Fast averaging

Abstract

Abstract-We are interested in the following question: given n numbers x1, . . . , xn, what sorts of approximation of average xave = 1 n (x1 + · · · + xn) can be achieved by knowing only r of these n numbers. Indeed the answer depends on the variation in these n numbers. As the main result, we show that if the vector of these n numbers satisfies certain regularity properties captured in the form of finiteness of their empirical moments (third or higher), then it is possible to compute approximation of xave that is within 1 ± ε multiplicative factor with probability at least 1 − δ by choosing, on an average, r = r(ε, δ, σ) of the n numbers at random with r is dependent only on ε, δ and the amount of variation σ in the vector and is independent of n.The task of computing average has a variety of applications such as distributed estimation and optimization, a model for reaching consensus and computing symmetric functions. We discuss implications of the result in the context of two applications: load-balancing in a computational facility running MapReduce, and fast distributed averaging.

Optimal Embedding of Functions for In-Network Computation: Complexity Analysis and Algorithms

“…(a) A schema to compute r(t) (b) Communication Network (c) Implementation 1 (d) Implementation 2 the nodes are in a √ n × √ n grid and communicate over wireline or wireless links, [14] obtains the time and number of transmissions required to compute a function. Randomized gossip algorithms [15], [16], where a random sequence of node-pairs exchange data and perform a specific computation is used in [17]- [19] for function computation. The interest is in time for all nodes to converge to the specified function.…”

“…Lines 16-20: Here the algorithm finally computes the total cost of the embedding the graph G by adding the cost of the edges between the vertices of last layer r, if any, when the vertices of last layer are placed at X i . And computes the optimal cost of embedding E, C(E), by choosing the placement of last layer which minimizes the overall cost (line [19][20]. The vector Z r stores the mapping of vertices at layer r under the embedding E.…”

“…In lines 12-13 the algorithm finds the optimal embedding cost of all the nodes and edges till V i conditioned on the mapping of vertices in V i ∩ V i+1 . Lines 16-18 compute the cost till the last layer and then the algorithm traces back from V r−1 to V 1 to get the final optimal cost and the corresponding embedding (lines [19][20][21][22][23]. Note that the time required to compute the cost till vertex V i depends on the size of this vertex (lines 8, 9 define that value).…”