Worst-case Time Complexity Research Articles

On the complexity of coordinated display of multimedia objects

A multimedia presentation can be represented as a collection of objects with temporal constraints that define when the objects are rendered. The display of a presentation is termed coordinated when the display of its objects respects the pre-specified temporal constraints. Otherwise, the display might suffer from failures that translate into meaningless scenarios. For example, a chase scene between a dinosaur and a jeep becomes meaningless if the system fails to render the dinosaur when displaying the scene. A previous study (M.L. Escobar-Molano et al., IEEE Trans. Knowledge Data Eng. 8 (3) (1996) 508–511) introduced a resource scheduling technique that guarantees a coordinated display of a presentation for single-disk architectures. This technique minimizes both latency and memory requirements and has a worst case time complexity O(n lg n) . This paper extends the previous study to multi-disk architectures and demonstrates the following: (1) the computation of a resource schedule that supports a coordinated display and yields the minimum latency is NP-hard, and (2) the computation of the minimum memory requirements to support a coordinated display within a pre-specified latency is NP-hard.

Navigating through triangle meshes implemented as linear quadtrees

Techniques are presented for navigating between adjacent triangles of greater or equal size in a hierarchical triangle mesh where the triangles are obtained by a recursive quadtree-like subdivision of the underlying space into four equilateral triangles. These techniques are useful in a number of applications, including finite element analysis, ray tracing, and the modeling of spherical data. The operations are implemented in a manner analogous to that used in a quadtree representation of data on the two-dimensional plane where the underlying space is tessellated into a square mesh. A new technique is described for labeling the triangles, which is useful in implementing the quadtree triangle mesh as a linear quadtree (i.e., a pointer-less quadtree); the navigation can then take place in this linear quadtree. When the neighbors are of equal size, the algorithms have a worst-case constant time complexity. The algorithms are very efficient, as they make use of just a few bit manipulation operations, and can be implemented in hardware using just a few machine language instructions. The use of these techniques when modeling spherical data by projecting it onto the faces of a regular solid whose faces are equilateral triangles, which are represented as quadtree triangle meshes, is discussed in detail. The methods are applicable to the icosahedron, octahedron, and tetrahedron. The difference lies in the way transitions are made between the faces of the polyhedron. However, regardless of the type of polyhedron, the computational complexity of the methods is the same.

Fast parallel shrinking to residues

The problem of shrinking the connected components of binary images to small residues is addressed. A new parallel algorithm is presented with the lowest known worst case time complexity as measured by iteration counts, i.e., ?(nc+nr)/2? iterations where the foreground of the image is contained within a rectangular region of size ncxnr pixels. The new algorithm also demonstrates superior average iteration counts in comparison with other shrinking algorithms on test images. The new algorithm uses a fully parallel application of an operator with an 8 pixel support. It reduces all connected components in any two-dimensional (2D) binary image to certain small 1, 2, or 3 pixel residues. It is potentially useful for component counting, as a fundamental step in connected component labeling algorithms implemented with fine grain realizations in 2D mesh computing architectures, and in other applications requiring preservation of foreground topology (connectivity).

An output encoding problem and a solution technique

We present a new output-encoding problem as follows. We are given a specification table, such as a truth table or a finite state machine (FSM) state table, where some of the outputs are specified in terms of ones, zeroes, and don't cares, and others are specified symbolically. The number of bits for encoding the output symbols may also be specified. We have to determine a binary code for each symbol of the symbolically specified output column such that the total number of output functions to be implemented after encoding the symbolic outputs and compacting the output columns is minimum. In this paper, we develop an exact algorithm to solve the above problem, analyze the worst case time complexity of the algorithm, and present experimental data to validate the claim that our encoding strategy helps to reduce the area of a synthesized circuit. In addition, we have investigated the possibility of using simple logical combinations of the already specified output columns to facilitate further reduction in the number of output functions.

The two-machine stochastic flowshop problem with arbitrary processing time distributions

We treat the two-machine flowshop problem with the objective of minimizing the expected makespan when the jobs possess stochastic durations of arbitrary distributions. We make three contributions in this paper: (1) we propose an exact approach with exponential worst-case time complexity. We also propose approximations which are computationally modest in their requirements. Experimental results indicate that our procedure is within less than 1 % of the optimum; and (2) we provide a more elementary proof of the bounds on the project completion time based on the concepts of ‘control networks’; and (3) we extend the ‘reverse search’ procedure of Avis and Fukuda [1[ to the context of permutation schedules.

An O(n log n) Average Time Algorithm for Computing the Shortest Network under a Given Topology

In 1992 F. K. Hwang and J. F. Weng published an O(n2) time algorithm for computing the shortest network under a given full Steiner topology interconnecting n fixed points in the Euclidean plane. The Hwang—Weng algorithm can be used to improve substantially existing algorithms for the Steiner minimum tree problem because it reduces the number of different Steiner topologies to be considered dramatically. In this paper we present an improved Hwang—Weng algorithm. While the worst-case time complexity of our algorithm is still O(n2) , its average time complexity over all the full Steiner topologies interconnecting n fixed points is O (n log n ).

Intersection reporting on two collections of disjoint sets

In this paper we introduce a new quick-find data structure to represent disjoint sets, called binomial list, showing that it has a better single-operation worst case time complexity than any previously known approach in solving the set union and intersection problem, a variant of the classical disjoint set union problem.

Breaking through then3 barrier: Faster object type inference

Abadi and Cardelli [1] present a series of type systems for their object calculi, four of which are first-order. Palsberg [22] has shown how typability in each one of these systems can be decided in time O(n3) and space O(n2), where n is the size of an untyped object expression, using an algorithm based on dynamic transitive closure. In this paper we improve each one of the four type inference problems from O(n3) to the following time complexities: [see article pdf to view table] Furthermore, our algorithms improve the space complexity from O(n2) to O(n) in each case. The key ingredient that lets us “beat” the worst-case time and space complexity induced by general dynamic transitive closure or similar algorithmic methods is that object subtyping, in contrast to record subtyping, is invariant: an object type is a subtype of a “shorter” type with a subset of the method names if and only if the common components have equal types. © 1999 John Wiley & Sons, Inc.

A link trie structure of storing multiple attribute relationships for natural language dictionaries

Word relation is primitive knowledge and it is very useful for natural language processing systems. In the traditional systems, although each knowledge dictionary is constructed in separation, recent natural language applications become more complex by integrating the above multi-attribute relationships. This paper presents an efficient data structure by introducing a trie that can define the linkage among their leaves. The linkage enables us to share the basic words required for multi-attribute relationships. Theoretical observations show that the worst-case time complexity of retrieving multi-attribute relationships is a constant. From the simulation results, it is shown that the presented method is 1/3 smaller than the competitive methods.

A one-phase algorithm to detect distributed deadlocks in replicated databases

Replicated databases that use quorum-consensus algorithms to perform majority voting are prone to deadlocks. Due to the P-out-of-Q nature of quorum requests, deadlocks that arise are generalized deadlocks and are hard to detect. We present an efficient distributed algorithm to detect generalized deadlocks in replicated databases. The algorithm performs reduction of a distributed wait-for-graph (WFG) to determine the existence of a deadlock. If sufficient information to decide the reducibility of a node is not available at that node, the algorithm attempts reduction later in a lazy manner. We prove the correctness of the algorithm. The algorithm has a message complexity of 2e messages and a worst-case time complexity of 2d+2 hops, where e is the number of edges and d is the diameter of the WFG. The algorithm is shown to perform significantly better in both time and message complexity than the best known existing algorithms. We conjecture that this is an optimal algorithm, in time and message complexity, to detect generalized deadlocks if no transaction has complete knowledge of the topology of the WFG or the system and the deadlock detection is to be carried out in a distributed manner.

An Incremental Distributed Algorithm for Computing Biconnected Components in Dynamic Graphs

This paper describes a distributed algorithm for computing the biconnected components of a dynamically changing graph. Our algorithm has a worst-case communication complexity of O(b+c) messages for an edge insertion and O(b'+c) messages for an edge removal, and a worst-case time complexity of O(c) for both operations, where c is the maximum number of biconnected components in any of the connected components during the operation, b is the number of nodes in the biconnected component containing the new edge, and b' is the number of nodes in the biconnected component just before the deletion.

A fast retrieving algorithm of hierarchical relationships using trie structures

Case structures are useful for natural language systems, such as word selection of machine translation systems, query understanding of natural language interfaces, meaning disambiguation of sentences and context analyses and so on. The case slot is generally constrained by hierarchical concepts because they are simple knowledge representations. With growing hierarchical structures, they are deeper and the number of concepts to be corresponded to one word increases. From these reasons, it takes a lot of cost to determine whether a concept for a given word is a sub-concept for concepting the case slot or not. This paper presents a faster method to determine the hierarchical relationships by using trie structures. The worst-case time complexity of determining relationships by the presented method could be remarkably improved for the one of linear (or sequential) searching, which depends on the number of concepts in the slot. From the simulation result, it is shown that the presented algorithm is 6 to 30 times faster than linear searching, while keeping the smaller size of tries.

A Homological Characterization of Q-Matrices

A real square matrix M is said to be a Q-matrix if the linear complementarity problem (q, M) has a solution for every vector q. There is, as yet, no characterization of Q-matrices which makes it easy to determine whether or not a given matrix is Q. Ideas from topology, in particular degree theory, have previously been used to obtain sufficient conditions for when a matrix is Q. In this paper we will apply some other ideas from topology to give a homological characterization of Q-matrices. Continuing to borrow from topology, we define the nerve of a matrix which, along with our characterization, leads to an algorithm for checking whether or not a matrix is Q. This algorithm has smaller bounds on its worst-case time complexity than previous methods.

Lower Bounds for Dynamic Algebraic Problems

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form "change input xi to value v" or "what is the value of output yi?". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent<br />algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal Omega(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an <br />Omega(sqrt(n)) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and<br />Tate's O(sqrt(n log n)) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Omega(sqrt(n)) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting<br />of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an ((log n)2= log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields.

On Rectilinear Distance-Preserving Trees

Given a set of terminals on the plane N={s,ν1,…,νn}, with a source terminal s, a Rectilinear Distance-Preserving Tree (RDPT) T(V, E) is defined as a tree rooted at s, connecting all terminals in N. An RDPT has the property that the length of every source to sink path is equal to the rectilinear distance between that source and sink. A Min- Cost Rectilinear Distance-Preserving Tree (MRDPT) minimizes the total wire length while maintaining minimal source to sink linear delay, making it suitable for high performance interconnect applications.This paper studies problems in the construction of RDPTs, including the following contributions. A new exact algorithm for a restricted version of the problem in one quadrant with O(n2) time complexity is proposed. A novel heuristic algorithm, which uses optimally solvable sub-problems, is proposed for the problem in a single quadrant. The average and worst-case time complexity for the proposed heuristic algorithm are O(n3/2) and O(n3), respectively. A 2-approximation of the quadrant merging problem is proposed. The proposed algorithm has time complexity O(α2T(n)+α3) for any constant α > 1, where T(n) is the time complexity of the solution of the RDPT problem on one quadrant. This result improves over the best previous quadrant merging solution which has O(n2T(n)+n3) time complexity.We test our algorithms on randomly uniform point sets and compare our heuristic RDPT construction against a Minimum Cost Rectilinear Steiner (MRST) tree approximation algorithm. Our results show that RDPTs are competitive with Steiner trees in total wire-length when the number of terminals is less than 32. This result makes RDPTs suitable for VLSI routing applications. We also compare our algorithm to the Rao-Shor RDPT approximation algorithm obtaining improvements of up to 10% in total wirelength. These comparisons show that the algorithms proposed herein produce promising results.

Efficient algorithms for single- and two-layer linear placement of parallel graphs

This paper outlines an algorithm for optimum linear ordering (OLO) of a weighted parallel graph with O( n log k) worst-case time complexity, and O(n + k log( n k ) log k) expected-case time complexity, where n is the total number of nodes and k is the number of chains in the parallel graph. Next, the two-layer OLO problem is considered, where the goal is to place the nodes linearly in two routing layers minimizing the total wire length. The two-layer problem is shown to subsume the maxcut problem and a befitting heuristic algorithm is proposed. Experimental results on randomly generated samples show that the heuristic algorithm runs very fast and outputs optimum solutions in more than 90% instances.

MOF-tree: A spatial access method to manipulate multiple overlapping features

In this paper we investigate the manipulation of large sets of 2-dimensional data representing multiple overlapping features (e.g. semantically distinct overlays of a given region), and we present a new access method, the MOF-tree. We perform an analysis with respect to the storage requirements and a time analysis with respect to window query operations involving multiple features (e.g. to verify if a constraint defined on multiple overlays holds or not inside a certain region). We examine both the pointer-based as well as the pointerless MOF-tree representations, using as space complexity measure the number of bits used in main memory and the number of disk pages in secondary storage respectively. In particular, we show that the new structure is space competitive in the average case, both in the pointer version and in the linear version, with respect to multiple instances of a region quadtree and a linear quadtree respectively, where each instance represents a single feature. Concerning the time performance of the new structure, we analyze the class of window (range) queries, posed on the secondary memory implementation. We show that the I/O worst-case time complexity for processing a number of window queries in the given image space, is competitive with respect to multiple instances of a linear quadtree, as confirmed by experimental results. Finally, we show that the MOF-tree can efficiently support spatial join processing in a spatial DBMS.

Integer Isotone Optimization

Consider the following integer isotone optimization problem. Given an n-vector x find an n-vector y with integer components so as to minimize $\max\{w_j|x_j - y_j| : 1 \leq j \leq n\}$ subject to $y_1 \leq y_2 \leq \cdots \leq y_n$, where each weight $w_j > 0$. In this article, the dual of this problem is defined, a strong duality theorem is established, and the set of all optimal solutions is shown to be all monotonic integer vectors lying in a vector interval. In addition, algorithms are obtained for computation of optimal solutions having the worst-case time complexity $O(n^2)$, when $w_j$ are arbitrary, and $O(n)$, when $w_j = 1$ for all j. The problem considered is of isotonic regression type and has practical applications, for example, to estimation and curve fitting. It is also of independent mathematical interest. The problem and the results can be easily extended to a partially ordered set.

On Testing a Bivariate Polynomial for Analytic Reducibility

LetKbe an algebraically closed field of characteristic zero. We present an efficient algorithm for determining whether or not a given polynomialf(x,y) inK[x,y] is analytically reducible overKat the origin. The algorithm presented is based upon an informal method sketched by Kuo (1989) which is in turn derived from ideas of Abhyankar (1988). The presentation contained herein emphasises the proofs of the algorithm's correctness and termination, and is suitable for computer implementation. A polynomial worst case time complexity bound is proved for a partial version of the algorithm.

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( nlog n), 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 log n) worst-case time algorithm is presented. For the input sequences of size from 4 to 10 5, 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.