In-Network Estimation of Frequency Moments

TL;DR

This paper introduces a distributed randomized algorithm for estimating scaled frequency moments of data over large alphabets in various network types, extending beyond linear functions to non-linear functions.

Contribution

It presents a novel distributed algorithm capable of estimating non-linear frequency moments in large alphabet data across different network models.

Findings

Algorithm achieves $(1-\delta)$-probability $\e$-approximation.

Time complexity is $O(M^{1-rac{1}{k-1}} T)$ for each network type.

Transmission cost is $O(M^{1-rac{1}{k-1}} rac{ ext{log} N ext{log} (rac{1}{\delta})}{\epsilon^2})$ bits per step.

Abstract

We consider the problem of estimating functions of distributed data using a distributed algorithm over a network. The extant literature on computing functions in distributed networks such as wired and wireless sensor networks and peer-to-peer networks deals with computing linear functions of the distributed data when the alphabet size of the data values is small, O(1). We describe a distributed randomized algorithm to estimate a class of non-linear functions of the distributed data which is over a large alphabet. We consider three types of networks: point-to-point networks with gossip based communication, random planar networks in the connectivity regime and random planar networks in the percolating regime both of which use the slotted Aloha communication protocol. For each network type, we estimate the scaled $k$-th frequency moments, for $k \geq 2$. Specifically, for every $k \geq 2,$ we give a distributed randomized algorithm that computes, with probability $(1-\delta),$ an $\epsilon$-approximation of the scaled $k$-th frequency moment, $F_k/N^k$, using time $O(M^{1-\frac{1}{k-1}} T)$ and $O(M^{1-\frac{1}{k-1}} \log N \log (\delta^{-1})/\epsilon^2)$ bits of transmission per communication step. Here, $N$ is the number of nodes in the network, $T$ is the information spreading time and $M=o(N)$ is the alphabet size.