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59] Using the CM Sketch, the space required to answer point queries with error ε||A||1 with probability at least 1 − δ is given by O(ε− min{1,1/z} ln 1/δ) for a Zipf distribution with parameter z. The proof of above theorem uses the idea of separating the inﬂuence of certain heavy items and arguing that, with signiﬁcant probability, only the inﬂuence of the remainder of the item counts, and bounding the latter using Zipﬁan properties. This result improves the “golden standard” of O(1/ε) space used in data stream algorithms for skewed data, z > 1.

Since Carole is not allowed to communicate with Paul, any protocol to solve this problem, even probabilistically, requires Ω(N ) bits of communi1 1 ] of length N = 2ε bits and set cation . Take a bitstring B[1 · · · 2ε A[i] = 2 if B[i] = 1; otherwise, set A[i] = 0 and add 2 to A. Whatever the value of B, ||A||1 = 1/ε. If we can answer point queries with accuracy ε||A||1 = 1, we can test any A[i] and determine the value of B[i] by reporting 1 if the estimated value of A[i] is above ε||A||1 and 0 otherwise.

For integer j, 0 ≤ j < log(N ) and integer k, 0 ≤ k < 2j , we deﬁne a proper wavelet φ(x)[j, k] by − 2j /N for x ∈ [kN/2j , kN/2j + N/2j+1 ), + 2j /N for x ∈ [kN/2j + N/2j+1 , (k + 1)N/2j ) and 0 otherwise. Additionally, deﬁne a√wavelet φ, also known as a scaling function, that takes the value +1/ N over the entire domain [0, N ). Number the φ(x)[j, k]’s as ψ0 , ψ1 , . . , ψN −2 in some arbitrary order and φ as ψN −1 . Wavelets can be used to represent signals. Assume N is a power of 2. , A, ψi ψi .