Showing papers on "Disjoint sets published in 1998"

Convexity of quadratic transformations and its use in control and optimization

Abstract: Quadratic transformations have the hidden convexity property which allows one to deal with them as if they were convex functions. This phenomenon was encountered in various optimization and control problems, but it was not always recognized as consequence of some general property. We present a theory on convexity and closedness of a 3D quadratic image of ℝn, n≥3, which explains many disjoint known results and provides some new ones.

Utilization Bounds for N-Processor Rate MonotoneScheduling with Static Processor Assignment

Abstract: We consider the schedulability of a set of independent periodic tasks under fixed priority preemptive scheduling on homogeneous multiprocessor systems. Assuming there is no task migration between processors and each processor schedules tasks preemptively according to fixed priorities assigned by the Rate Monotonic policy, the scheduling problem reduces to assigning the set of tasks to disjoint processors in such a way that the Monotonic policy, the scheduling problem reduces to assigning the set of tasks to disjoint processors in such a way that the schedulability of the tasks on each processor can be guaranteed. In this paper we show that the worst case achievable utilization for such systems is between n(2^{1/2}-1) and (n+1)/(1+2^{1/(n+1)}), where n stands for the number of processors. The lower bound represents 41 percent of the total system capacity and the upper bound represents 50 to 66 percent depending on n. Practicality of the lower bound is demonstrated by proving it can be achieved using a First Fit scheduling algorithm.

Some consequences of cryptographical conjectures for S 1 2 and EF

Abstract: We show that there is a pair of disjoint NP sets, whose disjointness is provable in S 2 1 and which cannot be separated by a set in P/poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 2 1 admitting an NP -definition of primes about which it can prove that no number satisfying the definition is composite.

On Local Search for Weighted K -Set Packing

Abstract: Given a collection of sets of cardinality at most k, with weights for each set, the maximum weighted packing problem is that of finding a collection of disjoint sets of maximum total weight. We study the worst case behavior of the t-local search heuristic for this problem proving a tight bound of k - 1 + 1/t . As a consequence, for any given r < 1/(k -1) we can compute in polynomial time a solution whose weight is at least r times the optimal.,

Computer system and process for training of analytical models using large data sets

Abstract: A database often contains sparse, i.e., under-represented, conditions which might be not represented in a training data set for training an analytical model if the training data set is created by stratified sampling. Sparse conditions may be represented in a training set by using a data set which includes essentially all of the data in a database, without stratified sampling. A series of samples, or “windows,” are used to select portions of the large data set for phases of training. In general, the first window of data should be a reasonably broad sample of the data. After the model is initially trained using a first window of data, subsequent windows are used to retrain the model. For some model types, the model is modified in order to provide it with some retention of training obtained using previous windows of data. Neural networks and Kohonen networks may be used without modification. Models such as probabilistic neural networks, generalized regression neural networks, Gaussian radial basis functions, decision trees, including K-D trees and neural trees, are modified to provide them with properties of memory to retain the effects of training with previous training data sets. Such a modification may be provided using clustering. is Parallel training models which partition the training data set into disjoint subsets are modified so that the partitioner is trained only on the first window of data, whereas subsequent windows are used to train the models to which the partitioner applies the data in parallel.

On the Existence of Disjoint Cycles in a Graph

Abstract: is a graph of order at least 3k with . Then G contains k vertex-disjoint cycles.

Method and system for mining long patterns from databases

Abstract: A method and apparatus for mining generally long patterns from a database of data records of items. An initial set C of candidates is first generated, each candidate c having disjoint sets of items c.head and c.tail. Frequent candidates from the set C are extracted and put into a set F, where the frequent candidates are those whose set {c.head ∪ c.tail} is an itemset having a minimum support. Non-frequent candidates in C are used to generate new candidates, which are added to the set C. After any candidates having a superset in the set Fare removed from C and F, the method steps are repeated on the new candidate set C, until C is empty. The candidates remaining in the working set Fare returned as the desired patterns.

A Tverberg-type result on multicolored simplices

Abstract: Let P1, P2,…, Pd+1 be pairwise disjoint n-element point sets in general position in d-space. It is shown that there exist a point O and suitable subsets Qi ⊆ Pi (i = 1, 2,…, d + 1) such that Qi ≥ cdPi, an every d-dimensional simplex with exactly one vertex in each Qi contains Q in its interior. Here cd is a positive constant depending only on d.

Kinetic binary space partitions for intersecting segments and disjoint triangles

Abstract: We describe randomized algorithms for e ciently maintaining a binary space partition of continuously moving, possibly intersecting, line segments in the plane, and of continuously moving but disjoint triangles in space. Our two-dimensional BSP has depth O(log n) and size O(n log n+ k) and can be constructed in expected O(n log n+ k log n) time, where k is the number of intersecting pairs. We can detect combinatorial changes to our BSP caused by the motion of the segments, and we can update our BSP in expectedO(log n) time per change. Our three-dimensional BSP has depth O(log n), size O(n log n+k), construction time O(n log n+k log n), and update time O(log n) (all expected), where k is the number of intersections between pairs of edges in the xyprojection of the triangles. Under reasonable assumptions about the motion of the segments or triangles, the expected number of number of combinatorial changes to either BSP is O(mn s(n)), where m is the number of moving objects and s(n) is the maximum length of an (n; s) Davenport-Schinzel sequence for some constant s.

Decision algorithms for unsplittable flow and the half-disjoint paths problem

Abstract: We consider t.he bounded unsplittable flow problem: given t,erminal pairs in a network, with associated real-valued demands in bhe range [0, 41, find a single flow path for each pair so that no more than 1 unit of demand is routed t.hrough any vertex. Thus, the setting is not directly comparable to that, of 6he classical disjoint paths problem (when all demands are equal to 1) we must deal with connect.ions having varied, real-valued amounts of demand, but we impose the boundedness restriction t.hat each connection can consume at most half the capacity of any vertex Our main result is a polynomial-time algorithm for t,he bounded unsplittable flow problem, in an arbitrary graph, when the number of terminal pairs is a ilxed constant. Our algorithm is conceptually much simpler than Robert,son and Seymour’s corresponding algorithm for t.he disjoint paths problem witch a constant number of terminal pairs; and we can decide the routability of a non-t,rivially super-constant number of terminal pairs (up t,o Q((log log n)2/15)) in polynomial time. We also obtain polynomial-time algorithms for several natural opt.imizat$ion problems derived from the bounded unsplitt,able flow problem, when the number of terminal pairs is sufficiently small, and algorithms with better bounds for the case of planar graphs. The results all carry over to problems involving edge capacities. Our approach makes use of several of the ideas underlying bhe Robert.son-Seymour algorithm, together with some new algorithmic components. The resu1t.s add to a growing body of work suggest*Department of Computer Science, Cornell University, Ithaca NY 14863. Email: ldeinber&s.comell,edu. Supported in part by an Alfred P. Sloan Research Fellowship and by NSF Faculty Early Career Development Award CCR-9701399. ing that versions of our boundedness restriction while often relatively mild from the point of view of the underlying motivation can have very interesting qualitative effects on the tractability of basic routing problems.

Detecting Parallelism in C Programs with Recursive Darta Structures

Abstract: In this paper we present techniques to detect three common patterns of parallelism in C programs that use recursive data structures. These patterns include, function calls that access disjoint sub-pieces of tree-like data structures, pointer-chasing loops that traverse list-like data structures, and array-based loops which operate on an array of pointers pointing to disjoint data structures. We design dependence tests using a family of three existing pointer analyses, namely points-to, connection and shape analyses, with special emphasis on shape analysis. To identify loop parallelism, we introduce special tests for detecting loop-carried dependences in the context of recursive data structures. We have implemented the tests in the framework of our McCAT C compiler, and we present some preliminary experimental results.

The time-dependent shortest pair of disjoint paths problem: Complexity, models, and algorithms

Abstract: In this paper, we examine complexity issues, models, and algorithms for the problem of finding a shortest pair of disjoint paths between two nodes of a network such that the total travel delay is minimized, given that the individual arc delays are time-dependent. Such disjoint paths address the issue of network vulnerability by prescribing alternate routes for traffic flows. Applications include the dispatching of duplicate packets of data to improve the reliability in communication networks and the diverting of traffic during congestion to reduce the chances of bottlenecks in transportation networks, in the presence of time-dependent variations in travel delays in the network. We prove that this problem, and many variations of it, are NP-hard and we develop a 0-1 linear programming model that can be used to solve this problem. This model can accommodate various degrees of disjointedness of the pair of paths, from complete to partial with respect to specific arcs. We also present some computational results obtained by solving the above models using CPLEX-MIP.

Dependent random graphs and spatial epidemics

Abstract: We extend certain exponential decay results of subcritical percolation to a class of locally dependent random graphs, introduced by Kuulasmaa as models for spatial epidemics on $\mathbb{Z}^d$. In these models, infected individuals eventually die (are removed) and are not replaced. We combine these results with certain continuity and rescaling arguments in order to improve our knowledge of the phase diagram of a modified epidemic model in which new susceptibles are born at some positive rate. In particular, we show that, throughout an intermediate phase where the infection rate lies between two certain critical values, no coexistence is possible for sufficiently small positive values of the recovery rate. This result provides a converse to results of Durrett and Neuhauser and Andjel and Schinazi. We show also that such an intermediate phase indeed exists for every $d \geq 1$ (i.e., that the two critical values mentioned above are distinct). An important technique is the general version of the BK inequality for disjoint occurrence, proved in 1994 by Reimer.

Lower Bounds for Transversal Covers

Abstract: A transversal cover is a set of gk points in k disjoint groups of size g and a collection of b transversal subsets, called blocks, such that any pair of points not contained in the same group appears in at least one block. A central question is to determine, for given g, the minimum possible b for fixed k, or, alternatively, the maximum k for fixed b. The case g=2 was investigated and completely solved by Sperner sperner:28, Renyi renyi:71, Katona katona:73, and Kleitman and Spencer kleitman:73. For arbitrary g, asymptotic results are known but little is understood for small values of k. Constructions exist but these only produce upper bounds on b. The present article is concerned with lower bounds on b. We develop three general lower bounds on b for fixedg and k. The first one is proved using one of the principal constructions brett:97a, the second comes from the study of intersecting set-systems, and the third is shown by a set packing argument. In addition, we investigate upper bounds on k for small fixed b. This proves useful to reduce or eliminate the gap between lower and upper bounds on b for some transversal covers with small k.

PAC learning intersections of halfspaces with membership queries

Abstract: A randomized learning algorithm {POLLY} is presented that efficiently learns intersections of s halfspaces in n dimensions, in time polynomial in both s and n . The learning protocol is the PAC (probably approximately correct) model of Valiant, augmented with membership queries. In particular, {POLLY} receives a set S of m = poly(n,s,1/e,1/δ) randomly generated points from an arbitrary distribution over the unit hypercube, and is told exactly which points are contained in, and which points are not contained in, the convex polyhedron P defined by the halfspaces. {POLLY} may also obtain the same information about points of its own choosing. It is shown that after poly(n , s , 1/e , 1/δ , log(1/d) ) time, the probability that {POLLY} fails to output a collection of s halfspaces with classification error at most e , is at most δ . Here, d is the minimum distance between the boundary of the target and those examples in S that are not lying on the boundary. The parameter log(1/d) can be bounded by the number of bits needed to encode the coefficients of the bounding hyperplanes and the coordinates of the sampled examples S . Moreover, {POLLY} can be extended to learn unions of k disjoint polyhedra with each polyhedron having at most s facets, in time poly(n , k , s , 1/e , 1/δ , log(1/d) , 1/γ ) where γ is the minimum distance between any two distinct polyhedra.

A Layout Adjustment Problem for Disjoint Rectangles Preserving Orthogonal Order

Abstract: For a given set of n rectangles place on a plane, we consider a problem of finding the minimum area layout of the rectangles that avoids intersections of the rectangles and preserves the orthogonal order. Misue et al. proposed an O(n2)-time heuristic algorithm for the problem. We first show that the corresponding decision problem for this problem is NP-complete. We also present an O(n2)-time heuristic algorithm for the problem that finds a layout with smaller area than Misue's.

Some new disjoint Golomb rulers

Abstract: Let a Golomb ruler be a set {a/sub i/} of integers so that all the differences a/sub i/-a/sub j/, i/spl ne/j, are distinct. Let H(I.J) be the smallest n such that there are I disjoint Golomb rulers each containing J elements chosen from {1,2,...n}. In 1990, Klove gave a table of bounds on H(I,J). In this correspondence we improve and extend this table with results found by computer search.

Piercing d-Intervals

Abstract: A (homogeneous) d-interval is a union of d closed intervals in the line Using topological methods, Tardos and Kaiser proved that for any finite collection of d-intervals that contains no k + 1 pairwise disjoint members, there is a set of O(dk) points that intersects each member of the collection Here we give a short, elementary proof of this result A (homogeneous) d-interval is a union of d closed intervals in the line Let H be a finite collection of d-intervals The transversal number τ(H) of H is the minimum number of points that intersect every member of H The matching number ν(H) of H is the maximum number of pairwise disjoint members of H Gyarfas and Lehel [3] proved that τ ≤ O(νd!) and Kaiser [4] proved that τ ≤ O(d2ν) His proof is topological, applies the Borsuk-Ulam theorem and extends and simplifies a result of Tardos [5] Here we give a very short, elementary proof of a similar estimate, using the method of [2] Theorem 1 Let H be a finite family of d-intervals containing no k + 1 pairwise disjoint members Then τ(H) ≤ 2d2k Proof Let H′ be any family of d-intervals obtained from H by possibly duplicating some of its members, and let n denote the cardinality of H′ Note that H′ contains no k + 1 pairwise disjoint members Therefore, by Turan’s Theorem, there are at least n(n − k)/(2k) unordered intersecting pairs of members of H′ Each such intersecting pair supplies at least 2 ordered pairs (p, I), where p is an end point of one of the intervals in a member of H′, I is a different member of H′, and p lies in I Since there are altogether at most 2dn possible choices for p, there is such a point that lies in at least n(n−k) k2dn members of H ′ besides the one in which it is an endpoint of an interval, showing that there is a point that lies in at least n 2dk of the members of H ′ This implies that for any rational weights on the members of H there is a point that lies in at least a fraction 1 2dk of the total weight By the min-max theorem it follows that there is a collection of m points so that each member of H contains at least m/(2dk) of them, and thus contains an interval that contains at least m/(2d2k) of the points Order the points from left to right, and take the set of all points whose rank in this ordering is divisible by dm/(2d2k)e This is a set of at most 2d2k points that intersects each member of H, completing the proof 2 Remarks It may be possible to improve the constant factor in the above proof Kaiser’s estimate is indeed better by roughly a factor of 2; τ(H) ≤ (d2 − d+ 1)ν(H) It will be interesting to decide if the quadratic dependence on d is indeed best possible Higher dimensional extensions are possible, using the techniques in [2], [1] ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Research supported in part by a USA-Israeli BSF grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University Email: noga@mathtauacil

Generalizing partial order and dynamic backtracking

Abstract: Recently, two new backtracking algorithms, dynamic backtracking (DB) and partial order dynamic backtracking (PDB) have been presented. These algorithms have the property to be additive on disjoint subproblems and yet use only polynomial space. Unlike DB, PDB only imposes a partial search order and therefore appears to have more freedom than DB to explore the search space. However, both algorithms are not directly comparable in terms of flexibility. In this paper we present new backtracking algorithms that are obtained by relaxing the ordering conditions of PDB. This gives them additional flexibility while still being additive on disjoint subproblems. In particular, we show that our algorithms generalize both DB and PDB.

Parametrization of the orbits of cubic surfaces

Abstract: The aim of the paper is the study of the orbits of the action of PGL4 on the space ℙ3 of the cubic surfaces of ℙ3, i.e., the classification of cubic surfaces up to projective motions. A varietyQ⊂ℙ19 is explicitely constructed as the union of 22 disjoint irreducible components which are either points or open subsets of linear spaces. More precisely, each orbit of the above action intersects one componentX ofQ in a finite number of points and the action of PGL4 restricted on each componentX is equivalent to the action of a finite groupG X onX which can be explicitely computed. Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms. The results are obtained with computational techniques and with the aid of some computer algebra systems like CoCoA, Macaulay and Maple.

Partitioning a sequence into few monotone subsequences

Abstract: In this paper we consider the problem of finding sets of long disjoint monotone subsequences of a sequence of \(n\) numbers. We give an algorithm that, after \(O(n \log n)\) preprocessing time, finds and deletes an increasing subsequence of size \(k\) (if it exists) in time \(O(n + k^2)\). Using this algorithm, it is possible to partition a sequence of \(n\) numbers into \(2 \lfloor \sqrt n \rfloor\) monotone subsequences in time \(O(n^{1.5})\). Our algorithm yields improvements for two applications: The first is constructing good splitters for a set of lines in the plane. Good splitters are useful for two dimensional simplex range searching. The second application is in VLSI, where we seek a partitioning of a given graph into subsets, commonly refered to as the pages of a book, where all the vertices can be placed on the spine of the book, and each subgraph is planar.

Splittings, Robustness, and Structure of Complete Sets

Abstract: We investigate the structure of EXP-complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. We study for various types of reductions the question of whether the set difference A-S for a hard set A and a sparse set S is still hard. We also address the question of which complete sets A can be split into sets A1 and A2 such that $A\equiv^P_r A_1\equiv^P_r A_2$ for reduction type r, i.e., which complete sets are mitotic. We obtain both positive and negative answers to these questions depending on the reduction type and the structure of the sparse set.

Learning Sub-classes of Monotone DNF on the Uniform Distribution

Abstract: In this paper, we give learning algorithms for two new subclass of DNF formulas: poly-disjoint One-read-once Monotone DNF; and Read-once Factorable Monotone DNF, which is a generalization of Read-once Monotone DNF formulas. Our result uses Fourier analysis to construct the terms of the target formula based on the Fourier coefficients corresponding to these terms. To facilitate this result, we give a novel theorem on the approximation of Read-once Factorable Monotone DNF formulas, in which we show that if a set of terms of the target formula have polynomially small mutually disjoint satisfying sets, then the set of terms can be approximated with small error by the greatest common factor of the set of terms. This approximation theorem may be of independent interest.