QuickHeapsort: Modifications and Improved Analysis

QuickHeapsort is a combination of Quicksort and Heapsort. We show that the expected number of comparisons for QuickHeapsort is always better than for Quicksort if a usual median-of-constant strategy is used for choosing pivot elements. In order to obtain the result we present a new analysis for QuickHeapsort splitting it into the analysis of the partition-phases and the analysis of the heap-phases. This enables us to consider samples of non-constant size for the pivot selection and leads to better theoretical bounds for the algorithm. Furthermore, we introduce some modifications of QuickHeapsort. We show that for every input the expected number of comparisons is at most \(n\log _{2}n - 0.03n + o(n)\) for the in-place variant. If we allow n extra bits, then we can lower the bound to \( n\log _{2} n -0.997 n+ o (n)\). Thus, spending n extra bits we can save more that 0.96n comparisons if n is large enough. Both estimates improve the previously known results. Moreover, our non-in-place variant does essentially use the same number of comparisons as index based Heapsort variants and Relaxed-Weak-Heapsort which use \( n\log _{2}n -0.9 n+ o (n)\) comparisons in the worst case. However, index based Heapsort variants and Relaxed-Weak-Heapsort require \({\Theta }(n\log n)\) extra bits whereas we need n bits only. Our theoretical results are upper bounds and valid for every input. Our computer experiments show that the gap between our bounds and the actual values on random inputs is small. Moreover, the computer experiments establish QuickHeapsort as competitive with Quicksort in terms of running time.     

A Simple Parallel Algorithm for the Single-Source Shortest Path Problem on Planar Digraphs

We present a simple parallel algorithm for the single-source shortest path problem in planar digraphs with nonnegative real edge weights. The algorithm runs on the EREW PRAM model of parallel computation in O((n^2ε+n^1−ε) log n) time, performing O(n^1+ε log n) work for any 0<ε<1/2. The strength of the algorithm is its simplicity, making it easy to implement and presumable quite efficient in practice. The algorithm improves upon the work of all previous parallel algorithms. Our algorithm is based on a region decomposition of the input graph and uses a well-known parallel implementation of Dijkstra's algorithm. The logarithmic factor in both the work and the time can be eliminated by plugging in a less practical, sequential planar shortest path algorithm together with an improved parallel implementation of Dijkstra's algorithm.     

Direct and indirect algorithms for on-line learning of disjunctions

It is easy to design on-line learning algorithms for learning k out of n variable monotone disjunctions by simply keeping one weight per disjunction. Such algorithms use roughly O(n^k) weights which can be prohibitively expensive. Surprisingly, algorithms like Winnow require only n weights (one per variable or attribute) and the mistake bound of these algorithms is not too much worse than the mistake bound of the more costly algorithms. The purpose of this paper is to investigate how exponentially many weights can be collapsed into only O(n) weights. In particular, we consider probabilistic assumptions that enable the Bayes optimal algorithm's posterior over the disjunctions to be encoded with only O(n) weights. This results in a new O(n) algorithm for learning disjunctions which is related to the Bylander's BEG algorithm originally introduced for linear regression. Besides providing a Bayesian interpretation for this new algorithm, we are also able to obtain mistake bounds for the noise free case resembling those that have been derived for the Winnow algorithm. The same techniques used to derive this new algorithm also provide a Bayesian interpretation for a normalized version of Winnow.     

On Independent Sets and Bicliques in Graphs

Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner’s upper bound on the number of maximal bicliques (Combinatorica, 20, 109–117, 2000) and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3 ^n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642 ⁿ ), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.     

Two simple and efficient algorithms for Jordan sorting and polygon cutting and clipping

This paper deals with the Jordan sorting problem: Given n intersection points of a Jordan curve with the x-axis in the order in which they occur along the curve, sort these points into the order in which they occur along the x-axis. The worst-case time complexity of this problem is θ(n). Unfortunately, the known O(n) time algorithms are too complicated, which causes that they are difficult to implement and slow for the inputs of sizes that are of practical interest. In this paper, two algorithms for Jordan sorting are presented. The first algorithm is extremely simple. Although its worst-case time complexity is O(nlogn), it is shown that the worst time is achieved only for special inputs. For most inputs, a better performance can be expected. Furthermore, an improved O(nlog logn) worst-case time algorithm is presented. For the input sequences of size from 4 to 10⁵, the algorithms are compared with Quicksort, with the algorithm based on splay trees and with the O(n) time algorithm proposed by Fung et al. The results show that our algorithms are faster. The relevant implementation details are given.     

Spin-the-Bottle Sort and Annealing Sort: Oblivious Sorting via Round-Robin Random Comparisons

We study sorting algorithms based on randomized round-robin comparisons. Specifically, we study Spin-the-bottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a temperature parameter. Both algorithms are simple, randomized, data-oblivious sorting algorithms, which are useful in privacy-preserving computations, but, as we show, Annealing sort is much more efficient. We show that there is an input permutation that causes Spin-the-bottle sort to require Ω(n ²logn) expected time in order to succeed, and that in O(n ²logn) time this algorithm succeeds with high probability for any input. We also show there is a specification of Annealing sort that runs in O(nlogn) time and succeeds with very high probability.     

Maintaining bridge-connected and biconnected components on-line

We consider the twin problems of maintaining the bridge-connected components and the biconnected components of a dynamic undirected graph. The allowed changes to the graph are vertex and edge insertions. We give an algorithm for each problem. With simple data structures, each algorithm runs inO(n logn +m) time, wheren is the number of vertices andm is the number of operations. We develop a modified version of the dynamic trees of Sleator and Tarjan that is suitable for efficient recursive algorithms, and use it to reduce the running time of the algorithms for both problems toO(mα(m,n)), where α is a functional inverse of Ackermann's function. This time bound is optimal. All of the algorithms useO(n) space.     

Practical Algorithms for Generating a Random Ordering of the Elements of a Weighted Set

This paper discusses the problem of efficiently generating a random ordering of the elements of an n-element weighted set, where the elements’ weights are interpreted as relative probabilities. It is assumed that only the following randomness primitives are available: flipping a biased coin, and selecting a random non-negative integer smaller than n. We review the existing literature on this problem, focusing on several simple algorithms whose running times are O(n ²) and O(nlogn). Asymptotically faster algorithms do exist, but they are mostly either very complicated, or require a stronger computational model. The main contribution of this paper is a self-contained specification of and correctness proof for an O(nlogn) mergesort-like folklore algorithm that is often implemented incorrectly. The key piece of the correct algorithm can be viewed as the weighted generalization of the riffle merge operation which appears in mathematical models of card shuffling. Empirical results are also presented which suggest that this algorithm might be faster on a real computer than other simple algorithms which use the same randomness primitives, because its pattern of memory accesses results in fewer cache misses. Finally, three fancier but still implementable algorithms are described, analyzed, and simulated, which are asymptotically faster than O(nlogn), but require that the input weights to be partially sorted.     

Nonadaptive quantum query complexity

We study the power of nonadaptive quantum query algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum query algorithm that computes a total boolean function depending on n variables must make Ω(n) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using queries to solve the same problem. Thus, in the nonadaptive setting, quantum algorithms for these tasks can achieve at most a very limited speed-up over classical query algorithms.     

An efficient certifying algorithm for the Hamiltonian cycle problem on circular-arc graphs

A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is an evidence that can be used to authenticate the correctness of the answer. A Hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once. The Hamiltonian cycle problem is to determine whether or not a graph contains a Hamiltonian cycle. The best result for the Hamiltonian cycle problem on circular-arc graphs is an O(n²logn)-time algorithm, where n is the number of vertices of the input graph. In fact, the O(n²logn)-time algorithm can be modified as a certifying algorithm although it was published before the term certifying algorithms appeared in the literature. However, whether there exists an algorithm whose time complexity is better than O(n²logn) for solving the Hamiltonian cycle problem on circular-arc graphs has been opened for two decades. In this paper, we present an O(Δn)-time certifying algorithm to solve this problem, where Δ represents the maximum degree of the input graph. The certificates provided by our algorithm can be authenticated in O(n) time.     

Packet Routing and PRAM Emulation on Star Graphs and Leveled Networks

We consider the problem of permutation routing on a star graph, an interconnection network which has better properties than the hypercube. In particular, its degree and diameter are sublogarithmic in the network size. We present optimal randomized routing algorithms that run in O(D) steps (where D is the network diameter) for the worst-case input with high probability. We also show that for the n-way shuffle network with N = nⁿ nodes, there exists a randomized routing algorithm which runs in O(n) time with high probability. Another contribution of this paper is a universal randomized routing algorithm that could do optimal routing for a large class of networks (called leveled networks) which includes the star graph. The associative analysis is also network-independent. In addition, we present a deterministic routing algorithm, for the star graph, which is near optimal. All the algorithms we give are oblivious. As an application of our routing algorithms, we also show how to emulate a PRAM optimally on this class of networks.     

New solutions for disjoint paths in P systems

We propose four fast synchronous distributed message-based algorithms, to identify maximum cardinality sets of edge- and node-disjoint paths, between a source node and a target node in a digraph. Previously, Dinneen et?al. presented two algorithms, based on the classical distributed depth-first search (DFS), which run in O(mf) steps, where m is the number of edges and f is the number of disjoint paths. Combining Cidon??s distributed DFS and our new result, Theorem 3, we propose two improved DFS-based algorithms, which run in O(nf) steps, where n is the number of nodes. We also present improved versions of our two breadth-first search (BFS) based algorithms, with the same complexity upperbound, O(nf). Empirically, for a large set of randomly generated digraphs, our DFS-based edge-disjoint algorithm is 39?% faster than Dinneen et?al.??s edge-disjoint algorithm and our BFS-based edge-disjoint algorithm is 80?% faster. All these improved algorithms have been inspired and guided by a P system modelling exercise, but are suitable for any distributed implementation.     

Efficient parallel algorithms forr-dominating set andp-center problems on trees

We develop efficient parallel algorithms for ther-dominating set and thep-center problems on trees. On a concurrent-read exclusive-write PRAM, our algorithm for ther-dominating set problem runs inO(logn log logn) time withn processors. The algorithm for thep-center problem runs inO(log² n log logn) time withn processors.     

Efficient DNA sticker algorithms for NP-complete graph problems

Adleman's successful solution of a seven-vertex instance of the NP-complete Hamiltonian directed path problem by a DNA algorithm initiated the field of biomolecular computing. We provide DNA algorithms based on the sticker model to compute all k-cliques, independent k-sets, Hamiltonian paths, and Steiner trees with respect to a given edge or vertex set. The algorithms determine not merely the existence of a solution but yield all solutions (if any). For an undirected graph with n vertices and m edges, the running time of the algorithms is linear in n+m. For this, the sticker algorithms make use of small combinatorial input libraries instead of commonly used large libraries. The described algorithms are entirely theoretical in nature. They may become very useful in practice, when further advances in biotechnology lead to an efficient implementation of the sticker model.     

Fast congruence closure and extensions

Congruence closure algorithms for deduction in ground equational theories are ubiquitous in many (semi-)decision procedures used for verification and automated deduction. In many of these applications one needs an incremental algorithm that is moreover capable of recovering, among the thousands of input equations, the small subset that explains the equivalence of a given pair of terms. In this paper we present an algorithm satisfying all these requirements. First, building on ideas from abstract congruence closure algorithms, we present a very simple and clean incremental congruence closure algorithm and show that it runs in the best known time O(n  log  n). After that, we introduce a proof-producing union-find data structure that is then used for extending our congruence closure algorithm, without increasing the overall O(n  log  n) time, in order to produce a k-step explanation for a given equation in almost optimal time (quasi-linear in k). Finally, we show that the previous algorithms can be smoothly extended, while still obtaining the same asymptotic time bounds, in order to support the interpreted functions symbols successor and predecessor, which have been shown to be very useful in applications such as microprocessor verification.     

Improving on-line construction of two-dimensional suffix trees for square matrices

The two-dimensional (2-D) suffix tree of an n×n square matrix A is a compacted trie that represents all square submatrices of A. We consider constructing 2-D suffix trees on-line, which means, instead of giving the whole matrix A in advance, A is separated and each part of A is given at different time as algorithms proceed. In general, developing an on-line algorithm is more difficult than developing an off-line algorithm. Moreover, the smaller the input grain size is, the harder it is to develop an on-line algorithm. In the case of 2-D suffix tree construction, dealing with a character at a time is harder than dealing with a row or a column at a time.In this paper we propose a randomized linear-time algorithm for constructing 2-D suffix trees on-line. This algorithm is superior to previous algorithms in two ways: (1) This is the first linear-time algorithm for constructing 2-D suffix trees on-line. Although there have been some linear-time algorithms for off-line construction, there were no linear-time algorithms for on-line construction. (2) We deal with the most fine-grain on-line case, i.e., our algorithm can construct a 2-D suffix tree even though only one character of A is given at a time, while previous on-line algorithms require at least a row and/or a column at a time.     

Efficient algorithmic learning of the structure of permutation groups by examples

This paper discusses learning algorithms for ascertaining membership, inclusion, and equality in permutation groups. The main results are randomized learning algorithms which take a random generator set of a fixed group G ≤ S_n as input. We discuss randomized algorithms for learning the concepts of group membership, inclusion, and equality by representing the group in terms of its strong sequence of generators using random examples from G. We present O(n³ log n) time sequential learning algorithms for testing membership, inclusion and equality. The running time is expressed as a function of the size of the object set. (G ≤ S_n can have as many as n! elements.) Our bounds hold for all input groups. We also introduce limited parallelism, and our lower processor bounds make our algorithms more practical.Finally, we show that learning two-groups is in class NC by reducing the membership, inclusion, and inequality problems to solving linear systems over GF(2). We present an O(log³ n) time learning algorithm using n^ω processors for learning two-groups from examples (where n × n matrix product takes logarithmic time using n^ω processors).     

A subexponential bound for linear programming

We present a simple randomized algorithm which solves linear programs withn constraints andd variables in expected $$\min \{ O(d^2 2^d n),e^{2\sqrt {dIn({n \mathord{\left/ {\vphantom {n {\sqrt d }}} \right. \kern-\nulldelimiterspace} {\sqrt d }})} + O(\sqrt d + Inn)} \}$$ time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfyingn given linear inequalities ind variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson's linear programming algorithm, this gives an expected bound of $$O(d^2 n + e^{O(\sqrt {dInd} )} ).$$ The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) ofn points ind-space, computing the distance of twon-vertex (orn-facet) polytopes ind-space, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).     

A recursive algorithm for the reliability of a circular connected-(r,s)-out-of-(m,n):F lattice system

A circular connected-(r, s)-out-of-(m, n):F lattice system consists of m×n components arranged in a cylindrical grid. Each of m circles has n components, and this system fails if and only if there exists a grid of size r ×s which all components are failed. A circular connected-(r, s)-out-of-(m, n):F lattice system might be used in reliability models for ‘Feelers for measuring temperature on reaction chamber’ and ‘TFT Liquid Crystal Display system with 360° wide area’.In this study, we proposed a new recursive algorithm for obtaining the reliability of a circular connected-(r, s)-out-of-(m, n):F lattice system. We evaluated our proposed algorithms in terms of computing time and memory capacity. Furthermore, a numerical experiment comparing our proposed algorithm with Yamamoto and Miyakawa's algorithm [Yamamoto, H., & Miyakawa, M. (1996). Reliability of circular connected-(r, s)-out-of-(m, n):F lattice system. Journal of the Operations Research Society of Japan, 39(3), 389–406] showed that our proposed algorithm is more effective for systems with a large n.     

Realization and synthesis of reversible functions

Reversible circuits play an important role in quantum computing. This paper studies the realization problem of reversible circuits. For any n-bit reversible function, we present a constructive synthesis algorithm. Given any n-bit reversible function, there are N distinct input patterns different from their corresponding outputs, where N≤2ⁿ, and the other (2ⁿ−N) input patterns will be the same as their outputs. We show that this circuit can be synthesized by at most 2n⋅N ‘(n−1)’-CNOT gates and 4n²⋅N NOT gates. The time and space complexities of the algorithm are Ω(n⋅4ⁿ) and Ω(n⋅2ⁿ), respectively. The computational complexity of our synthesis algorithm is exponentially lower than that of breadth-first search based synthesis algorithms.