Showing papers on "Average-case complexity published in 2000"

On the complexity of supervisory control design in the RW framework

Abstract: The time complexity of supervisory control design for a general class of problems is studied. It is shown to be very unlikely that a polynomial-time algorithm can be found when either (1) the plant is composed of m components running concurrently or (2) the set of legal behaviors is given by the intersection of n legal specifications. That is to say, in general, there is no way to avoid constructing a state space which has size exponential in m+n. It is suggested that, rather than discouraging future work in the field, this result should point researchers to more fruitful directions, namely, studying special cases of the problem, where certain structural properties possessed by the plant or specification lend themselves to more efficient algorithms for designing supervisory controls. As no background on the subject of computational complexity is assumed, we have tried to explain all the borrowed material in basic terms, so that this paper may serve as a tutorial for a system engineer not familiar with the subject.

A relationship between linear complexity and k-error linear complexity

Abstract: Linear complexity is an important cryptographic criterion of stream ciphers. The k-error linear complexity of a periodic sequence of period N is defined as the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period. This article shows a relationship between the linear complexity and the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.

A fast algorithm for determining the linear complexity of a sequence with period p/sup n/ over GF(q)

Abstract: A fast algorithm is presented for determining the linear complexity of a sequence with period p/sup n/ over GF (q), where p is an odd prime, and where q is a prime and a primitive root (mod p/sup 2/).

Parameter-less genetic algorithm: a worst-case time and space complexity analysis

Abstract: In this paper the worst case analysis of the time and space complexity of the parameter less genetic algorithm versus the genetic algorithm with an optimal population size is provided and the results of the analysis are discussed Since the assumptions in order for the analysis to be correct are very weak the result is applicable to a wide range of problems Various con gurations of the parameter less genetic algorithm are considered and the results of their time and space complexity are compared

Adaptive and efficient mutual exclusion (extended abstract)

Abstract: A distributed algorithm is adaptive if its performance depends on k, the number of processes that are concurrently active during the algorithm execution (rather than on n, the total number of processes). This paper presents adaptive algorithm for mutual exclusion using only read and write operations.The worst case step complexity cannot be a measure for the performance of mutual exclusion algorithms, because it is always unbounded in the presence of contention. Therefore, a number of different parameters are used to measure the algorithm's performance: The remote step complexity is the maximal number of steps performed by a process where a wait is counted as one step. The system response time is the time interval between subsequent entries to the critical section, where one time unit is the minimal interval in which every active process performs at least one step.The algorithm presented here has O(k) remote step complexity and O(log k) system response time, where k is the point contention. The space complexity of this algorithm is O(nN), where N is the range of processes' names.The space complexity of all previously known adaptive algorithms for various long-lived problems depends on N. We present a technique that reduces the space complexity of our algorithm to be a function of n, while preserving the other performance measures of the algorithm.

Characterization of non-deterministic quantum query and quantum communication complexity

Abstract: It is known that the classical and quantum query complexities of a total Boolean function f are polynomially related to the degree of its representing polynomial, but the optimal exponents in these relations are unknown. We show that the non-deterministic quantum query complexity of f is linearly related to the degree of a "non-deterministic" polynomial for f. We also prove a quantum-classical gap of 1 vs. N for non-deterministic query complexity for a total f. In the case of quantum communication complexity there is a (partly undetermined) relation between the complexity of f and the logarithm of the rank of its communication matrix. We show that the non-deterministic quantum communication complexity of f is linearly related to the logarithm of the rank of a non-deterministic version of the communication matrix and that it can be exponentially smaller than its classical counterpart.

Average-Case Quantum Query Complexity

Abstract: We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial [3]. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity.

Nondeterministic Quantum Query and Quantum Communication Complexities

Abstract: We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its ``nondeterministic polynomial'' degree. We also prove a quantum-vs-classical gap of 1 vs n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n+1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.

A perspective on Quicksort

Abstract: This article introduces the basic Quicksort algorithm and gives a flavor of the richness of its complexity analysis. The author also provides a glimpse of some of its generalizations to parallel algorithms and computational geometry.

The Computational Complexity of Densest Region Detection

Abstract: We investigate the computational complexity of the task of detecting dense regions of an unknown distribution from unlabeled samples of this distribution. We introduce a formal learning model for this task that uses a hypothesis class as it “anti-overfitting” mechanism. The learning task in our model can be reduced to a combinatorial optimization problem. We can show that for some constants, depending on the hypothesis class, these problems are NP-hard to approximate to within these constant factors. We go on and introduce a new criterion for the success of approximate optimization geometric problems. The new criterion requires that the algorithm competes with hypotheses only on the points that are separated by some margin ? from their boundaries. Quite surprisingly, we discover that for each of the two hypothesis classes that we investigate, there is a “critical value” of the margin parameter ?. For any value below the critical value the problems are NP-hard to approximate, while, once this value is exceeded, the problems become poly-time solvable.

A probabilistic result on multi-dimensional Delaunay triangulations, and its application to the 2D case

Abstract: This paper exploits the notion of “unfinished site”, introduced by Katajainen and Koppinen (1998) in the analysis of a two-dimensional Delaunay triangulation algorithm, based on a regular grid. We generalize the notion and its properties to any dimension k⩾2 : in the case of uniform distributions, the expected number of unfinished sites in a k -rectangle is O (N 1−1/k ) . This implies, under some specific assumptions, the linearity of a class of divide-and-conquer schemes based on balanced k-d trees. This general result is then applied to the analysis of a new algorithm for constructing Delaunay triangulations in the plane. According to Su and Drysdale (1995, 1997), the best known algorithms for this problem run in linear expected time, thanks in particular to the use of bucketing techniques to partition the domain. In our algorithm, the partitioning is based on a 2-d tree instead, the construction of which takes Θ(N log N) time, and we show that the rest of the algorithm runs in linear expected time. This “preprocessing” allows the algorithm to adapt efficiently to irregular distributions, as the domain is partitioned using point coordinates, as opposed to a fixed, regular basis (buckets or grid). We checked that even for the largest data sets that could fit in internal memory (over 10 million points), constructing the 2-d tree takes noticeably less CPU time than triangulating the data. With this in mind, our algorithm is only slightly slower than the reputedly best algorithms on uniform distributions, and is even the most efficient for data sets of up to several millions of points distributed in clusters.

A lower bound on the average-case complexity of shellsort

Abstract: We demonstrate an Ω(pn1+1/p ) lower bound on the average-case running time (uniform distribution) of p-pass Shellsort. This is the first nontrivial general lower bound for average-case Shellsort.

Linear complexity of a sequence obtained from a periodic sequence by either substituting, inserting, or deleting k symbols within one period

Abstract: A unified derivation of the bounds of the linear complexity is given for a sequence obtained from a periodic sequence over GF(q) by either substituting, inserting, or deleting k symbols within one period. The lower bounds are useful in case of n

Average-case analysis of lgorithms using Kolmogorov complexity

Abstract: Analyzing the average-case complexity of algorithms is a very practical but very difficult problem in computer science. In the past few years, we have demonstrated that Kolmogorov complexity is an important tool for analyzing the average-case complexity of algorithms. We have developed the incompressibility method. In this paper, several simple examples are used to further demonstrate the power and simplicity of such method. We prove bounds on the average-case number of stacks (queues) required for sorting sequential or parallel Queuesort or Stacksort.

Improvement of Perez and Vidal algorithm for the decomposition of digitized curves into line segments

Abstract: Perez and Vidal proposed (1994) an optimal algorithm for the decomposition of digitized curves into line segments. The number of segments is fixed a priori, and the error criterion is the sum of the square Euclidean distance from each point of the contour to its orthogonal projection onto the corresponding line segment. The complexity of Perez and Vidal algorithm is O(n/sup 2/.m) where n is the number of points and m is the number of segments. We propose improvements of the algorithm to reduce the complexity, using the A* algorithm. The optimality of the algorithm is preserved and its complexity is lower. Some comparative results are presented, showing that our method is systematically faster, in particular for a large number of segments.

A symbolic-numeric silhouette algorithm

Abstract: The silhouette algorithm developed by Canny (1988, 1993) is a general motion planning algorithm which is known to have the best complexity of all of the general and complete algorithms. The authors present a symbolic-numeric version of the algorithm. This version does not require the symbolic computation of the determinants of resultant matrices, and can work on floating point arithmetic. Though its combinatorial complexity remains the same, but its algebraic complexity has been improved significantly which is very important towards its implementation. Several numerical examples are also presented.

Computational properties of metaquerying problems

Abstract: Metaquerying is a datamining technology by which hidden dependencies among several database relations can be discovered. This tool has already been successfully applied to several real-world applications. Recent papers provide only very preliminary results about the complexity of metaquerying. In this paper we define several variants of metaquerying that encompass, as far as we know, all variants defined in the literature. We study both the combined complexity and the data complexity of these variants. We show that, under the combined complexity measure, metaquerying is generally intractable (unless P=NP), but we are able to single out some tractable interesting metaquerying cases (whose combined complexity is LOGCFL-complete). As for the data complexity of metaquerying, we prove that, in general, this is in P, but lies within AC0 in some interesting cases. Finally, we discuss the issue of equivalence between metaqueries, which is useful for optimization purposes.

A moment of perfect clarity II: consequences of sparse sets hard for NP with respect to weak reductions

Abstract: This issue's column, Part II of the article started in the preceding issue, is about progress on the question of whether NP has sparse hard sets with respect to weak reductions.Upcoming Complexity Theory Column articles include A. Werschulz on information-based complexity; J. Castro, R. Gavalda, and D. Guijarro on what complexity theorists can learn from learning theory; S. Ravi Kumar and D. Sivakumar on a to-be-announced topic; M. Holzer and P. McKenzie on alternating stack machines; and R. Paturi on the complexity of k-SAT.

A new faster sequence pair algorithm

Abstract: This paper introduces a new sequence pair algorithm which has complexity lower than O(M’.25), which is a significant improvement compared to the original O(M2) algorithm [l]. Furthermore, the new algorithm has complexity close to the theoretical lower bound. Experimental results, obtained with a straightforward implementation, confirm this improvement in complexity.

Complexity-distortion tradeoffs in image and video compression

Abstract: In this thesis we investigate variable complexity algorithms. The complexities of these algorithms are input-dependent, i.e., the type of input determines the complexity required to complete the operation. The key idea is to enable the algorithm to classify the inputs so that unnecessary operations can be pruned. The goal of the design of the variable complexity algorithm is to minimize the average complexity over all possible input types, including the cost of classifying the inputs. We study two of the fundamental operations in standard image/video compression, namely, the discrete cosine transform (DCT) and motion estimation (ME). We first explore variable complexity in inverse DCT by testing for zero inputs. The test structure can also be optimized for minimal total complexity for a given inputs statistics. In this case, the larger the number of zero coefficients, i.e., the coarser the quantization stepsize, the greater the complexity reduction. As a consequence, tradeoffs between complexity and distortion can be achieved. For direct DCT we propose a variable complexity fast approximation algorithm. The variable complexity part computes only DCT coefficients that will not be quantized to zeros according to the classification results (in addition the quantizer can benefit from this information by by-passing its operations for zero coefficients). The classification structure can also be optimized for a given input statistics. On the other hand, the fast approximation part approximates the DCT coefficients with much less complexity. The complexity can be scaled, i.e., it allows more complexity reduction at lower quality coding, and can be made quantization-dependent to keep the distortion degradation at a certain level. In video coding, ME is the part of the encoder that requires the most complexity and therefore achieving significant complexity reduction in ME has always been a goal in video coding research. We propose two fast algorithms based on fast distance metric computation or fast matching approaches. Both of our algorithms allow computational scalability in distance computation with graceful degradation in the overall image quality. The first algorithm exploits hypothesis testing in fast metric computation whereas the second algorithm uses thresholds obtained from partial distances in hierarchical candidate elimination. (Abstract shortened by UMI.)

Average-Case Quantum Query Complexity

Abstract: We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial [3]. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity.

Novel iterative division algorithm over GF(2/sup m/) and its semi-systolic VLSI realization

Abstract: Extends the binary algorithm invented by J. Stein [1967] and proposes two iterative division algorithms in finite field GF(2/sup m/). Algorithm EBg exhibits faster convergence while algorithm EBd has reduced complexity in each iteration. A (semi-)systolic array is designed for algorithm EBd, resulting in an area-time complexity better than the best result known to date based on the extended Euclid algorithm.

Complexity results for some eigenvector problems

Abstract: We consider the computation of eigenvectors over the integers, where each component x i satisfies for an integer b We address various problems in this context, and analyze their computational complexity We find that different problems are complete for the complexity classes Applying the results, finding bounded solutions of a Diophantine equation is shown to be intractable

Evolution, the criterion problem, and complexity

Abstract: Cultural and dual-inheritance models of evolution present ambiguities not typically present in biological evolution. Criteria and the ability to specify the adaptive value of a trait or cultural practice become less clear. When niche construction is added, additional challenges and ambiguities arise. Its dynamic nature increases the difficulty of identifying adaptations, tracing the causal path between a trait and its function, and identifying the links between environmental demands and the development of adaptations.

Complexity analysis for algorithmically simple strings

Abstract: Given a reference computer, Kolmogorov complexity is a well defined function on all binary strings. In the standard approach, however, only the asymptotic properties of such functions are considered because they do not depend on the reference computer. We argue that this approach can be more useful if it is refined to include an important practical case of simple binary strings. Kolmogorov complexity calculus may be developed for this case if we restrict the class of available reference computers. The interesting problem is to define a class of computers which is restricted in a {\it natural} way modeling the real-life situation where only a limited class of computers is physically available to us. We give an example of what such a natural restriction might look like mathematically, and show that under such restrictions some error terms, even logarithmic in complexity, can disappear from the standard complexity calculus. Keywords: Kolmogorov complexity; Algorithmic information theory.

Memory Complexity Analysis on Numerical Programs

Abstract: Memory systems become more and more complicated with so many efforts on bridging the large speed gap between processor and main memory. It is now difficult to gain high performance from a processor or a large parallel processing systems without considering the specific memory system features. Thus it becomes not enough just to use the time and space complexity to explain why different forms of one algorithm explore so different performance on one same platform. The complexity of memory systems must be incorporated into the analysis of algorithms. In 1996, Sun Jiachang first presented a new concept on memory complexity. It is believed that the complexity of an algorithm should consist of its computational complexity and memory complexity, among them, computational complexity consists of time complexity and space complexity, which are the basic characteristics of algorithm; while memory complexity is a varying characteristic, which will change with different implementations of the same algorithm and different platforms. The purpose of algorithmic optimization is to reduce the memory complexity, while the reduction of computational complexity needs new algorithmic research activity. In this paper, we try to analyze the different implementations of an algorithm and to predict the relative performance differences among them through combining the memory complexity analysis and the data movement/floating point operation ratio analysis. Further analysis with remote communication in parallel processing will be our future work.