Worst-case Behavior Research Articles

A simple proof of the inequality MFFD(L)≤71/60 OPT(L) + 1,L for the MFFD bin-packing algorithm

In 1985, Johnson and Garey[4] devised an algorithm which they call MFFD. Compared with other modifications of the famous FFD algorithm, their is apparently simpler in practical applications and substantially improves the worst case behavior of FFD. In fact, they proved that the inequality MFFD(L)≤71/60OPT(L)+31/6 holds for all the listsL. Their proof requires 40 pages. In this paper we give a proof for the inequality MFFD(L)≤71/60OPT(L)+1, ∀L. The proof is much simpler than theirs.

Deterministic rule based traffic descriptors for broadband ISDN: Worst case behaviour and its impact on connection acceptance controls

AbstractDeterministic rule based traffic descriptors in general, and leaky bucket based (LBB) traffic descriptors in particular, offer many advantages over statistical descriptors for specifying and monitoring traffic in the ATM based broadband ISDN (BISDN). However, these traffic descriptors do not specify the traffic pattern uniquely. Thus, acceptance of a connection request with specific traffic descriptor parameter values implies that the network has agreed to guaranteeing the quality of service (QOS) for the worst case traffic compliant with the traffic descriptor (and hence allowed by the monitor). This poses challenges which have not been addressed adequately. Two of the important ones are: (i) definition of the worst case behaviour and characterization of the worst case traffic compliant with the traffic descriptor; (ii) connection acceptance controls (CAC) for the worst case traffic sources compliant with the traffic descriptors. In this paper we address these questions for leaky bucket based (LBB) traffic descriptors. The focus is on the first problem. The implications on the second problem are discussed briefly. The performance measure and QOS requirements are in terms of the cell loss ratios.We first define the worst case behaviour for homogeneous and heterogeneous environments in terms of collective cell loss ratio (CLR)) from the multiplexer. We consider both the unbuffered and buffered cases. However, most of the results here are for the unbuffered case. FOTthe case of an unbuffered multiplexer and homogeneous source environment, we show that a simple on‐off source represents the worst case behaviour of an LBB source. In a heterogeneous source traffic environment, the on‐off behaviour is the worst only in a limited sense and the use of a collective measure results in anomalous relationships between the source behaviour and the CLR. A more appropriate measure should be based on individual cell loss ratios or on the maximum of the individual CLRs. We analyse LBB sources with respect to these measures and unbuffered resources, and obtain further characterization of the worst case behaviour. However, we also uncover some counter‐intuitive relationships. In particular, a seemingly ‘smooth’ source may be worse for the other sources than a seemingly ‘bursty’ source.For a buffered multiplexer, we show, via an example, that a three‐state source may be worse than an on‐off source even in a homogeneous source traffic environment. A complete characterization of the worst case behaviour for a buffered multiplexer remains an open challenge. Moreover, the LBB sources exhibit deterministic behaviour and their equivalent bandwidth need not be sub‐additive in sources. That is, adding the equivalent bandwidths of individual sources to obtain an ‘estimate’ of the equivalent bandwidth for a collection of such sources could be too optimistic in some situations. A fresh look at the connection acceptance control for such sources is therefore warranted.

A look-ahead heuristic for scheduling jobs with release dates on a single machine

The paper explores how limited look-ahead improves the performance of on-line heuristics. In particular, we consider the NP-complete single machine scheduling of independent jobs with release dates to minimize the total completion time. We present an on-line with look-ahead algorithm which foresees the next in-coming job. We study its worst-case behavior and prove that it outperforms most on-line and off-line heuristics.

Bandwidth allocation and access control in high-speed networks

The problem of bandwidth allocation and access regulation arises in the congestion control of Broadband ISDN networks. This paper assumes that a single user, described by an on-off fluid model, is connected to the network via a leaky bucket access control mechanism. The bandwidth allocated to this user and the leaky bucket parameters are to be selected so as to guarantee a negotiated level of delay probability at the access point and packet loss probability in the network which is modelled as an output buffer. The design problem is to minimize the allocated bandwidth subject to service guarantees and stability conditions for the input and output buffers. We provide a desirable feasible solution to the design problem. The paper studies the effect of non-conforming users on the network performance using the leaky bucket access control corresponding to this feasible solution. We provide expressions that quantify the impact of the leaky bucket parameters in access regulation and the worst-case queueing behavior at the output buffer. Finally, we discuss the extension of this methodology to the multiple leaky buckets case.

An analysis of shift class design problems

In this paper we consider a generalization of the Fixed Job Schedule Problem (FJSP) which appears in the aircraft maintenance process at an airport. A number of jobs must be carried out where each job requires processing from a fixed time to a fixed finish time. These jobs must be carried out by a number of machines which are available in specific shifts only. The jobs must be carried out in a non-preemptive way, although at the end of a shift preemption of a job is allowed sometimes. The problem is to choose the number of machines in each of the shifts in such a way that all jobs can be carried out and that the total costs of the machines or the total number of machines are minimum. In this paper we present an analysis of the computational complexity of these problems. We also analyse the worst case behaviour of the preemptive variant versus the non-preemptive variant.

Schedulability-analyzable exception handling for fault-tolerant real-time languages

Exceptions add a significant element of robustness and fault-tolerance to an application. In real-time systems, however, we require not only semantic correctness, robustness, and fault-tolerance, but also timely behavior. For restricted languages, schedulability analysis can be made efficient through use of polynomial-time analyses and transformations. In this paper, we show how a reasonable level of exception handling can be added to such a language. Specifically, we present an exception mechanism which reasonably provides for (1) guaranteed schedulability, (2) fine granularity of exception handling, (3) possibility of non-termination of the process that raises the exception, (4) reuse of exception handlers within the program, (5) information hiding in exception handlers, (6) exceptions as first-class objects, (7) casting or binding of exceptions, and (8) deadline modification. Furthermore, the mechanism allows for easy specification of multi-mode operation, and permits program component assignment using expected behavior while enabling schedulability analysis of worst-case (often exceptional) behavior. While adding features to exception handler semantics can create new analysis problems while solving expressivity problems, we show that the exception handling features described above are desirable, that the complexity of static schedulability analysis in the presence of these features is manageable, and that existing language provides all these desirable features.

Stochastic and worst case system identification are not necessarily incompatible

Stochastic and worst case approaches to system identification are different and are usually treated separately. In this communique we investigate the effect that a projection operator has on the worst case behavior of estimates derived by stochastic identification algorithms. We show that under certain assumptions the projections of the stochastic estimates are convergent in the worst case setting. We illustrate this result by applying it to least squares and maximum likelihood algorithms.

A calculus for the random generation of labelled combinatorial structures

A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving set, sequence, and cycle constructions. A general strategy is developed for solving the random generation problem with two closely related types of methods: for structures of size n, the boustrophedonic algorithms exhibit a worst-case behaviour of the form O(n log n); the sequential algorithms have worst case O(n2), while offering good potential for optimizations in the average case. The complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time O(n2).A companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification. Using optimizations dictated by the cost calculus, several random generation algorithms of the sequential type are developed; most of them have expected complexity 1/2n log n, and are thus only slightly superlinear. The approach is exemplified by the random generation of a number of classical combinatorial structures including Cayley trees, hierarchies, the cycle decomposition of permutations, binary trees, functional graphs, surjections, and set partitions.

Analysis of a Left-Shift Binary GCD Algorithm

We introduce a new left-shift binary algorithm, LSBGCD, for computing the greatest common divisor of two integers, and we provide an analysis of the worst-case behavior of this algorithm. The analysis depends on a theorem of Ramharter about the extremal behavior of certain continuants.

Approximating the permanent: A simple approach

AbstractIn this article, we present an efficient and very simple unbiased estimator for the Permanent of square (0–1) matrices. As the main result, we prove that our estimator, even though its worst case behavior is provably very bad, runs in time polynomial in the size of the input matrix for almost all matrices. We also generalize our technique and apply it to another enumeration problem, that of counting Hamilton cycles in directed graphs. The ease with which this was done suggests that the basic technique may have wide applicability. © 1994 John Wiley & Sons, Inc.

Skeletons from the Treecode Closet

We consider treecodes (N-body programs which use a tree data structure) from the standpoint of their worst-case behavior. That is, we derive upper bounds on the largest possible errors that are introduced into a calculation by use of various multipole acceptability criteria (MAC). We find that the conventional Barnes-Hut MAC can introduce potentially unbounded errors unless θ < 1/3, and that this behavior while rare, is demonstrable in astrophysically reasonable examples. We consider two other MACs closely related to the BH MAC. While they do not admit the same unbounded errors, they nevertheless require extraordinary amounts of CPU time to guarantee modest levels of accuracy. We derive new error bounds based on some additional, easily computed moments of the mass distribution. These error bounds form the basis for four new MACs which can be used to limit the absolute or relative error introduced by each multipole evaluation, or, with the introduction of some additional data structures, the absolute or rms error in the final acceleration of each particle. Using the Sum Squares MAC to analytically place a 1% bound on the rms error in a series of test models, we find that it significantly outperforms the θ = 0.65 BH MAC in terms of both accuracy (mean, rms, and maximum error) and performance (floating point operation count).

Implementing analysable hard real-time sporadic tasks in Ada 9X

In this paper we illustrate how systems containing hard real-time sporadic tasks can be analysed for their worst case behaviour. In order to undertake this schedulability analysis, it is necessary to define the minimum inter-arrival time and/or maximum arrival frequency of sporadic tasks. Furthermore, at run-time it is essential to ensure that sporadic tasks are not invoked more often than has been guaranteed by the analysis. We assume that sporadics are invoked by interrupts and that interrupts can be masked under software control.Sporadic tasks are often analysed using the notion of bandwidth preserving sporadic servers within the Rate Monotonic Scheduling Analysis scheme. At run-time this requires the underlying kernel to support complex execution time monitoring mechanisms. Unfortunately such mechanisms are not generally supported by Ada 9X. This paper shows that by using Deadline Monotonic Scheduling Analysis there is no need to resort to bandwidth preserving sporadic servers, and the facilities available in Ada 9X can be used.

Lower and upper bounds on time for multiprocessor optimal schedules

The lower and upper bounds on the minimum time needed to process a given directed acyclic task graph for a given number of processors are derived. It is proved that the proposed lower bound on time is not only sharper than the previously known values but also easier to calculate. The upper bound on time, which is useful in determining the worst case behavior of a given task graph, is presented. The lower and upper bounds on the minimum number of processors required to process a given task graph in the minimum possible time are also derived. It is seen with a number of randomly generated dense task graphs that the lower and upper bounds we derive are equal, thus giving the optimal time for scheduling directed acyclic task graphs on a given set of processors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An improved algorithm for the dynamic lot-sizing problem with learning effect in setups

The dynamic lot-sizing problem with learning in setups is a variation of the Wagner-Whitin lot-sizing problem where the setup costs are a concave, nondecreasing function of the cumulative number of setups. This problem has been a subject of some recent research. We extend the previously studied model to include nonstationary production costs and present an O(T2) algorithm to solve this problem. The worst-case complexity of our algorithm improves the worst-case behavior of the algorithms presently known in the literature. © 1993 John Wiley & Sons, Inc.

The Identification of Multiple Outliers

Abstract One approach to identifying outliers is to assume that the outliers have a different distribution from the remaining observations. In this article we define outliers in terms of their position relative to the model for the good observations. The outlier identification problem is then the problem of identifying those observations that lie in a so-called outlier region. Methods based on robust statistics and outward testing are shown to have the highest possible breakdown points in a sense derived from Donoho and Huber. But a more detailed analysis shows that methods based on robust statistics perform better with respect to worst-case behavior. A concrete outlier identifier based on a suggestion of Hampel is given.

The Identification of Multiple Outliers

Abstract One approach to identifying outliers is to assume that the outliers have a different distribution from the remaining observations. In this article we define outliers in terms of their position relative to the model for the good observations. The outlier identification problem is then the problem of identifying those observations that lie in a so-called outlier region. Methods based on robust statistics and outward testing are shown to have the highest possible breakdown points in a sense derived from Donoho and Huber. But a more detailed analysis shows that methods based on robust statistics perform better with respect to worst-case behavior. A concrete outlier identifier based on a suggestion of Hampel is given.

Graph theoretic analysis of PLA folding heuristics

Because the problem of folding a programmable logic array (PLA) to its smallest possible area is NP-complete, the design and analysis of PLA-folding heuristics is important. In this paper, we study PLA-folding heuristics from a theoretical perspective, using the folding ratio of the heuristic to characterize its worst case behavior. In the first portion of this paper, we obtain best possible bounds on the folding ratios of several practical and intuitively appealing heuristics. These results show that for arbitrary PLAs, each of the heuristics exhibits unbounded worst-case behavior. Moreover, the results of Ravi and Lloyd ( SIAM J. Comput. 17 (1988), 696–710) strongly suggest that no polynomial time approximation algorithm can have a constant folding ratio for arbitrary PLAs. Thus in the second portion of this paper, we analyze the performance of heuristics for certain restricted classes of PLAs characterized by special classes of intersection graphs.

Probabilistic Analysis of Packing and Related Partitioning Problems

In the last 10 years, there have been major advances in the average-case analysis of bin packing, scheduling and similar partitioning problems in one and two dimensions. These problems are drawn from important applications throughout industry, often under the name of stock cutting. This article briefly surveys many of the basic results, as well as the probabilistic methods used to obtain them. The impact of the research discussed here has been twofold. First, analysis has shown that heuristic solutions often perform extremely well on average and hence can be recommended in practice, even though worst-case behavior can be quite poor. Second, the techniques of applied probability that have developed for the analysis of bin packing have found application in completely different arenas, such as statistics and stochastic models.

Approximating constrained tetrahedrizations

The facets of a Delaunay tetrahedrization of a set of points in three-dimensional Euclidean space are determined except in the presence of cospherical points. Since it is not always possible to construct in three-dimensional Euclidean space a tetrahedrization that contains prespecified facets, we describe an algorithm to construct a tetrahedrization that incorporates specified convex, planar, polygonal regions, including triangles, as unions of facets of tetrahedra. Points added to the original set lie within the specified regions. The resulting tetrahedrization coincides with the original tetrahedrization except in the vicinity of the specified regions. The worst case behavior of the algorithm is linear in the number of original tetrahedra but exponential in the number of specified regions. Measurement of performance on data generated by applications awaits an implementation.

A binary algorithm for the Jacobi symbol

We present a new algorithm to compute the Jacobi symbol, based on Stein's binary algorithm for the greatest common divisor, and we determine the worst-case behavior of this algorithm. Our implementation of the algorithm runs approximately 7--25% faster than traditional methods on inputs of size 100--1000 decimal digits.