, |ai − bi| = 1, and aj = bj for all j �= i. In =-=[1]-=-, it is shown that the isoperimetric number of this grid graph is |F(S,S min S⊂V, 0<|S|≤n/2 c � � )| 1 = Θ = Θ |S| c � 1 n1/d � . By the definition of the edge set, the maximum degree of a node in the... |

8 |
A decentralized algorithm for spectral analaysis
- Kempe, McSherry
(Show Context)
Citation Context ..., then the average of the sensed values at the nodes gives a good estimate of the ambient temperature. For more sophisticated applications of a distributed averaging algorithm, we refer the reader to =-=[18]-=- and [19]. Averaging is used for the distributed computation of the top k eigenvectors of a graph in [18], while in [19] averaging is used in a throughput-optimal distributed scheduling algorithm in a... |

1 | Scale free aggregation in sensor networks,” in International Workshop on Algorithmic Aspects of Wireless Sensor Networks, 2004. Damon Mosk-Aoyama received the S.B. degree in computer science and engineering and the M.Eng. degree in electrical engineering - Enachescu, Goel, et al. - 2002 |