Approximate aggregation for tracking quantiles and range countings in wireless sensor networks

We consider the problem of tracking quantiles and range countings in wireless sensor networks. The quantiles and range countings are two important aggregations to characterize a data distribution. Let S(t)=(d1,…,dn)S(t)=(d1,…,dn) denote the multi-set of sensory data that have arrived until time t  , which is a sequence of data orderly collected by nodes s1,s2,…,sks1,s2,…,sk. One of our goals is to continuously track ϵ-approximate ϕ  -quantiles (0≤ϕ≤1)(0≤ϕ≤1) of S(t)S(t) for all ϕ  's with efficient total communication cost and balanced individual communication cost. The other goal is to track (ϵ,δ)(ϵ,δ)-approximate range countings satisfying the requirement of arbitrary precision specified by different users. In this paper, a deterministic tracking algorithm based on a dynamic binary tree is proposed to track ϵ-approximate ϕ  -quantiles, whose total communication cost is O(k/ϵ⋅log⁡n⋅log2⁡(1/ϵ))O(k/ϵ⋅log⁡n⋅log2⁡(1/ϵ)), where k is the number of the nodes in a network, n is the total number of the data, and ϵ   is the user-specified approximation error. For range countings, a Bernoulli sampling based algorithm is proposed to track (ϵ,δ)(ϵ,δ)-approximate range countings, whose total communication cost is O(2ϵ2ln⁡21−1−δ+nc), where δ   is the user-specified error probability, ncnc is the number of clusters.