The Data Stream Space Complexity of Cascaded Norms

Abstract— We consider the problem of estimating cascaded aggregates over a matrix presented as a sequence of updates in a data stream. A cascaded aggregate P ◦Q is defined by evaluating aggregate Q repeatedly over each row of the matrix, and then evaluating aggregate P over the resulting vector of values. This problem was introduced by Cormode and Muthukrishnan, PODS, 2005 [CM]. We analyze the space complexity of estimating cascaded norms on an n ×d matrix to within a small relative error. Let Lp denote the p-th norm, where p is a non-negative integer. We abbreviate the cascaded norm Lk ◦Lp by Lk,p. (1) For any constant k ≥ p ≥ 2, we obtain a 1-pass O(n1−2/kd1−2/p)-space algorithm for estimating Lk,p. This is optimal up to polylogarithmic factors and resolves an open question of [CM] regarding the space complexity of L4,2. We also obtain 1-pass space-optimal algorithms for estimating L∞,k and Lk,∞. (2) We prove a space lower bound of Ω(n1−1/k) on estimating Lk,0...

• Type: Conference
• Year: 2009
• Where: FOCS
• Authors: T. S. Jayram, David P. Woodruff
• Keywords: Data Streams, Estimating, FOCS 2009, Space Complexity, Theoretical Computer Science