Visibility problems for orthogonal objects in two-or three-dimensions

LetP be a set ofl points in 3-space, and letF be a set ofm opaque rectangular faces in 3-space with sides parallel tox- ory-axis. We present anO(n logn) time andO(n) space algorithm for determining all points inP which are visible from a viewpoint at (0,0,), wheren=l+m. We also present anO(n logn+k) time andO(n) space algorithm for the hidden-line elimination problem for the orthogonal polyhedra together with a viewpoint at (0,0,), wheren is the number of vertices of the polyhedra andk is the number of edge intersections in the projection plane.     

Theoretical comparisons of search strategies in branch-and-bound algorithms

Four known search strategies used in branch-and-bound algorithms-heuristic search, depth-first search, best-bound search, and breadth-first search-are theoretically compared from the viewpoint of the performance of the resulting algorithms. Heuristic search includes the other three as special cases. Since heuristic search is determined by a heuristic functionh, we first investigate how the performance of the resulting algorithms depends onh. In particular, we show that heuristic search is stable in the sense that a slight change inh causes only a slight change in its performance. The best and the worst heurstic functions are clarified, and also discussed is how the heuristic functionh should be modified to obtain a branch-and-bound algorithm with an improved performance. Finally, properties and limitations of depth-first search, best-bound search, and breadth-first search viewed as special cases of heuristic search are considered. In particular, it is shown that the stability observed for heuristic search no longer holds for depth-first search.     

Thiele Rational Interpolation for the Numerical Computation of the Reversible Randles–Sevcik Function in Electrochemistry

This paper uses Thiele rational interpolation to derive a simple method for computing the Randles–Sevcik function ^1/2(x), with relative error at most 1.9 × 10^–5 for – < x < . We develop a piecewise approximation method for the numerical computation of ^1/2(x) on the union (–, –10) [–10, 10] (10, ). This approximation is particularly convenient to employ in electrochemical applications where four significant digits of accuracy are usually sufficient. Although this paper is primarily concerned with the approximation of the Randles–Sevcik function, some examples are included that illustrate how Thiele rational interpolation can be employed to generate useful approximations to other functions of interest in scientific work.     

On Steiner minimal trees withLp distance

LetL _p be the plane with the distanced _p (A ₁ ,A ₂ ) = (¦x ₁ –x ₂¦ ^p + ¦y₁ –y ₂¦^p)^/1p wherex _i andy _i are the cartesian coordinates of the pointA _i . LetP be a finite set of points inL _p . We consider Steiner minimal trees onP. It is proved that, for 1 <p < , each Steiner point is of degree exactly three. Define the Steiner ratio _p to be inf{L _s (P)/L _m (P)¦PL _p } whereL _s (P) andL _m (P) are lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. Hwang showed ₁ = 2/3. Chung and Graham proved ₂ > 0.842. We prove in this paper that {} = 2/3 and (2/2)_{12 p} 3/2 for anyp.This work was supported in part by the National Science Foundation of China and the President Foundation of Academia Sinica.     

Robust H ∞ Control of Linear Uncertain Systems: Its Limiting Performance

The concept of a H norm is introduced for linear control systems with uncertain bounded parameters. A procedure for determining the lower estimate of the minimal robust H norm in the linear state feedback class is designed. An example of a system in which the estimate coincides with the minimal robust norm is given.     

Chebyshev domain truncation is inferior to fourier domain truncation for solving problems on an infinite interval

Domain truncation is the simple strategy of solving problems ony [-, ] by using a large but finite computational interval, [– L, L] Sinceu(y) is not a periodic function, spectral methods have usually employed a basis of Chebyshev polynomials,T _n(y/L). In this note, we show that becauseu(±L) must be very, very small if domain truncation is to succeed, it is always more efficient to apply a Fourier expansion instead. Roughly speaking, it requires about 100 Chebyshev polynomials to achieve the same accuracy as 64 Fourier terms. The Fourier expansion of a rapidly decaying but nonperiodic function on a large interval is also a dramatic illustration of the care that is necessary in applying asymptotic coefficient analysis. The behavior of the Fourier coefficients in the limitn for fixed intervalL isnever relevant or significant in this application.     

Lower bounds for set intersection queries

We consider the followingset intersection reporting problem. We have a collection of initially empty sets and would like to process an intermixed sequence ofn updates (insertions into and deletions from individual sets) andq queries (reporting the intersection of two sets). We cast this problem in thearithmetic model of computation of Fredman [F1] and Yao [Ya2] and show that any algorithm that fits in this model must take time (q+nq) to process a sequence ofn updates andq queries, ignoring factors that are polynomial in logn. We also show that this bound is tight in this model of computation, again to within a polynomial in logn factor, improving upon a result of Yellin [Ye]. Furthermore, we consider the caseq=O(n) with an additional space restriction. We only allow the use ofm memory locations, wherem n^3/2. We show a tight bound of (n²/m^1/3) for a sequence ofn operations, again ignoring the polynomial in logn factors.     

The mixing properties of certain processes related to Markov chains

Summary Let {Y _n ,– < n < } be either a function of a stationary Markov chain with countable state space, or a finitary process in the sense of Heller [3]. The purpose of this note is to prove that if {Y _n ,– < n < } is mixing, then it is aK-shift. (Definitions will be given below.)IfT is a measure-preserving transformation on a probability space, then the following implications relevant to the present paper are known: (1)T is aK-shift T is (r + 1)-mixing T isr-mixing T is totally ergodic T is ergodic, and (2)T is ergodic T is totally ergodic T isr-mixing T is aK-shift.It is not known if the classes ofr-mixing and (r + 1)-mixing transformations are distinct. (1-mixing is also called mixing.) The results of this note then imply that for the classes of transformation that we are consideringr-mixing and (r + 1)-mixing are equivalent.This research was partially supported by NSF Grant GP-19660.     

Weighted heuristic anytime search: new schemes for optimization over graphical models

Weighted heuristic search (best-first or depth-first) refers to search with a heuristic function multiplied by a constant w [31]. The paper shows, for the first time, that for optimization queries in graphical models the weighted heuristic best-first and weighted heuristic depth-first branch and bound search schemes are competitive energy-minimization anytime optimization algorithms. Weighted heuristic best-first schemes were investigated for path-finding tasks. However, their potential for graphical models was ignored, possibly because of their memory costs and because the alternative depth-first branch and bound seemed very appropriate for bounded depth. The weighted heuristic depth-first search has not been studied for graphical models. We report on a significant empirical evaluation, demonstrating the potential of both weighted heuristic best-first search and weighted heuristic depth-first branch and bound algorithms as approximation anytime schemes (that have sub-optimality bounds) and compare against one of the best depth-first branch and bound solvers to date.     

Reduced-order controllers inH ∞-Optimal synthesis methods of the first kind

It has often been believed that inH -optimal synthesis methods, the resulting controllers have order much greater than the order of the plant. Recently, Limebeer and Hung [9] have shown that inH -optimal synthesis problems of the first kind both optimal as well as suboptimal controllers have order which is no greater than the order of the plant. However, those authors do not provide any explicit algorithm for computing these controllers. In this paper we derive an explicit procedure for computing suboptimal controllers in problems of the first kind. These controllers have order no greater than the order of the plant and can be computed by solving only two Riccati and two Lyapunov equations.This research was supported by the National Science Foundation under Grant No. ECS-8810-578.     

Enhancing fault-tolerance in rate-monotonic scheduling

In this paper, we address the problem of supporting timeliness and dependability at the level of task scheduling. We consider the problem of scheduling a set of tasks, each of which, for fault-tolerance purposes, has multiple versions, onto the minimum number of processors. On each individual processor, the tasks are guaranteed their deadlines by the Rate-Monotonic algorithm. A simple online allocation heuristic is proposed. It is proven thatN2.33N ₀+, whereN is the number of processors required to feasibly schedule a set of tasks by the heuristic,N ₀ is the minimum number of processors required to feasibly schedule the same set of tasks, and is the maximum redundancy degree a task can have. The bound is also shown to be a tight upper bound. The average-case performance of the heuristic is studied through simulation. It is shown that the heuristic performs surprisingly well on the average.     

Thek-server dual and loose competitiveness for paging

Weighted caching is a generalization ofpaging in which the cost to evict an item depends on the item. We study both of these problems as restrictions of the well-knownk-server problem, which involves moving servers in a graph in response to requests so as to minimize the distance traveled.We give a deterministic on-line strategy for weighted caching that, on any sequence of requests, given a cache holdingk items, incurs a cost within a factor ofk/(k–h+1) of the minimum cost possible given a cache holdingh items. The strategy generalizes least recently used, one of the best paging strategies in practice. The analysis is a primal-dual analysis, the first for an on-line problem, exploiting the linear programming structure of thek-server problem.We introduceloose competitiveness, motivated by Sleator and Tarjan's complaint [ST] that the standard competitive ratios for paging strategies are too high. Ak-server strategy isloosely c(k)-competitive if, for any sequence, foralmost all k, the cost incurred by the strategy withk serverseither is no more thanc(k) times the minimum costor is insignificant.We show that certain paging strategies (including least recently used, and first in first out) that arek-competitive in the standard model are looselyc(k)-competitive providedc(k)/Ink and bothk/c(k) andc(k) are nondecreasing. We show that the marking algorithm, a randomized paging strategy that is (Ink)-competitive in the standard model, is looselyc(k)-competitive providedk–2 In Ink and both 2 Ink–c(k) andc(k) are nondecreasing.This paper is the journal version of On-line Caching as Cache Size Varies, which appeared in theProceedings of the 2nd Annual ACM-SIAM Symposium on Discrete Algorithms (1991). Details beyond those in this paper may be found in Competitive Paging and Dual-Guided Algorithms for Weighted Caching and Matching, which is the author's thesis and is available as Technical Report CS-TR-348-91 from the Computer Science Department at Princeton University.This research was performed while the author was at the Computer Science Department, Princeton University, Princeton, NJ 08544, USA, and was supported by the Hertz Foundation.     

On a class of timer hybrid systems reducible to finite state automata

A hybrid timer system with different timer rates and idling timer feature operating in dense time is modeled as an event-driven nondeterministic automaton and it is shown that the system is weak bisimulation equivalent to a finite state nondeterministic automaton. Our original model is an event driven infinite state automaton as in Dill (1989) and an explicit representation for the bisimulation equivalent finite state automaton whose state set consists of an index set of active timers andn pairs of bounded nonnegative integers or the symbol + is derived wheren is the number of clocks. The reduced model is simpler than Dill's difference bound matrix model—and similar to the model used in Alur et al. (1990)—for the finite system automaton since all difference inequalities are represented by a single order vector of integers.     

Gain Scheduled Output Feedback Control Based on LTI Controller Interpolation that Preserves LPV H ∞ Performance

A new approach to the design of a gain scheduled output feedback controller based on linear time-invariant (LTI) controller interpolation that preserves H performance without a varying-parameter rate feedback is proposed. After the controller design is translated into parameterized linear matrix inequalities about parameter matrices based on a parameter-dependent Lyapunov function, the sufficient condition is given for the partition of the varying-parameter set based on the concept of H performance covering. The varying-parameter set is thus partitioned into sufficiently small subsets. After the constant matrices are found for each of the subsets, the required continuous parameter matrices are obtained by using interpolation. The proposed controller overcomes the drawback that the gain scheduled controller may not be found by using the existing gain scheduled linear parameter varying (LPV) controller synthesis. Moreover, the varying-parameter rate feedback is eliminated and the conservatism of the controller design is reduced by means of limiting the upper bound of the varying-parameter rate. Simulation and experiment results prove the effectiveness of the proposed controller.     

Central Controllers in Problems with H ∞-Criteria: Existence and Extreme Possibilities

A functional approach to the problems of existence of solutions of the matrix Riccati equations that are used to design controllers in the problems of control with H -criterion was presented. New existence conditions for these solutions were obtained. The lower bound of the criterion parameter characterizing the degree of perturbation damping was estimated in terms of the plant frequency characteristics.     

An optimal parallel algorithm for planar cycle separators

We present an optimal parallel algorithm for computing a cycle separator of ann-vertex embedded planar undirected graph inO(logn) time onn/logn processors. As a consequence, we also obtain an improved parallel algorithm for constructing a depth-first search tree rooted at any given vertex in a connected planar undirected graph in O(log² n) time on n/logn processors. The best previous algorithms for computing depth-first search trees and cycle separators achieved the same time complexities, but withn processors. Our algorithms run on a parallel random access machine that permits concurrent reads and concurrent writes in its shared memory and allows an arbitrary processor to succeed in case of a write conflict.A preliminary version of this paper appeared as Improved Parallel Depth-First Search in Undirected Planar Graphs in theProceedings of the Third Workshop on Algorithms and Data Structures, 1993, pp. 407–420.Supported in part by NSF Grant CCR-9101385.     

Multi-Step H∞ Generalized Predictive Control

An H generalized control law is derived which has properties in common with both the GH and GPC control laws. The stability and robustness properties are the same as for a GH optimal controller but a cost on future predictive error and control action is dealt with in the same manner as for the GPC or model based predictive control laws. This type of long range predictive control algorithm has stability and robustness problems which are much improved through the H formalism. The general two-block problem solution is considered first and this is then reduced to a novel one-block multi-step GH control law which is very attractive computationally. This latter algorithm has all the benefits of one-block H controllers and yet the equations can be written in a form where GPC type constrained solutions can be obtained.     

Horizontal axis wind turbine control: Comparison of classical,LQG andH ∞ designs

A comparison between classical, LQG andH design procedures is presented for the design of the power regulator in a fixed speed horizontal axis wind turbine. The relative merits of the different approaches are discussed regarding both the procedures and the resulting controllers.     

Directed explicit-state model checking in the validation of communication protocols

The success of model checking is largely based on its ability to efficiently locate errors in software designs. If an error is found, a model checker produces a trail that shows how the error state can be reached, which greatly facilitates debugging. However, while current model checkers find error states efficiently, the counterexamples are often unnecessarily lengthy, which hampers error explanation. This is due to the use of naive search algorithms in the state space exploration.In this paper we present approaches to the use of heuristic search algorithms in explicit-state model checking. We present the class of A^* directed search algorithms and propose heuristics together with bitstate compression techniques for the search of safety property violations. We achieve great reductions in the length of the error trails, and in some instances render problems analyzable by exploring a much smaller number of states than standard depth-first search. We then suggest an improvement of the nested depth-first search algorithm and show how it can be used together with A^* to improve the search for liveness property violations. Our approach to directed explicit-state model checking has been implemented in a tool set called HSF-SPIN. We provide experimental results from the protocol validation domain using HSF-SPIN.