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Bounds on the OBDDSize of Integer Multiplication via Universal Hashing
, 2005
"... Bryant [5] has shown that any OBDD for the function MULn−1,n, i.e. the middle bit of the nbit multiplication, requires at least 2 n/8 nodes. In this paper a stronger lower bound of essentially 2 n/2 /61 is proven by a new technique, using a universal family of hash functions. As a consequence, one ..."
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Bryant [5] has shown that any OBDD for the function MULn−1,n, i.e. the middle bit of the nbit multiplication, requires at least 2 n/8 nodes. In this paper a stronger lower bound of essentially 2 n/2 /61 is proven by a new technique, using a universal family of hash functions. As a consequence, one cannot hope anymore to verify e.g. 128bit multiplication circuits using OBDDtechniques because the representation of the middle bit of such a multiplier requires more than 3 · 10 17 OBDDnodes. Further, a first nontrivial upper bound of 7/3 · 2 4n/3 for the OBDDsize of MULn−1,n is provided.
Element Distinctness, Frequency Moments, and Sliding Windows
"... Abstract — We derive new timespace tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. In particular, we develop a randomized algorithm for the ele ..."
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Abstract — We derive new timespace tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. In particular, we develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T ∈ Õ(n3/2/S1/2), smaller than previous lower bounds for comparisonbased algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and spaceefficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows [18], where the task is to take an input of length 2n − 1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a timespace tradeoff lower bound of T ∈ Ω(n2/S) for randomized multiway branching programs, and hence standard RAM and wordRAM models, to compute the number of distinct elements, F0, over sliding windows. The same lower bound holds for computing the loworder bit of F0 and computing any frequency moment Fk for k 6 = 1. This shows that frequency moments Fk 6 = 1 and even the decision problem F0 mod 2 are strictly harder than element distinctness. We provide even stronger separations on average for inputs from [n]. We complement this lower bound with a T ∈ Õ(n2/S) comparisonbased deterministic RAM algorithm for exactly computing Fk over sliding windows, nearly matching both our general lower bound for the slidingwindow version and the comparisonbased lower bounds for a single instance of the problem. We also consider the computations of order statistics over sliding windows.
On the OBDD complexity of the most significant bit of integer multiplication
 In Proc. of 5th TAMC
"... Abstract. Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for boolean functions. Among the many areas of application a ..."
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Abstract. Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for boolean functions. Among the many areas of application are verification, model checking, computeraided design, relational algebra, and symbolic graph algorithms. In this paper it is shown that the OBDD complexity of the most significant bit of integer multiplication is exponential answering an open question posed by Wegener (2000).
New results on the complexity of the middle bit of multiplication
, 2004
"... It is well known that the hardest bit of integer multiplication is the middle bit, i.e. MULn−1,n. This paper contains several new results on its complexity. First, the size s of randomized readk branching programs, or, equivalently, its space (log s) is investigated. A randomized algorithm for MULn ..."
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It is well known that the hardest bit of integer multiplication is the middle bit, i.e. MULn−1,n. This paper contains several new results on its complexity. First, the size s of randomized readk branching programs, or, equivalently, its space (log s) is investigated. A randomized algorithm for MULn−1,n with k = O(log n) (implying time O(n log n)), space O(log n) and error probability n −c for arbitrarily chosen constants c is presented. Second, the size of general branching programs and formulas is investigated. Applying Nechiporuk’s technique, lower bounds of Ω ` n 3/2 / log n ´ and Ω ` n 3/2 ´ , respectively, are obtained. Moreover, by bounding the number of subfunctions of MULn−1,n, it is proven that Nechiporuk’s technique cannot provide larger lower bounds than O(n 7/4 / log n) and O(n 7/4), respectively.
A Quadratic TimeSpace Tradeoff for Unrestricted Deterministic Decision Branching Programs
, 2006
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"... An asymptotically optimal lower bound on the OBDD size of the middle bit of multiplication for the pairwise ascending variable order Abstract. We prove that each OBDD (ordered binary decision diagram) for the middle bit of nbit integer multiplication with one of the asymptotically best known variab ..."
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An asymptotically optimal lower bound on the OBDD size of the middle bit of multiplication for the pairwise ascending variable order Abstract. We prove that each OBDD (ordered binary decision diagram) for the middle bit of nbit integer multiplication with one of the asymptotically best known variable orders, namely the pairwise ascending order x0, y0,..., xn−1, yn−1, requires size �(2 (6/5)n). This is asymptotically optimal due to an upper bound of the same order by Amano and Maruoka (2007). 1.
ON PERFECT HASHING OF NUMBERS WITH SPARSE DIGIT REPRESENTATION VIA MULTIPLICATION BY A CONSTANT
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Sliding Windows with Limited Storage
, 2013
"... The results of this paper are superceded by the paper at: http://arxiv.org/abs/1309.3690. We consider timespace tradeoffs for exactly computing frequency moments and order statistics over sliding windows [16]. Given an input of length 2n − 1, the task is to output the function of each window of len ..."
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The results of this paper are superceded by the paper at: http://arxiv.org/abs/1309.3690. We consider timespace tradeoffs for exactly computing frequency moments and order statistics over sliding windows [16]. Given an input of length 2n − 1, the task is to output the function of each window of length n, giving n outputs in total. Computations over sliding windows are related to direct sum problems except that inputs to instances almost completely overlap. • We show an average case and randomized timespace tradeoff lower bound of T · S ∈ Ω(n2) for multiway branching programs, and hence standard RAM and wordRAM models, to compute the number of distinct elements, F0, in sliding windows over alphabet [n]. The same lower bound holds for computing the loworder bit of F0 and computing any frequency moment Fk for k 6 = 1. We complement this lower bound with a T · S ∈ Õ(n2) deterministic RAM algorithm for exactly computing Fk in sliding windows. • We show timespace separations between the complexity of slidingwindow element distinctness and that of slidingwindow F0 mod 2 computation. In particular for alphabet [n] there is a very simple errorless slidingwindow algorithm for element distinctness that runs in O(n) time on average and uses O(log n) space. • We show that any algorithm for a single element distinctness instance can be extended to an algorithm for the slidingwindow version of element distinctness with at most a polylogarithmic increase in the timespace product. ar X iv