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CellProbe Lower Bounds for Succinct Partial Sums
, 2009
"... The partial sums problem in succinct data structures asks to preprocess an array A[1.. n] of bits into a data structure using as close to n bits as possible, and answer queries of the form Rank(k) = ∑ k A[i]. The problem i=1 has been intensely studied, and features as a subroutine in a number of s ..."
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The partial sums problem in succinct data structures asks to preprocess an array A[1.. n] of bits into a data structure using as close to n bits as possible, and answer queries of the form Rank(k) = ∑ k A[i]. The problem i=1 has been intensely studied, and features as a subroutine in a number of succinct data structures. We show that, if we answer Rank(k) queries by probing t cells of w bits, then the space of the data structure must be at least n+n/wO(t) bits. This redundancy/probe tradeoff is essentially optimal: Patrascu [FOCS’08] showed how to achieve n + n / (w/t) Ω(t) bits. We also extend our lower bound to the closely related Select queries, and to the case of sparse arrays.
The communication complexity of addition
, 2011
"... Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai ..."
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Suppose each of k ≤ no(1) players holds an nbit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a publiccoin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give a matching (publiccoin) Ω(k lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Ω(k lg lg k) for protocols that are oneway, like ours. We give a protocol to determine if ∑ xi> s with error 1 % and communication O(k lg k) lg n. For k ≥ 3 this improves on Nisan’s O(k lg 2 n) bound. A similar improvement holds for computing degree(k − 1) polynomialthreshold functions in the numberonforehead model. We give a (publiccoin, 2player, tight) Ω(lg n) lower bound to determine if x1> x2. This improves on the Ω ( √ lg n) bound by Smirnov (1988).