On the worst case of a minimal spanning tree algorithm for euclidean space

This paper concerns the worst case running time of the minimal spanning tree algorithm presented by Bentley and Friedman. For a set ofN points ink-dimensional Euclidean space the worst case performance of the algorithm is shown to beΘ(N ² logN), fork≧2 andΘ(N ²), fork=1.     

First fit decreasing scheduling on uniform multiprocessors

Independent tasks are nonpreemptively scheduled on m≥2 processors which are assumed to have different speeds. The purpose of this paper is to show that the worst case ratio of the multifit algorithm MF, which is based on the bin-packing method FFD (first fit decreasing), depends on the order of the processors and that the MF has a better worst case behavior than the well-known LPT algorithm for certain processor configurations.     

The expected number of extreme points of a random linear program

There has been increasing attention recently on average case algorithmic performance measures since worst case measures can be qualitatively quite different. An important characteristic of a linear program, relating to Simplex Method performance, is the number of vertices of the feasible region. We show 2 ⁿ to be an upper bound on the mean number of extreme points of a randomly generated feasible region with arbitrary probability distributions on the constraint matrix and right hand side vector. The only assumption made is that inequality directions are chosen independently in accordance with a series of independent fair coin tosses.We would like to thank the Institute of Pure and Applied Mathematics in Rio de Janeiro for supporting the authors' collaboration that led to this paper.     

A numerical exploration of compressed sampling recovery

This paper explores numerically the efficiency of ?¹ minimization for the recovery of sparse signals from compressed sampling measurements in the noiseless case. This numerical exploration is driven by a new greedy pursuit algorithm that computes sparse vectors that are difficult to recover by ?¹ minimization. The supports of these pathological vectors are also used to select sub-matrices that are ill-conditioned. This allows us to challenge theoretical identifiability criteria based on polytopes analysis and on restricted isometry conditions. We evaluate numerically the theoretical analysis without resorting to Monte-Carlo sampling, which tends to avoid worst case scenarios.     

Solution of an equiweighted minimax location problem on a hemisphere

A particular continuous single facility minimax location problem on the surface of a hemisphere is discussed. We assume that all the demand points are equiweighted. An algorithm, based on spherical trigonometry, for finding the minimax point is presented. The minimax point thus obtained is unique and the algorithm is O(n ²) in the worst case.     

An Ω(n 5/4) lower bound on the randomized complexity of graph properties

A decision tree algorithm determines whether an input graph withn nodes has a property by examining the entries of the graph's adjacency matrix and branching according to the information already gained. All graph properties which are monotone (not destroyed by the addition of edges) and nontrivial (holds for somes but not all graphs) have been shown to require (n ²) queries in the worst case.In this paper, we investigate the power of randomness in recognizing these properties by considering randomized decision tree algorithms in which coins may be flipped to determine the next entry to be examined. The complexity of a randomized algorithm is the expected number of entries that are examined in the worst case. The randomized complexity of a property is the minimum complexity of any randomized decision tree algorithm which computes the property. We improve Yao's lower bound on the randomized complexity of any nontrivial monotone graph property from (n log^1/12 n) to (n ^5/4).     

Universal Algorithms for Learning Theory. Part II: Piecewise Polynomial Functions

This paper is concerned with estimating the regression function f_ρ in supervised learning by utilizing piecewise polynomial approximations on adaptively generated partitions. The main point of interest is algorithms that with high probability are optimal in terms of the least square error achieved for a given number m of observed data. In a previous paper [1], we have developed for each β > 0 an algorithm for piecewise constant approximation which is proven to provide such optimal order estimates with probability larger than 1- m^-β. In this paper we consider the case of higher-degree polynomials. We show that for general probability measures ρ empirical least squares minimization will not provide optimal error estimates with high probability. We go further in identifying certain conditions on the probability measure ρ which will allow optimal estimates with high probability.     

A hybrid algorithm for finding thekth smallest ofn elements in O(n) time

The problem of finding thekth smallest ofn elements can be solved either with O(n) algorithms or with O(n ²) algorithms. Although they require a higher number of operations in the worst case, O(n ²) algorithms are generally preferred to O(n) algorithms because of their better average performance. We present a hybrid algorithm which is O(n) in the worst case and efficient in the average case.     

超尺寸物品装箱问题

本文给出一类新的装箱问题，超尺寸物品装箱问题。就实际解决该问题所普遍彩的两步法，证明了当采用经典目标函数并且拆分次数不超过2时，第二步采用FFDLR的渐进最坏比为3/2。进而针对超尺寸物品装箱问题的算法提出了一个评价效率更高的目标函数。证明了在此目标函数下，当不限制物品的最大尺寸时，第二步采用最优装法两步法的渐近最坏比为2。最后，给出渐近最坏与拆分次数的关系。     

A randomized algorithm for the min-max selecting items problem with uncertain weights

This paper deals with the min-max version of the problem of selecting p items of the minimum total weight out of a set of n items, where the item weights are uncertain. The discrete scenario representation of uncertainty is considered. The computational complexity of the problem is explored. A randomized algorithm for the problem is then proposed, which returns an O(ln K)-approximate solution with a high probability, where K is the number of scenarios. This is the first approximation algorithm with better than K worst case ratio for the class of min-max combinatorial optimization problems with unbounded scenario set.     

Quadripartite Sort

This paper presents an efficient and practical sorting algorithm, called Quadripartite Sort. It lies between MergeSort and QuickSort. This algorithm sortsnelements using bounded workspace andn log n + 1.75ncomparisons in the worst case. By empirical testing, we conjecture that this algorithm needs approximatelyn log n − ncomparisons on average. When usingm-way merging strategy, wheremis a larger constant, this algorithm becomes an in-place sorting algorithm whose comparison plus exchange total is absolutely minimum among known constant workspace algorithms. For example, using a 256-way merging, the comparison plus exchange total required is approximately 1.2495n log n + O(n) in the worst case.     

A cumulative not-first/not-last filtering algorithm in O(n 2log(n))

In cumulative and disjunctive constraint-based scheduling, the resource constraint is enforced by several filtering rules. Among these rules, we have (extended) edge-finding and not-first/not-last rules. The not-first/not-last rule detects tasks that cannot run first/last relatively to a set of tasks and prunes their time bounds. In this paper, it is presented a sound O (n ² log n) algorithm for the cumulative not-first/not-last rule where n is the number of tasks. This algorithm reaches the same fix point as previous not-first/not-last algorithms, although it may take additional iterations to do so. The worst case complexity of this new algorithm for the maximal adjustment is the same as our previous complete O (n ²|H| log n) not-first/not-last algorithm [7] where |H| is the maximum between the number of distinct earliest completion and latest start times of tasks. But, experimental results on benchmarks from the Project Scheduling Problem Library (PSPLib) and the Baptiste and Le Pape data set (BL) suggest that the new not-first/not-last algorithm has a substantially reduced runtime. Furthermore, the results demonstrate that in practice the new algorithm rarely requires more propagations than previous not-first/not-last algorithms.     

Robustness of Discrete Decisions to the Worst Case Distribution

This paper proposes a robustness measure for discrete decisions subject to uncertainty in the state probabilities. The robustness measure is the worst case difference in expected payoffs between the chosen action (the decision) and all other actions, as the state probabilities are allowed to vary simultaneously over prescribed tolerance ranges. In contrast to the more difficult to assess single distribution, the current approach requires only the assessment of an interval for each state probability; i.e. it works with a more easily assessed spectrum of distributions. It is shown that the robustness can be determined with an algorithm that is much simpler than the usual linear programming techniques. Its computational simplicity makes this algorithm amenable to manual calculation.     

超尺寸物品装箱问题及其算法

本文探讨一类新装箱问题-超尺寸物品装箱问题。针对实际解决该问题的两涉法，我们提出了一个评价效率更高的目标函数，证明了在此目标函数下两步法的渐近最坏比不小于2，并给出了渐近量坏比与拆分次数的关系。最后本文提出了一种不同于两步法的新在线算法MA，证明了在新目标函数下其渐近最坏比不超过7/4。     

Q2‖Cmax的对偶近似算法

本文讨论两台同类平行机排序问题,首先给出Ｍｕｌｔｉｆｉｔ算法在不同迭代初值下的紧界,然后利用一个新设计的对偶贪婪子过程构造出线性时间６／５－复合近似算法。     

Algorithms for coloring semi-random graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require n^δ (δ>0) colors even for bounded chromatic (k-colorable for fixed k) n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19 , 204–234 (1995)] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work as long as p≥n^−α(k)+ϵ where α(k)=(2k)/((k−1)(k+2)) and ϵ is a positive constant. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 125–158 (1998)     

Deadlock avoidance with a modified banker's algorithm

There are three methods for handling deadlocks in resource allocation systems: deadlock prevention, deadlock avoidance and deadlock detection combined with recovery. Of these three methods deadlock avoidance is preferable in many cases but seldom used on account of its high cost. We present a simple modification of a known deadlock avoidance algorithm, the banker's algorithm, which has a running time (mn ²) in a system consisting ofn processes andm different types of resources. Our modified algorithm gives an amortized worst case running time ofO(mn) under certain likely conditions and in that way it can be considered a competitive method for handling deadlocks. At worst, our algorithm is twice as fast as the banker's algorithm.This work was partly supported by The National Swedish Board for Technical Development (STUF) under contract number 85-3127.     

Uniform machine scheduling with machine available constraints

1.IntroductionIntheclassicalparallelmachineschedulingareaweassumethatmachinesarealwaysavailable.However,aspointedin[1],inrealindustrysettingsthisassumptionmaynotbetrue.Forexample,machinesmaynotalwaysbeavailablebecauseoftheirpreventivemaintenanceduringtheschedulingperiod.Thatistosay,eachmachineiisunavailablefromsibuntilrib(05sib5rib),where0SkSm,withmbeingthenumberofunavailabilityperiodsformachineiduringtheplanninghorizon.Inotherwords,somepapersstatethatmachinesareavailableintimewindows,whichi…     

Finding the Shortest Watchman Route in a Simple Polygon

We present the first polynomial time algorithm that finds the shortest route in a simple polygon such that all points of the polygon are visible from the route. This route is called the shortest watchman route, and we do not assume any restrictions on the route or on the simple polygon. Our algorithm runs in worst case O(n ⁶ ) time, but it is adaptive, making it run faster on polygons with a simple structure. Received December 12, 1997, and in revised form September 30, 1998.     

Competitive clustering algorithms based on ultrametric properties

We propose in this paper two new competitive unsupervised clustering algorithms: the first algorithm deals with ultrametric data, it has a computational cost of O(n). The second algorithm has two strong features: it is fast and flexible on the processed data type as well as in terms of precision. The second algorithm has a computational cost, in the worst case, of O(n²), and in the average case, of O(n). These complexities are due to exploitation of ultrametric distance properties. In the first method, we use the order induced by an ultrametric in a given space to demonstrate how we can explore quickly data proximity. In the second method, we create an ultrametric space from a sample data, chosen uniformly at random, in order to obtain a global view of proximities in the data set according to the similarity criterion. Then, we use this proximity profile to cluster the global set. We present an example of our algorithms and compare their results with those of a classic clustering method.