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Patrascu M.,AT and T Labs. | Viola E.,Northeastern University
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2010

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) = Σi k=1 A[i]. The problem 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 trade-off 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. Copyright © by SIAM.

Chan T.M.,University of Waterloo | Patrascu M.,AT and T Labs.
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2010

We give an O(n√lg n)-time algorithm for counting the number of inversions in a permutation on n elements. This improves a long-standing previous bound of O(n lg n/ lg lg n) that followed from Dietz's data structure [WADS'89], and answers a question of Andersson and Petersson [SODA'95]. As Dietz's result is known to be optimal for the related dynamic rank problem, our result demonstrates a significant improvement in the offline setting. Our new technique is quite simple: we perform a "vertical partitioning" of a trie (akin to van Emde Boas trees), and use ideas from external memory. However, the technique finds numerous applications: for example, we obtain • in d dimensions, an algorithm to answer n offline orthogonal range counting queries in time O(n lg d-2+1/dn); • an improved construction time for online data structures for orthogonal range counting; • an improved update time for the partial sums problem; • faster Word RAM algorithms for finding the maximum depth in an arrangement of axis-aligned rectangles, and for the slope selection problem. As a bonus, we also give a simple (1 + ε)-approximation algorithm for counting inversions that runs in linear time, improving the previous O(n lg lg n) bound by Andersson and Petersson. Copyright © by SIAM.

Patrascu M.,AT and T Labs. | Williams R.,IBM
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2010

We describe reductions from the problem of determining the satisfiability of Boolean CNF formulas (CNF-SAT) to several natural algorithmic problems. We show that attaining any of the following bounds would improve the state of the art in algorithms for SAT: • an O(nk-ε) algorithm for k-DOMINATING SET, for any k ≥ 3, • a (computationally efficient) protocol for 3-party set disjointness with o(m) bits of communication, • an no(d) algorithm for d-SUM, • an O(n2-ε) algorithm for 2-SAT formulas with m = n1+o(1) clauses, where two clauses may have unrestricted length, and • an O((n + m) k-ε) algorithm for HornSat with k unrestricted length clauses. One may interpret our reductions as new attacks on the complexity of SAT, or sharp lower bounds conditional on exponential hardness of SAT. Copyright © by SIAM.

Zhou X.,AT and T Labs | Zhou X.,Google | Nelson L.,AT and T Labs
Journal of Lightwave Technology | Year: 2014

This paper presents a systematic review of several digital signal processing (DSP)-enabled technologies recently proposed and demonstrated for high spectral efficiency (SE) 400 Gb/s-class and beyond optical networks. These include 1) a newly proposed SE-adaptable optical modulation technology - time-domain hybrid quadrature amplitude modulation (QAM), 2) two advanced transmitter side digital spectral shaping technologies - Nyquist signaling (for spectrally-efficient multiplexing) and digital pre-equalization (for improving tolerance toward channel narrowing effects), and 3) a newly proposed training-assisted two-stage carrier phase recovery algorithm that is designed to address the detrimental cyclic phase slipping problem with minimal training overhead. Additionally, this paper presents a novel DSP-based method for mitigation of equalizer-enhanced phase noise impairments. It is shown that performance degradation caused by the interaction between the long-memory chromatic dispersion compensating filter/equalizer and local oscillator laser phase noise can be effectively mitigated by replacing the commonly used fast single-tap phase-rotation-based equalizer (for typical carrier phase recovery) with a fast multi-tap linear equalizer. Finally, brief reviews of two high-SE 400 Gb/s-class WDM transmission experiments employing these advanced DSP algorithms are presented. © 1983-2012 IEEE.

Iacono J.,New York University | Patrascu M.,AT and T Labs.
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2012

We consider the dictionary problem in external memory and improve the update time of the well-known buffer tree by roughly a logarithmic factor. For any λ ≥ max{lg lg n, log M/B(n/B)}, we can support updates in time O(λ/B) and queries in sublogarithmic time, O(log λ n). We also present a lower bound in the cell-probe model showing that our data structure is optimal. In the RAM, hash tables have been use to solve the dictionary problem faster than binary search for more than half a century. By contrast, our data structure is the first to beat the comparison barrier in external memory. Ours is also the first data structure to depart convincingly from the indivisibility paradigm. Copyright © SIAM.

Patrascu M.,AT and T Labs | Roditty L.,Bar - Ilan University
Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS | Year: 2010

We give the first improvement to the space/approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC'01]. For unweighted graphs, our distance oracle has size O(n5/3) = O(n 1.66⋯) and, when queried about vertices at distance d, returns a path of length 2d + 1. For weighted graphs with m = n2/α edges, our distance oracle has size O(n2/3√α) and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space Ω(n2/√α). For unweighted graphs, this implies a Ω(n1.5) space lower bound to achieve approximation 2d + 1. © 2010 IEEE.

Patrascu M.,AT and T Labs. | Thorup M.,AT and T Labs.
Proceedings of the Annual ACM Symposium on Theory of Computing | Year: 2011

Randomized algorithms are often enjoyed for their simplicity, but the hash functions used to yield the desired theoretical guarantees are often neither simple nor practical. Here we show that the simplest possible tabulation hashing provides unexpectedly strong guarantees. The scheme itself dates back to Carter and Wegman (STOC'77). Keys are viewed as consisting of c characters. We initialize c tables T 1, ..., T c mapping characters to random hash codes. A key x = (x 1, ..., x c) is hashed to T 1[x 1] ⊕ ... ⊕ T c[x c], where ⊕ denotes xor. While this scheme is not even 4-independent, we show that it provides many of the guarantees that are normally obtained via higher independence, e.g., Chernoff-type concentration, min-wise hashing for estimating set intersection, and cuckoo hashing. © 2011 ACM.

Patrascu M.,AT and T Labs
SIAM Journal on Computing | Year: 2011

We show that a large fraction of the data-structure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for (i) high-dimensional problems, where the goal is to show large space lower bounds; (ii) constant-dimensional geometric problems, where the goal is to bound the query time for space O(n polylogn); and (iii) dynamic problems, where we are looking for a trade-off between query and update time. (In the last case, our bounds are slightly weaker than the originals, losing a lglgn factor.) Our reductions also imply the following new results: (i) an Ω(lg n/ lg lg n) bound for four-dimensional range reporting, given space O(n polylogn) (this is quite timely, since a recent result [Y. Nekrich, in Proceedings of the 23rd ACM Symposium on Computational Geometry (SoCG), 2007, pp. 344-353] solved three-dimensional reporting in O(lg2 lg n) time, raising the prospect that higher dimensions could also be easy); (ii) a tight space lower bound for the partial match problem, for constant query time; and (iii) the first lower bound for reachability oracles. In the process, we prove optimal randomized lower bounds for lopsided set disjointness. © 2011 Society for Industrial and Applied Mathematics.

Patrascu M.,AT and T Labs.
Proceedings of the Annual ACM Symposium on Theory of Computing | Year: 2010

We consider a number of dynamic problems with no known poly-logarithmic upper bounds, and show that they require nΩ(1) time per operation, unless 3SUM has strongly subquadratic algorithms. Our result is modular: (1) We describe a carefully-chosen dynamic version of set disjointness (the "multiphase problem"), and conjecture that it requires nΩ(1) time per operation. All our lower bounds follow by easy reduction. (2) We reduce 3SUM to the multiphase problem. Ours is the first nonalgebraic reduction from 3SUM, and allows 3SUM-hardness results for combinatorial problems. For instance, it implies hardness of reporting all triangles in a graph. (3) It is plausible that an unconditional lower bound for the multiphase problem can be established via a number-on-forehead communication game. © 2010 ACM.

Rosenberg E.,AT And T Labs
Community Ecology | Year: 2016

We extend Kenkel's model for determining the minimal allowable box size s∗ to be used in computing the box counting dimension of a self-similar geometric fractal. This minimal size s∗ is defined in terms of a specified parameter ϵ which is the deviation of a computed slope from the box counting dimension. We derive an exact implicit equation for s∗ for any e. We solve the equation using binary search, compare our results to Kenkel's, and illustrate how s∗ varies with e. A listing of the Python code for the binary search is provided. We also derive a closed form estimate for s∗ having the same functional form as Kenkel's empirically obtained expression.

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