Linear Complexity Algorithm Research Articles

Iterative Converging Algorithms for Computing Bounds on Durations of Activities in Pert and Pert-Like Models

Since its invention in 1958, Program Evaluation and Review Technique (PERT) has been widely used during the planning, design, and implementation of projects. Pert models the activities of a project as a single source-single sink directed acyclic graph where nodes represent events (end or beginning of activities) and arcs activities. The maximum amount by which an activity can be delayed without delaying the overall project is called the slack. Critical tasks have zero slack whereas all noncritical tasks have positive slacks. Pert is a valuable tool in the management of large projects since it allows to compute the slack of each activity of the project. Such information may be crucial in avoiding cost overruns that would be caused by delays to critical activities and/or excessive delays to noncritical activities. What Pert fails to provide is how one should go about distributing remaining slack on noncritical activities while taking into consideration properties of the activities as well as precedence relationships among them, so as to have reasonable upper bounds on duration of all activities, critical or noncritical. In this paper we propose several algorithms for the distribution of slack on non-critical activities. We show that if one desires to distribute the remaining slack proportionally to the initially assigned activity durations then the problem is in P, and propose an algorithm of linear time complexity. However if one desires to use distribution functions other than the initial durations of activities, then the problem of slack distribution becomes NP-complete. Finding the maximal bounds corresponding to zero-slack solution at the sink requires iterative application of exponential algorithm. For that case we introduce an approximation algorithm of linear time complexity on each iteration. The algorithm iteratively increases bounds on durations of activities and converges to the zero-slack solution on all paths from the source node to the sink node in the Pert-like graph. The algorithms described in this paper were successfully applied to solving timing bounds problems in VLSI design.

Integrated and hierarchical sortmap-relabel image segmentation methods

We propose two image segmentation algorithms: the inte- grated sortmap relabel with adjacent-region merging (ISARM), and the self-guided sortmap relabel with adjacent-region merging (SGSARM). Due to the integration of noise reduction and fast merging, ISARM pro- vides a 25% improvement in processing time, as compared to leading existing algorithms such as the region adjacency graph (RAG) algorithm, on a variety of test images. ISARM also provides better segmentation accuracy than the RAG algorithm, by a measure combining the mean squared error and the number of regions obtained. SGSARM is designed for use with large images (say 102431024 or larger). It incorporates two levels of processing: an edge detection algorithm of linear complexity, which is applied to large images to detect regions of interest (ROIs), followed by ISARM for finer segmentation of each ROI. SGSARM there- fore has significant advantages in speed and accuracy when used in large images. Simulation results are provided to demonstrate the perfor- mance of both algorithms. © 2002 Society of Photo-Optical Instrumentation Engi- neers. (DOI: 10.1117/1.1511246)

An Efficient Algorithm for Topologically Correct Segmentation of the Cortical Sheet in Anatomical MR Volumes

Polygon-mesh representations of the cortices of individual subjects are of anatomical interest, aid visualization of functional imaging data and provide important constraints for their statistical analysis. Due to noise and partial volume sampling, however, conventional segmentation methods rarely yield a voxel object whose outer boundary represents the folded cortical sheet without topological errors. These errors, called handles, have particularly deleterious effects when the polygon mesh constructed from the segmented voxel representation is inflated or flattened. So far handles had to be removed by cumbersome manual editing, or the computationally more expensive method of reconstruction by morphing had to be used, incorporating the a priori constraint of simple topology into the polygon-mesh model. Here we describe a linear time complexity algorithm that automatically detects and removes handles in presegmentations of the cortex obtained by conventional methods. The algorithm's modifications reflect the true structure of the cortical sheet. The core component of our method is a region growing process that starts deep inside the object, is prioritized by the distance-to-surface of the voxels considered for inclusion and is selftouching-sensitive, i.e., voxels whose inclusion would add a handle are never included. The result is a binary voxel object identical to the initial object except for “cuts” located in the thinnest part of each handle. By applying the same method to the inverse object, an alternative set of solutions is determined, correcting the errors by addition instead of deletion of voxels. For each handle separately, the solution more consistent with the intensities of the original anatomical MR scan is chosen. The accuracy of the resulting polygon-mesh reconstructions has been validated by visual inspection, by quantitative comparison to an expert's manual corrections, and by crossvalidation between reconstructions from different scans of the same subject's cortex.

An optimal algorithm for solving the 1-median problem on weighted 4-cactus graphs

The median problem has been extensively studied in the last three decades. On general graphs, the problem can be solved in O( n 3), where n is the number of vertices in a graph. For tree networks, however, more efficient algorithms can be devised to find the medians in O( n). In this paper, we study weighted 4-cactus (W4C) graphs which provide a less restrictive network structure than trees and show that median problem can be solved just as efficiently on W4C graphs as on trees. Many examples are provided demonstrating the rudiments of our algorithm and a linear time complexity algorithm is developed.

On a new class of structured matrices

In this paper we continue the study of structured matrices which admit a linear complexity inversion algorithm. The new class which is studied here appears naturally as the class of matrices of input output operators for discrete time dependent descriptor linear systems. The algebra of such operators is analyzed. Multiplication and inversion algorithms of linear complexity are presented and their implementation is illustrated.

Linear-complexity algorithms for QoS support in input-queued switches with no speedup

We present several fast, practical linear-complexity scheduling algorithms that enable provision of various quality-of-service (QoS) guarantees in an input-queued switch with no speedup. Specifically, our algorithms provide per-virtual-circuit transmission rate and cell delay guarantees using a credit-based bandwidth reservation scheme. Our algorithms also provide approximate max-min fair sharing of unreserved switch capacity. The novelties of our algorithms derive from judicious choices of edge weights in a bipartite matching problem. The edge weights are certain functions of the amount and waiting times of queued cells and credits received by a virtual circuit. By using a linear-complexity variation of the well-known stable-marriage matching algorithm, we present theoretical proofs and demonstrate by simulations that the edge weights are bounded. This implies various QoS guarantees or contracts about bandwidth allocations and cell delays. Network management can then provide these contracts to the clients. We present several different algorithms of varied complexity and performance (as measured by the usefulness of each algorithm's contract). While most of this paper is devoted to the study of soft guarantees, a few hard guarantees can also be proved rigorously for some of our algorithms. As can be expected, the provable guarantees are weaker than the observed performance bounds in simulations. Although our algorithms are designed for switches with no speedup, we also derive upper bounds on the minimal buffer requirement in the output queues necessary to prevent buffer overflow when our algorithms are used in switches with speedup larger than one.

An Algorithm for the k-Error Linear Complexity of Sequences over GF(pm) with Period pn, p a Prime

An algorithm is given for the k-error linear complexity of sequences over GF(pm) with period pn, p a prime. The algorithm is derived by the generalized Games–Chan algorithm for the linear complexity of sequences over GF(pm) with period pn and by using the modified cost different from that used in the Stamp–Martin algorithm for sequences over GF(2) with period 2n. A method is also given for computing an error vector which gives the k-error linear complexity.

Linear complexity algorithm for semiseparable matrices

A new linear complexity algorithm for general nonsingular semiseparable matrices is presented. For symmetric matrices whose semiseparability rank equals to 1 this algorithm leads to an explicit formula for the inverse matrix.

Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices

Matrices represented as a sum of diagonal and semiseparable ones are considered here. These matrices belong to the class of structured matrices which arises in numerous applications. Fast O( N) algorithms for their inversion were developed before under additional restrictions which were a source of instability. Our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms. In this paper, the case of semiseparable matrices of order one is considered.

Path covering problems and testing of printed circuits

This work deals with a problem, arising in the context of printed circuit testing, that will be formulated as a path covering problem on trees. For this problem, which consists in minimizing the number of continuity testing, an exact algorithm of linear complexity is here proposed for the first time. A general classification of path covering problems is also presented and a brief review of the literature is given.

Language Stability and Stabilizability of Discrete Event Dynamical Systems

This paper studies the stability and stabilizability of discrete event dynamical systems (DEDSs) modeled by state machines. Stability and stabilizability are defined in terms of the behavior of the DEDSs, i.e. the language generated by the state machines (SMs). This generalizes earlier work where they were defined in terms of legal and illegal states rather than strings. The notion of reversal of languages is used to obtain algorithms for determining the stability and stabilizability of a given system. The notion of stability is then generalized to define the stability of infinite or sequential behavior of a DEDS modeled by a Büchi automaton. The relationship between the stability of finite and stability of infinite behavior is obtained and a test for stability of infinite behavior is obtained in terms of the test for stability of finite behavior. An algorithm of linear complexity for computing the regions of attraction is presented, which is used for determining the stability and stabilizability of a given system defined in terms of legal states. This algorithm is then used to obtain efficient tests for checking sufficient conditions for language stability and stabilizability.

An algorithm for the k-error linear complexity of binary sequences with period 2/sup n/

Certain applications require pseudo-random sequences which are unpredictable in the sense that recovering more of the sequence from a short segment must be computationally infeasible. It is shown that linear complexity is useful in determining such sequences. A generalized linear complexity that has application to the security of stream ciphers is proposed, and an efficient algorithm is given for the case in which the sequence is binary with period 2/sup n/. This algorithm generalizes an algorithm presented by R.A. Games and A.H. Chan (1983). >

Stability of Discrete Event System Behavior

Abstract This paper studies the stability and stabilizability of Discrete Event Dynamical Systems (DEDS’s) in the framework of Ramadge and Wonham. We define stability and stabilizability in terms of the behavior of the DEDS’s, i. e. the language generated by the state machines (SM’s) used to model the DEDS. The notions of stability and stabilizability of DEDS’s are presented in a more general setting than in earlier works where they were defined in terms of legal and illegal states rather than strings. The notion of revarsal of languages is used to obtain algorithms for determining the stability and stabilizability of a given system. We also present an algorithm of linear complexity for computing the regions of attraction which is used for determining the stability and stabilizability of a given system defined in terms of legal states. This algorithm is then used to obtain more efficient tests of sufficiency of stability and stabilizability of systems defined in terms of languages.

New algorithm for testing random access memories

A linear time complexity algorithm to test RAMs is presented. The algorithm requires only 7n read/write operations and provides an extensive fault coverage. The main advantage of the proposed scheme is that a small test time can be achieved.

Properties and performance bounds for closed free choice synchronized monoclass queueing networks

It is shown that many monoclass queuing networks (QN) with synchronizations can naturally be modeled with a subclass of Petri nets (PN) called free-choice nets (FC), for which a wide gamut of qualitative behavioral and structural results have been derived. Some of these net theoretic results are used to characterize the ergodicity, boundedness, and liveness of closed free-choice synchronized QNs. Upper and lower throughput bounds are defined based on the mean value of the service times, without any assumption on the probability distributions (thus including both the deterministic and the stochastic cases). It is shown that monotonicity properties exist between the throughput bounds and the parameters of the model in terms of population and service times. Proposed are (theoretically polynomial and practically linear complexity) algorithms for the computation of these bounds, based on linear programming problems defined on the incidence matrix of the underlying FC net. Using classical laws from queuing theory, bounds are provided for mean queue lengths and response time.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A note on locating a central vertex of a 3-cactus graph

We extend linear time algorithms for a variety of vertex location measures on trees to include 3-cactus graphs. To do this the original 3-cactus graph is transformed into a tree. The behavior of median and variance measures under this transformation is studied and linear (in the number of vertices) time complexity algorithms for the minimum weighted vertex variance problem and the vertex restricted stochastic queue median are developed.

Parallel testing for pattern-sensitive faults in semiconductor random-access memories

A design strategy is presented for efficient and comprehensive parallel testing of high-density, MOS random-access memories (RAMs). Parallel test algorithms for RAMs have been developed on the basis of this design-for-testability approach for a broad class of pattern-sensitive faults. Two algorithms which are significantly more efficient than previous approaches are examined. The first algorithm detects the static and dynamic pattern-sensitive faults over a neighborhood of five cells. The second algorithm tests the symmetric pattern-sensitive faults over a neighborhood of nine cells. It tests an n-bit RAM organized as a square root n* square root n array in 97 square root n memory cycles. The design-for-testability approach modifies the existing RAM architecture very little, so that it can be implemented very easily. The additional overhead is only about 2 square root n transistors. The low overhead allows high reliability, and the additional circuit for each bit line can fit within the 3 lambda -to-6 lambda pitch width in a high-density, single-transistor dynamic RAM. Although the algorithm is designed to detect pattern-sensitive faults, the modified architecture can be readily used to speed up other conventional algorithms of linear complexity by a factor of O( square root n).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Linear complexity algorithms for hierarchical routing

A hierarchical procedure for net routing on l*m*n grid-graphs, where l is the number of layers, m is the number of rows, and n is the number of columns, is presented. The hierarchy reduces the overall problem to a sequence of subproblems, where each subproblem works on an l*m'*n portion of the overall grid. For each subproblem, an initial constructive placement (CP) of as many nets as possible is used; then a generalized dynamic programming (DP) algorithm to solve the Steiner problem of an l*2*n grid-graph is used for any remaining nets. The CP procedure uses simplistic routing assumptions to route quickly as many nets as possible. If all nets are routed, the subproblem solution is (locally) optimal and the corresponding branch of the binary recursion tree generated by the hierarchy is pruned. For any unrouted nets, the generalized DP procedure is called to route each net, one at a time, with a run time complexity of O(K*n) where n is the number of columns and K is a function of the grid cross section and layer/wiring restrictions. There are no a priori layer restrictions or limitations on the number of layers used; three-layer cases and even some four-layer cases are feasible for the PYRAMID 90-X. >

Linear complexity parallel algorithm for discrete-time Wiener filters with optimum lag

A parallel algorithm of linear complexity for efficient updating of the solution of a Toeplitz system of equations with a shifted right hand side is presented. If in the initial system, the right-hand side is f=f(0), . . ., f(N), the whole computation will take 6N+1 multiplications/divisions and 3N+1 additions/subtractions per processor. >

A note on diagonal innovation matrices

By noting that a recently introduced class of so-called "diagonal innovation matrices" has a form previously encountered in state-space estimation problems, new linear complexity algorithms are presented for their factorization and inversion.