Linear Time Solution Research Articles

Reproducing works of Calder

Many fine art pieces have been reproduced in digital form. The digital reproductions have been used to store and transmit the original work. In contrast, mobiles, or moving sculptures, such as those designed by Alexander Calder cannot be reproduced realistically by photographs and/or static images. The real characteristics of mobiles come from the motions generated by interactive external forces applied to their structures. Hence people could not fully enjoy them through static images or even static three-dimensional models. We present a virtual mobile system where users can easily control the mobile and can feel the impressions that the artist originally intended to provide. Virtual winds are generated by blowing on a microphone which then exert external forces to the mobile. This microphone interface lets users control the mobile while they are watching it through a monitor. We introduce a linear time solution for the constraint dynamics and an improved impulse dynamics to speed up the simulation. Using these techniques, we achieve a real-time simulation of the mobile on personal computers. The techniques presented can easily be extended to simulate other interactive dynamics systems. Copyright ©2001 John Wiley & Sons, Ltd.

Reproducing works of Calder

AbstractMany fine art pieces have been reproduced in digital form. The digital reproductions have been used to store and transmit the original work. In contrast, mobiles, or moving sculptures, such as those designed by Alexander Calder cannot be reproduced realistically by photographs and/or static images. The real characteristics of mobiles come from the motions generated by interactive external forces applied to their structures. Hence people could not fully enjoy them through static images or even static three‐dimensional models. We present a virtual mobile system where users can easily control the mobile and can feel the impressions that the artist originally intended to provide. Virtual winds are generated by blowing on a microphone which then exert external forces to the mobile. This microphone interface lets users control the mobile while they are watching it through a monitor. We introduce a linear time solution for the constraint dynamics and an improved impulse dynamics to speed up the simulation. Using these techniques, we achieve a real‐time simulation of the mobile on personal computers. The techniques presented can easily be extended to simulate other interactive dynamics systems. Copyright ©2001 John Wiley & Sons, Ltd.

Connectivity of Planar Graphs

We give here three simple linear time algorithms on planar graphs: a 4-connexity test for maximal planar graphs, an algorithm enumerating the triangles and a 3-connexity test. Although all these problems got already linear-time solutions, the presented algorithms are both simple and efficient. They are based on some new theoretical results.

An optimal algorithm for the construction of the system dependence graph

Program slicing can be used to aid in a variety of software maintenance activities including code understanding, code testing, debugging, and program reengineering. Program slicing (as well as other program analysis functions including forward slicing) can be efficiently performed on an internal program representation called a system dependence graph (SDG). The construction of the SDG depends primarily on the calculation of the transitive dependences which in turn depends in the calculation of the data dependences. In this paper we demonstrate the correctness and the optimality (with respect to the number of iterations required) of our method of calculating the transitive data dependences. We make a worst-case analysis of our algorithm, and find that is faster than the Horwitz, Reps, and Binkley (HRB) algorithm, and comparable to the Reps, Horwitz, Sagiv, and Rosay (RHSR) algorithm. Furthermore, our method requires neither the (explicit) calculation of the GMOD and GREF sets nor the construction of a linkage grammar and the corresponding subordinate characteristic graphs of the linkage grammar’s non-terminals. Unlike both the HRB and RHSR algorithms, our algorithm treats recursive procedures separately, permitting a faster (linear time) solution of non-recursive procedures. Additionally, a beneficial side effect of this method is that it provides us with a new method for performing interprocedural, flow-sensitive data flow analysis.

A system to place observers on a polyhedral terrain in polynomial time

The Art Gallery Problem deals with determining the number of observers necessary to cover an art gallery room such that every point is seen by at least one observer. This problem is well known and has a linear time solution for the 2D case, but little is known in the 3D case. In this paper we present a polynomial time solution for the 3D version of the Art Gallery Problem. Because the problem is NP-hard, the solution presented is an approximation, and we present the bounds to our solution. The solution uses techniques from: (i) computational geometry to first build a terrain hierarchy keeping the overall terrain's shape and to compute the visibility map for each observer; (ii) graph coloring to make a first placement of observers on the terrain; and (iii) set coverage to reduce the number of observers based on their visibility map. A complexity analysis is presented for each step and an analysis of the overall quality of the solution is given.

Parallel Implementations of the Selection Problem: A Case Study

The selection problem has been studied extensively on sequential machines. A linear average time solution and a linear worst-case solution are considered as the standard by most researchers. Theoretical work is also available on parallel models, but it has not been widely implemented on parallel machines. This paper presents an in-depth analysis of the implementation of the standard algorithms, on a number of multiprocessors and supercomputers from the entire spectrum of Flynn's classification, using both an imperative (C based languages with vendor specific parallel extensions) and a functional (SISAL) language. Very interesting results were obtained for all of the experiments performed, leading us to the conclusion that the selection problem has very efficient parallel implementations. Hand-tuned C programs with parallel extensions provided good efficiency but were time-consuming in terms of development. On the other hand, the SISAL code is fully portable and the same program was used on all the machines. The performances of SISAL implementations were comparable to the ones of the hand-tuned C implementations. On all the tests, the routines were able to sustain good speed-up and reasonable efficiency, even with a large number of processors. In two cases (one machine using SISAL, and one using a C-based language), we were able to obtain an efficiency higher than 80% with a configuration close or equal to the maximum number of processors.

A unified linear-time temporal logic solution to the steam-boiler control specification problem

The TLL XYZ/E is a formal language able to represent the dynamic semantics and the static semantics in a unified framework. It supports the whole process of program development, i. e. from the abstract specification to the efficiently executable program in a formal, precise and convenient way. the steam boiler control specification problem, a large case study in the fields of real time, hybrid and communication systems, is discussed with XYZ/E. The approach covers physical model construction, formal specification, stepwise refinement, verification, executable program and visual user interface programming.

Cartographic line simplification and polygon CSG formulæ in [formula omitted] time

A constructive solid geometry (CSG) conversion for a polygon takes a list of vertices and produces a formula representing the polygon as an intersection and union of primitive halfspaces. The cartographers' favorite line simplification algorithm recursively selects from a list of data points those to be used to represent a linear feature, such as a coastline, on a map. By using a data structure that maintains convex hulls of polygonal lines under splits, both were known to have O( n log n) time solutions in the worst-case. This paper shows that both are easier than sorting by presenting an O(n log ∗ n) algorithm for maintaining convex hulls under splits at extreme points. It opens the question of whether there are practical, linear-time solutions to these problems.

Single-layer channel routing and placement with single-sided nets

This paper considers the optimal offset, feasible offset, and optimal placement problems for a more general form of single-layer VLSI channel routing than has usually been considered in the past. Most prior works require that every net has exactly one terminal on each side of the channel. As long as only one side of the channel contains multiple terminals of the same net, we provide linear-time solutions to all three problems. Such results are implausible, if the placement of terminals is entirely unrestricted; in fact, the size of the output for the feasible offset problem may be ω( n 2). The linear-time results also hold with a ragged boundary on the side of the channel with multiple connections to the same net.

Monte Carlo radiosity

The fast radiosity-type methods for very complex diffuse environments, introduced herein, present a nearly linear-time solution. The outlined procedures rely on recursive algorithms with stochastic convergence for solving the radiosity equation system. Approximations of gathering and shooting at very low computational cost—rather than the exact matrix of a single reflection—are used. The efficiency of the methods will be increased by applying variance reduction techniques.

Efficient decoding of compressed data

In this article, we discuss the problem of enhancing the speed of Huffman decoding. One viable solution to this problem is the multibit scheme which uses the concept of k-bit trees to decode up to k bits at a time. A linear-time optimal solution for the mapping of 2-bit trees into memory is presented. The optimal solution is derived by formulating the memory mapping problem as a binary string mapping problem and observing that at most four different 4-bit patterns can occur within any 2-bit Huffman tree. In addition to reducing the processing time of decoding, the optimal scheme is storage efficient, does not require changes to the encoding process, and is suitable for hardware implementations

Serial and Parallel Algorithms for ( k,2)-Partite Graphs

We introduce a class of layered graphs which we call ( k,2)-partite and which we argue are an interesting class because of several important applications. We show that testing for ( k,2)-partiteness can be done efficiently both on sequential and parallel machines, by showing that membership is in NSPACE(log n) and in NC 2. We show that ( k,2)-partite graphs have bounded path width. We then show that a particular NP-complete problem, namely Maximum Independent Set, is solvable in linear time on bounded pathwidth graphs if the path decomposition is included in the input. Finally, we show that the Maximum Independent Set problem is in NC 2 for ( k,2)-partite graphs. We note that linear time solutions for certain NP-complete problems have been shown for a wider class of graphs, namely partial k-trees. Our linear time algorithm is somewhat simpler in structure. We conjecture that our techniques can be used on many NP-complete problems to yield efficient algorithms for ( k,2)-partite graphs.

Deterministic Dcomposition of Recursive Graph Classes

The popular class of series-parallel graphs can be built recursively from single edges by combining smaller components via connections only at a fixed pair of vertices called terminals. This recursive construction property with a limited number of terminals is essential to the linear time solution of problems on these graphs. A second useful property of these graphs is that decomposition is deterministic with respect to the series-parallel rules. This implies that the parse-tree of decomposition (which is required by the algorithms) can be determined in a straightforward manner by repeatedly applying the decomposition rules. Subject to retaining these properties, we examine how far the series-parallel graphs can be generalized. Corollaries of our results yield the deterministic decomposition of the series-parallel and Halin graph classes.

A linear time algorithm for finding all farthest neighbors in a convex polygon

Aggarwal et al. [A. Aggarwal, M.M. Klawe, S. Moran, P. Shor and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica 2 (1987) 209–233] showed how to compute in O( n) time one farthest neighbor for every vertex of a convex n-gon. In this note we extend this result to obtain a linear time algorithm for finding all farthest neighbors for every vertex of a convex polygon. Our algorithm yields a linear time solution to the symmetric all-farthest neighbors problem for simple polygons, thereby settling an open question raised by Toussaint in 1983 [G.T. Toussaint, The symmetric all-farthest neighbor problem, Comp. Math. Appl. 9 (6) (1983) 747–753.]

Mechanical translation of set theoretic problem specifications into efficient RAM code-A case study

This paper illustrates a fully automatic top-down approach to program development in which formal problem specifications are mechanically translated into efficient RAM code. This code is guaranteed to be totally correct and an upper bound on its worst case asymptotic running time is automatically determined. The user is only required to supply the system with a formal problem specification, and is relieved of all responsibilities in the rest of the program development process. These results are obtained, in part, by greatly restricting the system to handle a class of determinate, set theoretic, tractable problems. The most essential transformational techniques that are used are fixed point iteration, finite differencing, and data structure selection. Rudimentary forms of these techniques have been implemented and used effectively in the RAPTS transformational programming system. This paper explains the conceptual underpinnings of our approach by considering the problem of attribute closure for relational databases and systematically deriving a program that implements a linear time solution.

On BF-orderable graphs

We introduce BF-orderable graphs as a generalization of acyclic graphs. BF-orderable graphs permit a linear time solution to the single source shortest path problem. We give a graph-theoretic characterization of BF-orderable graphs by forbidden subgraphs.

Minimizing the density of terminal assignments in layout design

We present a linear-time solution to the problem of assigning each of n given entry terminals positioned at the upper row of a channel to one of m given exit terminals positioned at the lower row, so as to minimize the density of the resulting channel routine problem.

A linear time solution to the single function coarsest partition problem

The problem of finding the coarsest partition of a set S with respect to another partition of S one or more functions on S has several applications, one of which is the state minimization of finite state automata. In 1971, Hopcroft presented an algorithm to solve the many function coarsest partition problem for sets of n elements in O( n log n) time and O( n) space. In 1974, Aho, Hopcroft and Ullman presented an O( n log n) algorithm that solves the special case of this problem for only one function. Both these algorithms use a negative strategy that repeatedly refines the original partition until a solution is found. We present a new algorithm to solve the single function coarsest partition problem in O( n) time and space using a different, constructive approach. Our algorithm can be applied to the automated manufacturing of woven fabric.

A linear time solution to the single function coarsest partition problem

A linear time solution to the single function coarsest partition problem