Computing the subset partial order for dense families of sets

We give an algorithm to compute the subset partial order (called the subset graph) for a family F of sets containing k sets with N elements in total and domain size n. Our algorithm requires O(nk²/logk) time and space on a Pointer Machine. When F is dense, i.e. N=Θ(nk), the algorithm requires O(N²/log²N) time and space. We give a construction for a dense family whose subset graph is of size Θ(N²/log²N), indicating the optimality of our algorithm for dense families. The subset graph can be dynamically maintained when F undergoes set insertions and deletions in O(nk/logk) time per update (that is sub-linear in N for the case of dense families). If we assume words of b?k bits, allow bits to be packed in words, and use bitwise operations, the above running time and space requirements can be reduced by a factor of blog(k/b+1)/logk and b²log(k/b+1)/logk respectively.     

Log-logarithmic worst-case range queries are possible in space Θ(N)

Let S denote a set of N records whose keys are distinct nonnegative integers less than some initially specified bound M. This paper introduces a new data structure, called the y-fast trie, which uses Θ(N) space and Θ(log log M) time for range queries on a random access machine. We will also define a simpler but less efficient structure, called the x-fast trie.     

Minimal on-line labelling

The problem of on-line labelling is one of assigning integer labels in the range 1 to N to an input stream of at most N distinct items, drawn from a linearly ordered set, so that at each step the labels respect the ordering on the items. To maintain this constraint, items may have to be relabelled to accommodate new ones. With T(M,N) denoting the total number of relabellings that have to be performed for the first M inputs, it is known that for any given constant c in the range 0<c<1 there are exact bounds T(Nc,N)=Θ(NlogN) and T(cN,N)=Θ(Nlog²N). However, in the case c=1, when the labelling is called minimal, is known only that T(N,N)=O(Nlog³N). Existing algorithms for minimal on-line labelling are complicated, and our aim in this paper is to give a simplified and self-contained account of the problem.     

Practical constructive schemes for deterministic shared-memory access

We present three explicit schemes for distributingM variables amongN memory modules, whereM=Θ(N ^1.5),M = Θ(N ²), andM=Θ(N ³), respectively. Each variable is replicated into a constant number of copies stored in distinct modules. We show thatN processors, directly accessing the memories through a complete interconnection, can read/write any set ofN variables in worst-case timeO (N ^1/3),O(N ^1/2), andO(N ^2/3), respectively for the three schemes. The access times for the last two schemes are optimal with respect to the particular redundancy values used by such schemes. The address computation can be carried out efficiently by each processor without recourse to a complete memory map and requiring onlyO(1) internal storage.     

Approximation algorithms for the bandwidth minimization problem for a large class of trees

We present approximation algorithms for the bandwidth minimization problem (BMP) for a large class of trees. The BMP is NP-hard, even for trees of maximum node degree 3. The problem finds applications in many areas, including VLSI layout, multiprocessor scheduling, and matrix processing, and has been studied for both graphs and matrices. We study the problem on trees having the following property: given any tree nodev, the depth difference of any two nonempty subtrees rooted atv is bounded by a constantk. We call such treesh(k)trees orgeneralized height-balanced (GHB)trees. The above definition extends the class of balanced trees to trees with depthd=Θ(\N\). For any tree in the above defined class, anO (logd) times optimal algorithm is presented. Furthermore, we extend the application of the algorithm to trees that simulate theh(k) property, which we callh(k)-like trees, and also provide intuitive ideas for an approximation algorithm for general trees.     

An output sensitive algorithm for computing a maximum independent set of a circle graph

We present an output sensitive algorithm for computing a maximum independent set of an unweighted circle graph. Our algorithm requires O(nmin{d,α}) time at worst, for an n vertex circle graph where α is the independence number of the circle graph and d is its density. Previous algorithms for this problem required Θ(nd) time at worst.     

Fast Sequential Importance Sampling to Estimate the Graph Reliability Polynomial

The reliability polynomial of a graph counts its connected subgraphs of various sizes. Algorithms based on sequential importance sampling (SIS) have been proposed to estimate a graph’s reliability polynomial. We develop an improved SIS algorithm for estimating the reliability polynomial. The new algorithm runs in expected time O(mlogn α(m,n)) at worst and ≈m in practice, compared to Θ(m ²) for the previous algorithm. We analyze the error bounds of this algorithm, including comparison to alternative estimation algorithms. In addition to the theoretical analysis, we discuss methods for estimating the variance and describe experimental results on a variety of random graphs.     

A novel constant degree and constant congestion DHT scheme for peer-to-peer networks

1 Introduction and related work In recent years, peer-to-peer computing has attracted significant attention from both industry field and academic field[1-3]. The core component of many proposed peer-to- peer systems is the distributed hash table (DHT) schemes[4,5] that use a hash table-like interface to publish and look up data objects. Many proposed DHT schemes[6-15] are based on some traditional interconnection to- pology: Chord[6], Tapestry[7,8], Pastry[9] are based on hypercube topolog…     

Ranking-Based Black-Box Complexity

Randomized search heuristics such as evolutionary algorithms, simulated annealing, and ant colony optimization are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime analysis results have appeared in the last 20 years, a powerful complexity theory for such algorithms is yet to be developed. We enrich the existing notions of black-box complexity by the additional restriction that not the actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the black-box algorithm. Many randomized search heuristics belong to this class of algorithms. We show that the new ranking-based model can give more realistic complexity estimates. The class of all binary-value functions has a black-box complexity of O(logn) in the previous black-box models, but has a ranking-based complexity of Θ(n). On the other hand, for the class of all OneMax functions, we present a ranking-based black-box algorithm that has a runtime of Θ(n/logn), which shows that the OneMax problem does not become harder with the additional ranking-basedness restriction.     

A Linear Time Algorithm for Sequential Diagnosis in Hypercubes

This paper describes a new sequential diagnosis algorithm for hypercubes. The algorithm is based on the PMC model and it assumes the existence of a central observer for syndrome decoding. If we denote the total number of processors in a given hypercube by N, then the algorithm achieves Θ([formula]) degree of diagnosability using only O(N) tests over all iterations of diagnosis and repair. The aggregated syndrome decoding time is also shown to be O(N) for this algorithm. The number of iterations of diagnosis and repair needed by the algorithm is O(log N).     

Online computation with advice

We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. The advice is a function, defined by the online algorithm, of the whole request sequence. The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b=0 corresponds to the classical online model, and b=⌈log∣A∣⌉, where A is the algorithm’s action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones.In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k-server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1≤b≤Θ(logn), where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio Ω(log(n)/b) and we present a deterministic online algorithm for MTS with competitive ratio O(log(n)/b). For the k-server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k^O(1/b) for any choice of Θ(1)≤b≤logk.     

An improved algorithm for Boolean matrix multiplication

A new algorithm for computing the product of two arbitraryN×N Boolean matrices is presented. The algorithm requiresO (N ³/logN) bit operations and onlyO(N logN) bits of additional storage. This represents an improvement on the Four Russians' method which requires the same number of operations but usesO(N ³/logN) bits of additional storage.     

A fast pessimistic one-step diagnosis algorithm for hypercube multicomputer systems

This paper describes a system-level diagnosis algorithm for hypercube multicomputer systems. The algorithm is based on the PMC model and can isolate all faulty processors to within a set that contains at most one fault-free processor. If we denote by N the total number of processors in a hypercube system to be diagnosed, then, based on the judiciously designed data structures, the algorithm can run in O(Nlog₂N) time; whereas the best-known diagnosis algorithm, the YML algorithm, runs in O(N^2.5) time. Consequently, the new algorithm is remarkably superior to the YML algorithm in terms of the time cost.     

On measuring the distance between histograms

A distance measure between two histograms has applications in feature selection, image indexing and retrieval, pattern classification and clustering, etc. We propose a distance between sets of measurement values as a measure of dissimilarity of two histograms. The proposed measure has the advantage over the traditional distance measures regarding the overlap between two distributions; it takes the similarity of the non-overlapping parts into account as well as that of overlapping parts. We consider three versions of the univariate histogram, corresponding to whether the type of measurement is nominal, ordinal, and modulo and their computational time complexities are Θ(b), Θ(b) and O(b²) for each type of measurements, respectively, where b is the number of levels in histograms.     

A Unified Approach to the Parallel Construction of Search Trees

We present a unified parallel algorithm for constructing various search trees. The tree construction is based on a unified scheme, called bottom-level balancing, which constructs a perfectly balanced search tree having a uniform distribution of keys. The algorithm takes O(log log N) time using N/log log N processors on the EREW PRAM model, and O(1) time with N processors on the CREW PRAM model, where N is the number of keys in the tree.     

Efficient region segmentation on compressed gray images using quadtree and shading representation

Image segmentation, which partitions the image into homogeneous regions, is a fundamental operation in image processing. Suppose the input gray image with size N×N has been compressed into the compressed image via quadtree and shading representation. Assume that the number of blocks in the representation is B, commonly B<N² due to the compression effect. This paper first derives some closed forms for computing the mean/variance of any block and for calculating the two statistical measures of any merged region in O(1) time. It then presents an efficient O(Bα(B))-time algorithm for performing region segmentation on the compressed image directly where α(B) is the inverse of the Ackerman's function and is a very slowly growing function. With the same time complexity, our results extend the pioneering results by Dillencourt and Samet from the map image to the gray image. In addition, with four real images, experimental results show that our proposed algorithm has about 55.4% execution time improvement ratio when compared to the previous fastest region-segmentation algorithm by Fiorio and Gustedt whose O(N²)-time algorithm is run on the original N×N gray image.     

A filtering algorithm for k-mismatch with don't cares

We present a filtering based algorithm for the k-mismatch pattern matching problem with don't cares. Given a text t of length n and a pattern p of length m with don't care symbols in either p or t (but not both), and a bound k, our algorithm finds all the places that the pattern matches the text with at most k mismatches. The algorithm is deterministic and runs in Θ(nm1/3k1/3log2/3m) time.     

An adaptive global reduction algorithm for wormhole-routed 2D meshes

This paper presents a global reduction algorithm for wormhole-routed 2D meshes. Well-known reduction algorithms that are optimized for short vectors have complexity O(M log N), where N = n × n is the number of nodes, and M the vector length. Algorithms suitable for long vectors have complexity O(√N + M). Previously known asymptotically optimal algorithms with complexity O(log N + M) incur inherent network contention among constituent messages. The proposed algorithm adapts to the given vector length, resulting in complexities O(M log N) for short vectors, O(log N + M) for medium-sized vectors, and O(√N + M) for sufficiently long vectors. The O(√N + M) version is preferred to the O(log N + M) version for long vectors, due to its small coefficient associated with M, the dominating factor for such vectors. The algorithm is contention-free in a synchronous environment. Under asynchronous execution models, depth contention (contention among message-passing steps) may occur. However, simulation studies show that the effect of depth contention on the actual performance is negligible.     

Boolean-width of graphs

We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods-Boolean sums of neighborhoods-across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n-vertex graph given with a decomposition tree of boolean-width k, we solve Maximum Weight Independent Set in time O(n²k2^2k) and Minimum Weight Dominating Set in time O(n²+nk2^3k). With an additional n² factor in the runtime, we can also count all independent sets and dominating sets of each cardinality.Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of GF(2)-sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ(n²) vertices has boolean-width Θ(logn) and rank-width, clique-width, tree-width, and branch-width Θ(n).     

Unimodal regression via prefix isotonic regression

This paper gives algorithms for determining real-valued univariate unimodal regressions, that is, for determining the optimal regression which is increasing and then decreasing. Such regressions arise in a wide variety of applications. They are shape-constrained nonparametric regressions, closely related to isotonic regression. For unimodal regression on n weighted points our algorithm for the L₂ metric requires only Θ(n) time, while for the L₁ metric it requires time. For unweighted points our algorithm for the L_∞ metric requires only Θ(n) time. All of these times are optimal. Previous algorithms were for the L₂ metric and required Ω(n²) time. All previous algorithms used multiple calls to isotonic regression, and our major contribution is to organize these into a prefix isotonic regression, determining the regression on all initial segments. The prefix approach reduces the total time required by utilizing the solution for one initial segment to solve the next.