Sorting in constant number of row and column phases on a mesh

An algorithm for sorting on a mesh by alternately transforming the rows and columns is presented. The algorithm runs in a constant number of row- and column-transformation phases (sixteen phases), an improvement over the previous best upper bound ofO(log logm) phases,m being the number of rows in the mesh. A corresponding lower bound of five phases is also shown.     

Describing an algorithm by Hopcroft

We give an algorithm, its correctness proof, and its proof of execution time bound, for finding the sets of equivalent states in a deterministic finite state automaton. The time bound is K · m · n · log n where K is a constant, m the number of input, symbols, and n the number of states. Hopcroft [3] has already published such an algorithm. The main reason for this paper is to illustrate the use of communicating an algorithm to others using a structured, top-down approach. We have also been able to improve on Hopcroft's algorithm by reducing the size of the algorithm and correspondingly complicating the proof of the running time bound.This research was supported by NSF Grant No. GJ-28176.     

An efficient algorithm for checking the robust stability of a polytope of polynomials

An efficient algorithm for checking the robust stability of a polytope of polynomials is proposed. This problem is equivalent to a zero exclusion condition at each frequency. It is shown that such a condition has to be checked at only afinite number of frequencies. We formulate this problem as aparametric linear program which can be solved by the Simplex procedure, with additional computations between steps consisting of polynomial evaluations and calculation of positive polynomial roots. Our algorithm requires a finite number of steps (corresponding to frequency checks) and in the important case when the polytope of parameters is a hypercube, this number is at most of orderO(m ³ n ²), wheren is the degree of the polynomials in the family andm is the number of parameters. Supported by NASA under Contract No. NCC2-477 and by a Charles Powell Foundation Grant.     

Algorithmic approach to the consecutive retrieval property

Ghosh's consecutive retrieval property (CR property) not only represents an elegant file organization, but also raises the problem of how to construct such a file with this property. Ghosh used ann ×m 0–1 incidence matrix, wheren is the number of records andm is the number of queries, as a tool for constructing a file with the CR property. In this paper the rows and columns of the incidence matrix are restricted to unique 0–1 patterns. It is found that such a unique incidence matrix cannot have the CR property if the number of rows is greater than 2m–1. This upper bound can be used to generatem(2m–1) columns, which form all the matrices with the CR property that may correspond to the given matrix. Two algorithms are presented which map the columns of the given incidence matrix to these columns with consecutive l's. A proposed implementation in terms of data base design is also presented.     

An Improved (and Practical) Parameterized Algorithm for the Individual Haplotyping Problem MFR with Mate-Pairs

The Individual Haplotyping MFR problem is a computational problem that, given a set of DNA sequence fragment data of an individual, induces the corresponding haplotypes by dropping the minimum number of fragments. Bafna, Istrail, Lancia, and Rizzi proposed an algorithm of time O(2^2k m ² n+2^3k m ³) for the problem, where m is the number of fragments, n is the number of SNP sites, and k is the maximum number of holes in a fragment. When there are mate-pairs in the input data, the parameter k can be as large as 100, which would make the Bafna-Istrail-Lancia-Rizzi algorithm impracticable. The current paper introduces a new algorithm PM-MFR of running time , where k ₁ is the maximum number of SNP sites that a fragment covers (k ₁ is smaller than n), and k ₂ is the maximum number of fragments that cover a SNP site (k ₂ is usually about 10). Since the time complexity of the algorithm PM-MFR is not directly related to the parameter k, the algorithm solves the Individual Haplotyping MFR problem with mate-pairs more efficiently and is more practical in real biological applications. This research was supported in part by the National Natural Science Foundation of China under Grant Nos. 60433020 and 60773111, the Program for New Century Excellent Talents in University No. NCET-05-0683, the Program for Changjiang Scholars and Innovative Research Team in University No. IRT0661, and the Scientific Research Fund of Hunan Provincial Education Department under Grant No. 06C526.     

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(m(m,n)), where is a functional inverse of Ackermann's function. This time bound is optimal. All of the algorithms useO(n) space.Research at Princeton University supported in part by National Science Foundation Grant DCR-86-05962 and Office of Naval Research Contract N00014-91-J-1463.This work was partially done while the author was at the Department of Computer Science, Princeton University, Princeton, NJ 08544, USA.     

Area minimization for hierarchical floorplans

In this paper we study the area-minimization problem for hierarchical floorplans. We settle an open problem on the complexity of the area-minimization problem for hierarchical floorplans by showing it to be NP-complete (even for balanced hierarchical floorplans). We then present a new algorithm for determining the nonredundant realizations of a wheel. The algorithm has time costO(k ² logk) and space cost0(k ²) if each block in a wheel has at mostk realizations. Based on the new algorithm for a wheel, we design a new pseudopolynomial area-minimization algorithm for hierarchical floorplans of order-5. The time and space costs of the algorithm are0((nM)²log(nM) and0(n ² M), respectively, wheren is the number of basic blocks andM is an upper bound on the dimensions of the realizations of the basic blocks. The area-minimization algorithm was implemented. Experimental results show that it is very fast.The research of Peichen Pan and C. L. Liu was partially supported by the NSF under Grant MIP-9222408. The research of Weiping Shi was partially supported by the NSF under Grant MIP-9309120.     

Multiple filtration and approximate pattern matching

Given a text of lengthn and a query of lengthq, we present an algorithm for finding all locations ofm-tuples in the text and in the query that differ by at mostk mismatches. This problem is motivated by the dot-matrix constructions for sequence comparison and optimal oligonucleotide probe selection routinely used in molecular biology. In the caseq=m the problem coincides with the classicalapproximate string matching with k mismatches problem. We present a new approach to this problem based on multiple hashing, which may have advantages over some sophisticated and theoretically efficient methods that have been proposed. This paper describes a two-stage process. The first stage (multiple filtration) uses a new technique to preselect roughly similarm-tuples. The second stage compares thesem-tuples using an accurate method. We demonstrate the advantages of multiple filtration in comparison with other techniques for approximate pattern matching.This research was supported in part by the National Science Foundation under Grant No. DMS 90-05833 and the National Institute of Health under Grant No. GM-36230.     

Tight bounds on the number of minimum-mean cycle cancellations and related results

We prove a tight (min(nm log(nC), nm²)) bound on the number of iterations of the minimum-mean cycle-canceling algorithm of Goldberg and Tarjan [13]. We do this by giving the lower bound and by improving the strongly polynomial upper bound on the number of iterations toO(nm ²). We also give an improved version of the maximum-mean cut canceling algorithm of [7], which is a dual of the minimum-mean cycle-canceling algorithm. Our version of the dual algorithm runs in O(nm²) iterations.Partially supported by NSF Presidential Young Investigator Grant CCR-8858097 with matching funds provided by AT&T, IBM Faculty Development Award, and a grant from the 3M Corporation.     

Finding the closed partition of a planar graph

In this paper we consider the problem of finding aclosed partition in a directed graph. This problem has applications in concurrent probabilistic program verification. The best sequential algorithm known for this problem runs inO(mn) time wherem is the number of directed edges andn is the number of vertices in the given digraph. In this paper we present a linear-time sequential algorithm to solve the closed partition problem for planar digraphs that arecompact. We then build on this algorithm to obtain an O(n^1.5)-time sequential algorithm to solve the closed partition problem for a general planar digraph.This work was supported in part by NSF Grant CCR 89-10707.     

Second Attribute Algorithm Based on Tree Expression

One view of finding a personalized solution of reduct in an information system is grounded on the viewpoint that attribute order can serve as a kind of semantic representation of user requirements. Thus the problem of finding personalized solutions can be transformed into computing the reduct on an attribute order. The second attribute theorem describes the relationship between the set of attribute orders and the set of reducts, and can be used to transform the problem of searching solutions to meet user requirements into the problem of modifying reduct based on a given attribute order. An algorithm is implied based on the second attribute theorem, with computation on the discernibility matrix. Its time complexity is O（n^2 × m） （n is the number of the objects and m the number of the attributes of an information system）. This paper presents another effective second attribute algorithm for facilitating the use of the second attribute theorem, with computation on the tree expression of an information system. The time complexity of the new algorithm is linear in n. This algorithm is proved to be equivalent to the algorithm on the discernibility matrix.     

Interpolation by geometric algorithm

We present a novel geometric algorithm to construct a smooth surface that interpolates a triangular or a quadrilateral mesh of arbitrary topological type formed by n vertices. Although our method can be applied to B-spline surfaces and subdivision surfaces of all kinds, we illustrate our algorithm focusing on Loop subdivision surfaces as most of the meshes are in triangular form. We start our algorithm by assuming that the given triangular mesh is a control net of a Loop subdivision surface. The control points are iteratively updated globally by a simple local point-surface distance computation and an offsetting procedure without solving a linear system. The complexity of our algorithm is O(mn) where n is the number of vertices and m is the number of iterations. The number of iterations m depends on the fineness of the mesh and accuracy required.     

Detection of rotational and involutional symmetries and congruity of polyhedra

We propose a simple and efficient general algorithm for determining both rotational and involutional symmetries of polyhedra. It requiresO(m ²) time and usesO(m) space, wherem is the number of edges of the polyhedron. As this is the lower bound of the symmetry detection problem for the considered output form, our algorithm is optimal. We show that a slight modification of our symmetry detection algorithm can be used to solve the related conguity problem of polyhedra.     

On the constructions of constant-composition codes from perfect nonlinear functions

A new construction of constant-composition codes based on all known perfect nonlinear functions from F _q m to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F _q m to itself. It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1. Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al. in 2005, are equivalent to two special types of the new constant-composition codes. Supported in part by the National Natural Science Foundation of China (Grant Nos. 60573028, 60803156), the Open Research Fund of the National Mobile Communications Research Laboratory of Southeast University (Grant No. W200805) and in part by Singapore Ministry of Education Academic Research Fund (Grant No. T206B2204)     

Exact Algorithms for MAX-SAT

The maximum satisfiability problem (MAX-SAT) is stated as follows: Given a boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2ⁿ “steps”? Here, n is the number of distinct variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that a popular algorithm based on branch-and-bound is bounded by O(2ⁿ) in time, where n is the maximum number of occurrences of any variable in the input.When the input formula is in 2-CNF, that is, each clause has at most two literals, MAX-SAT becomes MAX-2-SAT and the decision version of MAX-2-SAT is still NP-complete. The best bound of the known algorithms for MAX-2-SAT is O(m2^m/5), where m is the number of clauses. We propose an efficient decision algorithm for MAX-2-SAT whose time complexity is bound by O(n2ⁿ). This result is substantially better than the previously known results. Experimental results also show that our algorithm outperforms any algorithm we know on MAX-2-SAT.     

An optimal approximation algorithm for the rectilinearm-center problem

Given a set ofn points on the plane, the rectilinearm-center problem is to findn rectilinear squares covering all thesen points such that the maximum side length of these squares is minimized. In this paper we prove that there is no polynomial-time algorithm with an error ratio < 2 for the rectilinearm-center problem unless NP = P. A polynomial-time approximation algorithm with an error ratio of 2 is also proposed.This research work was partially supported by a grant from the National Science Council of the Republic of China under Grant NSC-77-0408-E007-03.     

Utilization Bounds for Multiprocessor Rate-Monotonic Scheduling

In this paper, we extend Liu and Layland's utilization bound for fixed priority scheduling on uniprocessors to homogeneous multiprocessor systems under a partitioning strategy. Assuming that tasks are pre-emptively scheduled on each processor according to fixed priorities assigned by the Rate-Monotonic policy, and allocated to processors by the First Fit algorithm, we prove that the utilization bound is (n–1)(2^1/2–1)+(m–n+1)(2^1/(m–n+1)–1), where m and n are the number of tasks and processors, respectively. This bound is valid for arbitrary utilization factors. Moreover, if all the tasks have utilization factors under a value , the previous bound is raised and the new utilization bound considering is calculated. Finally, simulation provides the average-case behavior.     

Randomized competitive algorithms for the list update problem

We prove upper and lower bounds on the competitiveness of randomized algorithms for the list update problem of Sleator and Tarjan. We give a simple and elegant randomized algorithm that is more competitive than the best previous randomized algorithm due to Irani. Our algorithm uses randomness only during an initialization phase, and from then on runs completely deterministically. It is the first randomized competitive algorithm with this property to beat the deterministic lower bound. We generalize our approach to a model in which access costs are fixed but update costs are scaled by an arbitrary constantd. We prove lower bounds for deterministic list update algorithms and for randomized algorithms against oblivious and adaptive on-line adversaries. In particular, we show that for this problem adaptive on-line and adaptive off-line adversaries are equally powerful.A preliminary version of these results appeared in a joint paper with S. Irani in theProceedings of the 2nd Symposium on Discrete Algorithms, 1991 [17].This research was partially supported by NSF Grants CCR-8808949 and CCR-8958528.This research was partially supported by NSF Grant CCR-9009753.This research was supported in part by the National Science Foundation under Grant CCR-8658139, by DIMACS, a National Science Foundation Science and Technology center, Grant No. NSF-STC88-09648.     

An efficient query learning algorithm for ordered binary decision diagrams

In this paper, we propose a new algorithm that exactly learns ordered binary decision diagrams (OBDDs) with a given variable ordering via equivalence and membership queries. Our algorithm uses at most n equivalence queries and at most 2n (log₂ m + 3n) membership queries, where n is the number of nodes in the target-reduced OBDD and m is the number of variables. The upper bound on the number of membership queries is smaller by a factor of O(m) compared with that for the previous best known algorithm proposed by [R. Gavaldà, D. Guijarro, Learning Ordered Binary Decision Diagrams, Proceedings of the 6th International Workshop on Algorithmic Learning Theory, 1995, pp. 228–238].     

Near-optimal message routing and broadcasting in faulty hypercubes

A distributed routing algorithm for faulty hypercubes is described. This algorithm uses a directed depth-first approach to find a path between the sender and receiver of a message whenever at least one non-faulty path exists. We show that, when an arbitrary number of elements of the hypercube can be faulty, the algorithm always routes messages using fewer than 2N hops, whereN is the number of nodes in the hypercube. This performance is shown to be within a factor of two of the optimal worst-case routing efficiency. Through foult simulations, we show that, even when up to half of the elements in the cube are faulty, complete the analysis, we prove that our algorithm is deadlock-free. Finally, we present two extensions of the algorithm. The first uses local storage to reduce the overhead of the algorithm while the second allows reliable broadcasting in the presence of an arbitrary number of faults.Supported in part by the National Science Foundation under Grant CCR-9010547.Supported in part by the National Science Foundation Instrumentation Grant CDA-8820627.