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Pattern matching in polyphonic music as a weighted geometric translation problem
 In Proc. 5th International Conference on Music Information Retrieval
, 2004
"... We consider the music pattern matching problem—to find occurrences of a small fragment of music called the “pattern” in a larger body of music called the “score”—as a problem of translating a set of horizontal line segments in the plane to find the best match in a larger set of horizontal line segme ..."
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We consider the music pattern matching problem—to find occurrences of a small fragment of music called the “pattern” in a larger body of music called the “score”—as a problem of translating a set of horizontal line segments in the plane to find the best match in a larger set of horizontal line segments. Our contribution is that we use fairly general weight functions to measure the quality of a match, thus enabling approximate pattern matching. We give an algorithm with running time O(nm log m), where n is the size of the score and m is the size of the pattern. We show that the problem, in this geometric formulation, is unlikely to have a significantly faster algorithm because it is at least as hard as a basic problem called 3SUM that is conjectured to have no subquadratic algorithm. We present some examples to show the potential of this method for finding minor variations of a theme, and for finding polyphonic musical patterns in a polyphonic score. 1.
Optimal freespace management and routingconscious dynamic placement for reconfigurable devices
 IEEE Transactions on Computers
, 2007
"... We describe algorithmic results on two crucial aspects of allocating resources on arraybased hardware devices with partial reconfigurability. By using methods from the field of computational geometry, we derive a method that allows correct maintainance of free and occupied space of a set of n recta ..."
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We describe algorithmic results on two crucial aspects of allocating resources on arraybased hardware devices with partial reconfigurability. By using methods from the field of computational geometry, we derive a method that allows correct maintainance of free and occupied space of a set of n rectangular modules in time O(nlog n); previous approaches needed a time of O(n 2) for correct results and O(n) for heuristic results. We also show a matching lower bound of Ω(n log n), so our approach is optimal. We also show that finding an optimal feasible communicationconscious placement (which minimizes the total weighted Manhattan distance between the new module and existing demand points) can be computed with Θ(nlog n). Both resulting algorithms are practically easy to implement and show convincing experimental behavior. ACM Classification: C.1.3: Reconfigurable Hardware; F.2.2.c: Geometrical problems and computations
Threesomes, degenerates, and love triangles
 In Proc. 55th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS
, 2014
"... The 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial Opn2qtime algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies Ωpn2q lower bounds on numerous ..."
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The 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial Opn2qtime algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies Ωpn2q lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is Opn3{2?log nq and give two subquadratic 3SUM algorithms, a deterministic one running in Opn2{plog n { log log nq2{3q time and a randomized one running in Opn2plog log nq2 { log nq time with high probability. Our results lead directly to improved bounds for kvariate linear degeneracy testing for all odd k ě 3. The problem is to decide, given a linear function fpx1,..., xkq “ α0 `ř1ďiďk αixi and a set A Ă R, whether 0 P fpAkq. We show the decision tree complexity of this problem is Opnk{2?log nq. Finally, we give a subcubic algorithm for a generalization of the pmin,`qproduct over realvalued matrices and apply it to the problem of finding zeroweight triangles in weighted graphs. We give a depthOpn5{2?log nq decision tree for this problem, as well as an algorithm running in time Opn3plog log nq2 { log nq. 1
3sum hardness in (dynamic) data structures
 CoRR
"... We prove lower bounds for several (dynamic) data structure problems conditioned on the well known conjecture that 3SUM cannot be solved in O(n2−Ω(1)) time. This continues a line of work that was initiated by Pǎtraşcu [STOC 2010] and strengthened recently by Abboud and VassilevskaWilliams [FOCS 20 ..."
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We prove lower bounds for several (dynamic) data structure problems conditioned on the well known conjecture that 3SUM cannot be solved in O(n2−Ω(1)) time. This continues a line of work that was initiated by Pǎtraşcu [STOC 2010] and strengthened recently by Abboud and VassilevskaWilliams [FOCS 2014]. The problems we consider are from several subfields of algorithms, including text indexing, dynamic and fault tolerant graph problems, and distance oracles. In particular we prove polynomial lower bounds for the data structure version of the following problems: Dictionary Matching with Gaps, Document Retrieval problems with more than one pattern or an excluded pattern, Maximum Cardinality Matching in bipartite graphs (improving known lower bounds), dfailure Connectivity Oracles, Preprocessing for Induced Subgraphs, and Distance Oracles for Colors. Our lower bounds are based on several reductions from 3SUM to a special set intersection problem introduced by Pǎtraşcu, which we call Pǎtraşcu’s Problem. In particular, we provide a new reduction from 3SUM to Pǎtraşcu’s Problem which allows us to obtain stronger conditional lower bounds for (some) problems that have already been shown to be 3SUM hard, and for several of the problems examined here. Our other lower bounds are based on reductions from the Convolution3SUM problem, which was introduced by Pǎtraşcu. We also prove that up to a logarithmic factor, the Convolution3SUM problem is equivalent to 3SUM when the inputs are integers. A previous reduction of Pǎtraşcu shows that a subquadratic algorithm for Convolution3SUM implies a similarly subquadratic 3SUM algorithm, but not that the two problems are asymptotically equivalent or nearly equivalent. 1
A geometric approach to pattern matching in polyphonic music
, 2004
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii The music pattern matching problem involves finding ma ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii The music pattern matching problem involves finding matches of a small fragment of music called the “pattern ” into a larger body of music called the “score”. We represent music as a series of horizontal line segments in the plane, and reformulate the problem as finding the best translation of a small set of horizontal line segments into a larger set of horizontal line segments. We present an efficient algorithm that can handle general weight models that measure the musical quality of a match of the pattern into the score, allowing for approximate pattern matching. We give an algorithm with running time O(nm(d + log m)), where n is the size of the score, m is the size of the pattern, and d is the size of the discrete set of musical pitches used. Our algorithm compares favourably to previous approaches to the music pattern
Algorithms for Deterministic Call Admission Control of Prestored VBR Video Streams
"... Abstract — We examine the problem of accepting a new request for a prestored VBR video stream that has been smoothed using any of the smoothing algorithms found in the literature. The output of these algorithms is a piecewise constantrate schedule for a Variable BitRate (VBR) stream. The schedule ..."
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Abstract — We examine the problem of accepting a new request for a prestored VBR video stream that has been smoothed using any of the smoothing algorithms found in the literature. The output of these algorithms is a piecewise constantrate schedule for a Variable BitRate (VBR) stream. The schedule guarantees that the decoder buffer does not overflow or underflow. The problem addressed in this paper is the determination of the minimal time displacement of each new requested VBR stream so that it can be accommodated by the network and/or the video server without overbooking the committed traffic. We prove that this calladmission control problem for multiple requested VBR streams is NPcomplete and inapproximable within a constant factor, by reducing it from the VERTEX COLOR problem. We also present a deterministic morphologysensitive algorithm that calculates the minimal time displacement of a VBR stream request. The complexity of the proposed algorithm along with the experimental results we provide indicate that the proposed algorithm is suitable for realtime determination of the time displacement parameter during the call admission phase.
Call admission control algorithm for prestored VBR video streams
, 2008
"... We examine the problem of accepting a new request for a prestored VBR video stream that has been smoothed using any of the smoothing algorithms found in the literature. The output of these algorithms is a piecewise constantrate schedule for a Variable BitRate (VBR) stream. The schedule guarante ..."
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We examine the problem of accepting a new request for a prestored VBR video stream that has been smoothed using any of the smoothing algorithms found in the literature. The output of these algorithms is a piecewise constantrate schedule for a Variable BitRate (VBR) stream. The schedule guarantees that the decoder buffer does not overflow or underflow. The problem addressed in this paper is the determination of the minimal time displacement of each new requested VBR stream so that it can be accomodated by the network and/or the video server without overbooking the committed traffic. We prove that this calladmission control problem for multiple requested VBR streams is NPcomplete and inapproximable within a constant factor, by reducing it from the VERTEX COLOR problem. We also present a deterministic morphologysensitive algorithm that calculates the minimal time displacement of a VBR stream request. The complexity of the proposed algorithm make it suitable for realtime determination of the time displacement parameter during the call admission phase.
AT&T Labs
"... We consider a number of dynamic problems with no known polylogarithmic upper bounds, and show that they require n Ω(1) time per operation, unless 3SUM has strongly subquadratic algorithms. Our result is modular: 1. We describe a carefullychosen dynamic version of set disjointness (the multiphase p ..."
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We consider a number of dynamic problems with no known polylogarithmic upper bounds, and show that they require n Ω(1) time per operation, unless 3SUM has strongly subquadratic algorithms. Our result is modular: 1. We describe a carefullychosen dynamic version of set disjointness (the multiphase problem), and conjecture that it requires n Ω(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 3SUMhardness 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 numberonforehead communication game.
Scandinavian Thins on Top of Cake: New and improved algorithms for stacking and packing
"... We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest conve ..."
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We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a lineartime algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NPhard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.
Approximating the Maximum Overlap of Polygons under Translation
, 2014
"... Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present an (1 − ε)approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P. Our algorithm runs in O(cn) time, where c i ..."
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Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present an (1 − ε)approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P. Our algorithm runs in O(cn) time, where c is a constant that depends only on k and ε. This suggest that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.