Lecture 3 — September 1, 2016 2 Review of Last Lecture

1 Overview We continue discussing the distinct elements problem. Recall that in this problem, you receive as input a stream S = (x 1 , x 2 ,. . .), with each x i ∈ [n], and you are supposed to compute k = #{v ∈ [n] : v appears in S}. Topics for today: • Review of the algorithm from last lecture. • The LogLog algorithm, which is more space efficient. • Generalizing the algorithm from last lecture to work in the turnstile model. • Space complexity lower bounds via a connection with one-way communication complexity. Recall the algorithm from last lecture: 1. (a) For r = 1 to R ≤ O(log log n δ), in parallel: i. Pick a random hash function h t,r : [n] → [10t] from pairwise independent family H. ii. Record Y t,r = 1 if some v ∈ S satisfies h t,r (v) = 0 0 otherwise. (b) Compute the empirical averagê Y t = 1 R R r=1 Y t,r. (c) Set z t = 1 ifˆY t > 0.14 0 otherwise. (d) Output 2 t zt. Analysis We showed last time that if k < t, then P[z t = 0] ≥ 1 − δ log n , while if k > 2t, then P[z t = 1] ≥ 1 − δ log n. Therefore, by the union bound, the output of the algorithm is within a factor of 2 of k with probability at least 1 − δ. The space used by this algorithm is O((log 2 n) · (log log n + log(1/δ))).