Random classical codes have good error correcting properties, and yet they are notoriously hard to decode in practice. Despite many decades of extensive study, the fastest known algorithms still run in exponential time. The Learning Parity with Noise (LPN) problem, which can be seen as the task of decoding a random linear code in the presence of noise, has thus emerged as a prominent hardness assumption with numerous applications in both cryptography and learning theory. Is there a natural quantum analog of the LPN problem? In this work, we introduce the Learning Stabilizers with Noise (LSN) problem, the task of decoding a random stabilizer code in the presence of local depolarizing noise. We give both polynomial-time and exponential-time quantum algorithms for solving LSN in various depolarizing noise regimes, ranging from extremely low noise, to low constant noise rates, and even higher noise rates up to a threshold. Next, we provide concrete evidence that LSN is hard. First, we show that LSN includes LPN as a special case, which suggests that it is at least as hard as its classical counterpart. Second, we prove worst-case to average-case reductions for variants of LSN. We then ask: what is the computational complexity of solving LSN? Because the task features quantum inputs, its complexity cannot be characterized by traditional complexity classes. Instead, we show that the LSN problem lies in a recently introduced (distributional and oracle) unitary synthesis class. Finally, we identify several applications of our LSN assumption, ranging from the construction of quantum bit commitment schemes to the computational limitations of learning from quantum data.

We introduce various measures of forward classical communication for bipartite quantum channels. Since a point-to-point channel is a special case of a bipartite channel, the measures reduce to measures of classical communication for point-to-point channels. As it turns out, these reduced measures have been reported in prior work of Wang et al. on bounding the classical capacity of a quantum channel. As applications, we show that the measures are upper bounds on the forward classical capacity of a bipartite channel. The reduced measures are upper bounds on the classical capacity of a point-to-point quantum channel assisted by a classical feedback channel. Some of the various measures can be computed by semi-definite programming.

We introduce various measures of forward classical communication for bipartite quantum channels. Since a point-to-point channel is a special case of a bipartite channel, the measures reduce to measures of classical communication for point-to-point channels. As it turns out, these reduced measures have been reported in prior work of Wang et al. on bounding the classical capacity of a quantum channel. As an application, we show that the reduced measures are upper bounds on the classical capacity of a point-to-point quantum channel assisted by a classical feedback channel. Some of the various measures can be computed by semi-definite programming.

We prove that the classical capacity of an arbitrary quantum channel assisted by a free classical feedback channel is bounded from above by the maximum average output entropy of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve the classical capacity of a quantum erasure channel, and by taking into account energy constraints, we conclude the same for a pure-loss bosonic channel. The method for establishing the aforementioned entropy bound involves identifying an information measure having two key properties: 1) it does not increase under a one-way local operations and classical communication channel from the receiver to the sender and 2) a quantum channel from sender to receiver cannot increase the information measure by more than the maximum output entropy of the channel. This information measure can be understood as the sum of two terms, with one corresponding to classical correlation and the other to entanglement.

We provide polylog sparse quantum codes for correcting the Erasure channel arbitrarily close to the capacity. Specifically, we provide the [[n, k, d]] quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least 0.33 and with a generating set (S1, S2, ... Sn-k) such that |Si| ≤ log2+ζ(n) for all i and for any ζ > 0 with high probability. In this paper, we show that the result of Delfosse et al. is tight: one can construct capacity approaching codes with weight almost O(1).

In this paper, we investigate the additivity phenomenon in the quantum dynamic capacity region of a quantum channel for trading the resources of classical communication, quantum communication, and entanglement. Understanding such an additivity property is important if we want to optimally use a quantum channel for general communication purposes. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity region, and it is largely unknown if this could lead to a strictly superadditive double or triple resource capacity region, respectively. For example, if a channel has additive classical and quantum capacities, can the classical-quantum capacity region be strictly superadditive? In this paper, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple resource capacity region for a given channel.

n this work, we consider the following family of two prover one-round games. In the CHSH_q game, two parties are given x,y in F_q uniformly at random, and each must produce an output a,b in F_q without communicating with the other. The players' objective is to maximize the probability that their outputs satisfy a+b=xy in F_q. This game was introduced by Buhrman and Massar (PRA 2005) as a large alphabet generalization of the celebrated CHSH game---which is one of the most well-studied two-prover games in quantum information theory, and which has a large number of applications to quantum cryptography and quantum complexity.

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors.

Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one noisy channel to simulate another. For channels of nonzero capacity, this simulation is always possible, but for it to be efficient, auxiliary resources of the proper kind and amount are generally required. In the classical case, shared randomness between sender and receiver is a sufficient auxiliary resource, regardless of the nature of the source, but in the quantum case, the requisite auxiliary resources for efficient simulation depend on both the channel being simulated, and the source from which the channel inputs are coming. For tensor power sources (the quantum generalization of classical memoryless sources), entanglement in the form of standard ebits (maximally entangled pairs of qubits) is sufficient, but for general sources, which may be arbitrarily correlated or entangled across channel inputs, additional resources, such as entanglement-embezzling states or backward communication, are generally needed. Combining existing and new results, we establish the amounts of communication and auxiliary resources needed in both the classical and quantum cases, the tradeoffs among them, and the loss of simulation efficiency when auxiliary resources are absent or insufficient. In particular, we find a new single-letter expression for the excess forward communication cost of coherent feedback simulations of quantum channels (i.e., simulations in which the sender retains what would escape into the environment in an ordinary simulation), on nontensor-power sources in the presence of unlimited ebits but no other auxiliary resource. Our results on tensor power sources establish a strong converse to the entanglement-assisted capacity theorem.

Quantum money is a cryptographic protocol in which a mint can produce a quantum state, no one else can copy the state, and anyone (with a quantum computer) can verify that the state came from the mint. We present a concrete quantum money scheme based on superpositions of diagrams that encode oriented links with the same Alexander polynomial. We expect our scheme to be secure against computationally bounded adversaries.

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. In the binary case, Varshamov has shown that almost all good linear codes for the asymmetric channel are also good for the symmetric channel. Our results indicate that Varshamov's argument does not extend to the nonbinary case, i.e., one can find better linear codes for asymmetric channels than for symmetric ones.

A d-dimensional Keller graph has vertices which are numbered with each of the 4d possible d-digit numbers (d-tuples) which have each digit equal to 0, 1, 2, or 3. Two vertices are adjacent if their labels differ in at least two positions, and in at least one position the difference in the labels is two modulo four. Keller graphs are in the benchmark set of clique problems from the DIMACS clique challenge, and they appear to be especially difficult for clique algorithms. The dimension seven case was the last remaining Keller graph for which the maximum clique order was not known. It has been claimed in order to resolve this last case it might take a “high speed computer the size of a major galaxy”. This paper describes the computation we used to determine that the maximum clique order for dimension seven is 124.

We construct families of high performance quantum amplitude damping codes. All of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension. One family is built from nonlinear error-correcting codes for classical asymmetric channels, with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block lengthn≥ 8 exceptn=2r-1. We generalize this construction to employ classical codes overGF(3) with which we numerically obtain better performing codes up to length 14. Because the resulting codes are of the codeword stabilized (CWS) type, conceptually simple (though potentially computationally expensive) encoding and decoding circuits are available.

Unitary gates are interesting resources for quantum communication in part because they are always invertible and are intrinsically bidirectional. This paper explores these two symmetries: time-reversal and exchange of Alice and Bob. We present examples of unitary gates that exhibit dramatic separations between forward and backward capacities (even when the back communication is assisted by free entanglement) and between entanglement-assisted and unassisted capacities, among many others. Along the way, we will give a general time-reversal rule for relating the capacities of a unitary gate and its inverse that will explain why previous attempts at finding asymmetric capacities failed. Finally, we will see how the ability to erase quantum information and destroy entanglement can be a valuable resource for quantum communication.

We show how good quantum error-correcting codes can be constructed using generalized concatenation. The inner codes are quantum codes, the outer codes can be linear or nonlinear classical codes. Many new good codes are found, including both stabilizer codes as well as so-called non-additive codes.

The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for kges2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. *We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O tilde(radicn) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on Dinur's version of the PCP Theorem and is inherently non-relativizing. *We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all kges=2. *We give evidence that QMA(2) sube PSPACE, by showing that this would follow from "strong amplification" of QMA(2) protocols. *We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.

Error correction procedures are considered which are designed specifically for the amplitude damping channel. Amplitude damping errors are analyzed in the stabilizer formalism. This analysis allows a generalization of the[4,1]ldquoapproximaterdquo amplitude damping code. This generalization is presented as a class of[2(M+1),M] codes; quantum circuits for encoding and recovery operations are presented. A[7,3]amplitude damping code based on the classical Hamming code is presented. All of these are stabilizer codes whose encoding and recovery operations can be completely described with Clifford group operations. Finally, optimization options are described in which recovery operations may be further adapted according to the damping probabilitygamma.

In this article we present a theoretical analysis of the online Sum-of-Squares algorithm ( SS ) for bin packing along with several new variants. SS is applicable to any instance of bin packing in which the bin capacity B and item sizes s ( a ) are integral (or can be scaled to be so), and runs in time O ( nB ). It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear, SS also has sublinear expected waste. For any discrete distribution where the optimal expected waste is bounded, SS has expected waste at most O (log n ). We also discuss several interesting variants on SS , including a randomized O ( nB log B )-time online algorithm SS * whose expected behavior is essentially optimal for all discrete distributions. Algorithm SS * depends on a new linear-programming-based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution F , just what is the growth rate for the optimal expected waste.

Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: previous work has shown, however, that it not only requires less classical communication but also gives rise to a tradeoff between these two resources in the appropriate setting. We discuss this problem from first principles, including the various choices one may follow in the definitions of the actual resources. Our main result is a general method of remote state preparation for arbitrary states of many qubits, at a cost of 1 bit of classical communication and 1 bit of entanglement per qubit sent. In this "universal" formulation, these ebit and cbit requirements are shown to be simultaneously optimal by exhibiting a dichotomy. Our protocol then yields the exact tradeoff curve for memoryless sources of pure states (including the case of incomplete knowledge of the ensemble probabilities), based on the recently established quantum-classical tradeoff for visible quantum data compression. A variation of that method allows us to solve the even more general problem of preparing entangled states between sender and receiver (i.e., purifications of mixed state ensembles). The paper includes an extensive discussion of our results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding.

I examine the question of why so few classes of quantum algorithms have been discovered. I give two possible explanations for this, and some thoughts about what lines of research might lead to the discovery of more quantum algorithms.

The entanglement-assisted classical capacity of a noisy quantum channel (C/sub E/) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that the capacity C/sub E/ is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs /spl rho/, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of /spl rho/ after half of it has passed through the channel. We calculate entanglement-assisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglement-assisted communication - communication over a discrete memoryless channel (DMC) between parties who share prior random information - we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMCs of equal capacity can simulate one another with unit asymptotic efficiency.

In this paper we present a theoretical analysis of the deterministic on-line Sum of Squares algorithm (SS) for bin packing, introduced and studied experimentally in [8], along with several new variants.SS is applicable to any instance of bin packing in which the bin capacity B and item sizes s(a) are integral (or can be scaled to be so), and runs in time O(nB).It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear, SS also has sublinear expected waste.For any discrete distribution where the optimal expected waste is bounded, SS has expected waste at most O(log n).In addition, we present a randomized O(nB log B)-time on-line algorithm SS*, based on SS, whose expected behavior is essentially optimal for all discrete distributions.Algorithm SS* also depends on a new linear-programming-based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution F, just what is the growth rate for the optimal expected waste.An off-line randomized variant SS** performs well in a worst-case sense: For any list L of integer-sized items to be packed into bins of a fixed size B, the expected number of bins used by SS** is at most OPT(L) + ~.

We consider the one-dimensional bin packing problem with unit-capacity bins and item sizes chosen according to the discrete uniform distribution U{j,k}, $1 < j \leq k,$ where each item size in {1/k,2/k,. . .,j/k} has probability 1/j of being chosen. Note that for fixed j,k as $m\rightarrow\infty$ the discrete distributions U{mj,mk} approach the continuous distribution U(0,j/k], where the item sizes are chosen uniformly from the interval (0,j/k]. We show that average-case behavior can differ substantially between the two types of distributions. In particular, for all j,k with j < k-1, there exist on-line algorithms that have constant expected wasted space under U{j,k}, whereas no on-line algorithm has even o(n 1/2 ) expected waste under U(0,u] for any $0 < u \leq 1$. Our U{j,k} result is an application of a general theorem of Courcoubetis and Weber [C. Courcoubetis and R.R. Weber, Probab. Engrg. Inform. Sci., 4 (1990), pp. 447--460] that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of n items must be either $\Theta (n)$, $\Theta (n^{1/2} )$, or O(1), depending on whether certain"perfect" packings exist. The perfect packing theorem needed for the U{j,k} distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper.

We survey the field of quantum information theory. In particular, we discuss the fundamentals of the field, source coding, quantum error-correcting codes, capacities of quantum channels, measures of entanglement and quantum cryptography.

The problem of finding quantum error correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

A probabilistically checkable debate system (PCDS) for a language L consists of a probabilistic polynomial-time verifier V and a debate between Player 1, who claims that the input x is in L, and Player 0, who claims that the input x is not in L. It is known that there is a PCDS for L in which V flips O(log n) coins and reads O(1) bits of the debate if and only if L is in PSPACE [A. Condon, J. Feigenbaum, C. Lund, and P. Shor, Chicago J. Theoret. Comput. Sci., 1995, No. 4]. In this paper, we restrict attention to RPCDSs, which are PCDSs in which Player 0 follows a very simple strategy: On each turn, Player 0 chooses uniformly at random from the set of legal moves. We prove the following result. Theorem. L has an RPCDS in which the verifier flips O(log n) coins and reads O(1) bits of the debate if and only if L is in PSPACE. This new characterization of PSPACE is used to show that certain stochastic PSPACE-hard functions are as hard to approximate closely as they are to compute exactly. Examples of such functions include optimization versions of Dynamic Graph Reliability, Stochastic Satisfiability, Mah-Jongg, Stochastic Generalized Geography, and other "games against nature" of the type introduced in [C. Papadimitriou, J. Comput. System Sci., 31 (1985), pp. 288--301].

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1/log/sup c/t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1/t). We do this by showing that operations can be performed on quantum data encoded by quantum error-correcting codes without decoding this data.

A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a cost in computation time of at most a polynomial factor: It is not clear whether this is still true when quantum mechanics is taken into consideration. Several researchers, starting with David Deutsch, have developed models for quantum mechanical computers and have investigated their computational properties. This paper gives Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored. These two problems are generally considered hard on a classical computer and have been used as the basis of several proposed cryptosystems. We thus give the first examples of quantum cryptanalysis.>

Many complex proesses can be modeled by (countably) infinite, multidimensional Markov chains. Unfortunately, cnrnmt theoretical techniques for analyzing infinite Markov chains are for the most part limited to three or fewer dimensions. In this paper we propose a computer-aided approach to the analy-sis of higher-dimensional domains, using several open problems about the average-case behavior of the Best Fit bin packing algo-rithm as case studies. We show how to use dynamic and liiear programming to construct potential functions thal when applied to suitably modified multi-step versions of our original Markov chain, yield drifts that are bounded away fmm O. This enables us to completely classify the expected behavior of Best Fit under dis-crete uniform distributions U{J, K) when K is small. (Under U { J, K}, the allowed item sizes are i/K, 1 S i S J, with all J pos-sibilities equally likely.) In addition, we can answer yes to the long-standing open question of whether there exist distributions of this form for which Best Fit yields linearly-growing waste. The proof of the latter theorem relies on a 24-hour computation, and although its validity does not depend on the linear progra-mmingpackage we used, it does tely on the correctness of our dynamic progr smming code and of our computer’s implementation of the IEEE floating point standard.

Article Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions Share on Authors: Anne Condon View Profile , Joan Feigenbaum View Profile , Carsten Lund View Profile , Peter Shor View Profile Authors Info & Claims STOC '93: Proceedings of the twenty-fifth annual ACM symposium on Theory of ComputingJune 1993 Pages 305–314https://doi.org/10.1145/167088.167190Online:01 June 1993Publication History 21citation326DownloadsMetricsTotal Citations21Total Downloads326Last 12 Months2Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

X and Y are random variables. Person P/sub x/ knows X, Person P/sub y/ knows Y, and both know the underlying probability distribution of the random pair (X, Y). Using a predetermined protocol, they exchange messages over a binary, error-free, channel in order for P/sub y/ to learn X. P/sub x/ may or may not learn Y. C/sub m/ is the number of information bits that must be transmitted (by both persons) in the worst case if only m messages are allowed. C/sub infinity / is the corresponding number of bits when there is no restriction on the number of messages exchanged. We consider three aspects of this problem. C/sub 4/. It is known that one-message communication may require exponentially more bits than the minimum possible: for some random pairs, C/sub 1/=2/sup C infinity -1/. Yet just two messages suffice to reduce communication to almost the minimum: for all random pairs, C/sub 2/or=(2- in )C/sub infinity />or=c. Asymptotically, this is the largest possible discrepancy. Amortized complexity. The amortized complexity of (X,Y) is the limit, as k grows, of the number of bits required in the worst case for L independent repetitions of (X, Y), normalized by k. We show that the four-message amortized complexity of all random pairs is exactly log mu . Hence, when a random pair is repeated many times, no bits can be saved if P/sub x/ knows Y in advance.>

An O(n log n)-time online algorithm is given for packing items i.i.d. uniform on (0, 1) into bins of size 1 with expected wasted space Theta (n/sup 1/2/ log /sup 1/2/n). This matches the lowest bound that no online algorithm can achieve O(n/sup 1/2/ log /sup 1/2/ n) wasted space. It is done by analyzing another algorithm which involves putting balls into buckets online. The analysis of this second algorithm also gives bound on the stochastic rightward matching problem, which arises in analyzing not only the above online bin packing problem, but also a 2-D problem of packing rectangles into a half-infinite strip. The bounds on rightward matching thus give good bounds for the 2-D strip packing problem.>

We consider the average case behavior of onedmensional bin paekmg algorithms in the case where bins have unit capacity and item sizes are chosen according to the ' 'dficrete uniform" distribution U~; k), 1 s j < k, where each item size in the set {llk,21k,..., ji k) has probability 1/j of beiig chosen.Note that for fixed j,k the distributions U{?nj;mk]' approach the continuous distribution U(O, jlk] as m A W, where in U(O, jl k] the item sizes are chosen uniformly horn the half-open interval (O,jik].In this paper, we show that average case behavior can differ substantially under the two types of distributions.We show that for all j, k, j < k-1, there exist on-line algorithms that have constant expected waste under U~; k], whereas no on-line algorithm can have less than C2(n1'2) waste under U(O, U] for any u s 1. Conmariwise, although the First Fit Decreasing (off-line) algorithm has constant expected waste under U(O, u] for all u < 1/2,

The stochastic upright matching problem has had many important applications, most notably in statistics and the average-case analysis of algorithms. A problem instance is a set of n points chosen uniformly at random in the unit square. The points are labeled with signs; the signs are chosen independently and each is equally likely to be a plus or minus. An up-right matching of S is a matching of minus points to plus points such that if $( x,y )$ is a minus point matched to the plus point $( x',y' )$, then $x\leqq x'$ and $y\leqq y'$. The problem is to estimate the expected number of points left unmatched in a maximum upright matching of S. It is well known that if $U_n $ denotes the number of unmatched points, then $E[ U_n ] = \Theta ( \sqrt{n} \log^{3/4} n )$. Existing proofs of the upper bound $O( \sqrt{n} \log^{3/4} n )$ are quite long and difficult to follow. This paper presents a much simpler and more compact proof. A distinctive feature of the new proof is the use of Fourier expansions.

Paint one side of a rubber disk black and the other side white; stretch the disk any way you wish in three-dimensional space, subject to the condition that from any point in space, if you look down you see either the white side of the disk or nothing at all. Now make the stretched disk transparent but color its boundary black; project its boundary into a plane that lies below the disk. The resulting curve is self-overlapping. We show how to test whether a given curve is self-overlapping, and how to count how many essentially different stretchings of the disk could give rise to the same curve.

NC approximation algorithms are given for the unweighted and weighted set cover problems. The algorithms use a linear number of processors and give a cover that has at most log n times the optimal size/weight, thus matching the performance of the best sequential algorithms. The set cover algorithm is applied to learning theory, providing an NC algorithm for learning the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC dimension which has an NC hypothesis finder. In addition, a linear-processor NC algorithm is given for a variant of the set cover problem and used to obtain NC algorithms for several problems in computational geometry.>

We give a simple algorithmic technique for building geometric structures. The technique is randomized and incremental. As an application, we give an algorithm of this kind for computing the intersection of a set of halfspaces in three dimensions. (This intersection problem is linear-time equivalent to the computation of the convex hull of a point set.) The algorithm requires Ο(n log n) expected time, where the expectation is over the random behavior of the algorithm. A similar algorithm can be used to determine the intersection of a set of unit balls in E3, the problem of spherical intersection. This problem arises in the computation of the diameter of a point set in E3. For a set S of n points, the diameter of S is the greatest distance between two points in S. We give a randomized reduction from the problem of determining the diameter to the problem of computing spherical intersections, resulting in a Las Vegas algorithm for the diameter requiring Ο(n log n) expected time. The best algorithms previously known for this problem have worst-case time bounds no better than Ο(n √n log n) [Agg].

We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram of n points in the plane can be computed in Θ(n) time when these points form the vertices of a convex polygon in, say, counterclockwise order. This settles an outstanding open problem in computational geometry. Our techniques can also be used to obtain linear time algorithms for computing the farthest-point Voronoi diagram and the medial axis of a convex polygon and for deleting a vertex from a general planar Voronoi diagram.

Article Free Access Share on Geometric applications of a matrix searching algorithm Authors: A Aggarwal IBM T. J. Watson Center, Yorktown Heights IBM T. J. Watson Center, Yorktown HeightsSearch about this author , M Klawe IBM Almaden Research Center, San Jose IBM Almaden Research Center, San JoseView Profile , S Moran IBM T. J. Watson Center, Yorktown Heights IBM T. J. Watson Center, Yorktown HeightsSearch about this author , P Shor Math. Sciences Research Institute, Berkeley Math. Sciences Research Institute, BerkeleyView Profile , R Wilber IBM Almaden Research Center, San Jose IBM Almaden Research Center, San JoseView Profile Authors Info & Claims SCG '86: Proceedings of the second annual symposium on Computational geometryAugust 1986Pages 285–292https://doi.org/10.1145/10515.10546Published:01 August 1986Publication History 39citation1,255DownloadsMetricsTotal Citations39Total Downloads1,255Last 12 Months234Last 6 weeks35 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

The minimax grid matching problem is a fundamental combinatorial problem associated with the average case analysis of algorithms.The problem has arisen in a number of interesting and seemingly unrelated areas, including wafer-scale integration of systolic arrays, twodimensional discrepancy problems, and testing pseudorandom number generators.However, the minimax grid matching problem is best known for its application to the maximum up-right matching problem.The maximum up-right matching problem was originally defined by Karp, Luby and Marchetti-Spaccamela in association with algorithms for 2dimensional bin packing.More recently, the up-right matching problem has arisen in the average case analysis of on-line algorithms for 1-dimensional bin packing and dynamic allocation.In this paper, we solve both the minimax grid matching problem and the maximum up-right matching problem.As a direct result, we obtain tight upper bounds on the average case behavior of the best algorithms known for 2-dimensional bin packing, 1-dimensional on-line packing and on-line dynamic allocation.The results also solve a long-open question in mathematical statistics.

In this paper we give tighter bounds than were previously known for the performance of the bin packing algorithms Bets Fit and First Fit when the inputs are uniformly distributed on [0,1]. We also give a general lower bound for the performance of any on-line bin packing algorithm. These results are proven by analyzing problems concerning matching random points in a unit square. We give a new lower bound for upward right matching and grid matching.