Neal E. Young

Competitive paging algorithms

The paging problem is that of deciding which pages to keep in a memory of k pages in order to minimize the number of page faults. We develop the marking algorithm, a randomized on-line algorithm for the paging problem. We prove that its expected cost on any sequence of requests is within a factor of 2Hk of optimum. (Where Hk is the kth harmonic number, which is roughly In k.) The best such factor that can be achieved is Hk. This is in contrast to deterministic algorithms, which cannot be guaranteed to be within a factor smaller than k of optimum. An alternative to comparing an on-line algorithm with the optimum off-line algorithm is the idea of comparing it to several other on-line algorithms. We have obtained results along these lines for the paging problem. Given a set of on-line algorithms ‘Support was provided by a Weizmann fellowship. ‘Partial support was provided by the International Computer Science Institute, Berkeley, CA, and by NSF Grant CCR-8411954. 3Support was provided by the International Computer Science Institute and operating grant A8092 of the Natural Sciences and Engineering Research Council of Canada. Current

An Efficient Targeting Strategy for Multiobject Spectrograph Surveys: the Sloan Digital Sky Survey "Tiling" Algorithm

Large surveys using multiobject spectrographs require automated methods for deciding how to efficiently point observations and how to assign targets to each pointing. The Sloan Digital Sky Survey (SDSS) will observe around 10 6 spectra from targets distributed over an area of about 10,000 deg 2 , using a multiobject fiber spectrograph that can simultaneously observe 640 objects in a circular field of view (referred to as a ‘‘ tile ’’) 1=49 in radius. No two fibers can be placed closer than 55 00 during the same observation; multiple targets closer than this distance are said to ‘‘ collide.’’ We present here a method of allocating fibers to desired targets given a set of tile centers that includes the effects of collisions and that is nearly optimally efficient and uniform. Because of large-scale structure in the galaxy distribution (which form the bulk of the SDSS targets), a naive covering of the sky with equally spaced tiles does not yield uniform sampling. Thus, we present a heuristic for perturbing the centers of the tiles from the equally spaced distribution that provides more uniform completeness. For the SDSS sample, we can attain a sampling rate of greater than 92% for all targets, and greater than 99% for the set of targets that do not collide with each other, with an efficiency greater than 90% (defined as the fraction of available fibers assigned to targets). The methods used here may prove useful to those planning other large surveys.

Balancing minimum spanning and shortest path trees

We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and aγ>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+√2γ times the shortest-path distance, and yet the total weight of the tree is at most 1+√2/γ times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.

The k-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 Tarjans complaint [ST] that the standard competitive ratios for paging strategies are too high. Ak-server strategy islooselyc(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 Θ(Ink)-competitive in the standard model, is looselyc(k)-competitive providedk−2 In Ink→∞ and both 2 Ink−c(k) andc(k) are nondecreasing.

On-line file caching

Abstract. Consider the following file caching problem: in response to a sequence of requests for files, where each file has a specified size and retrieval cost , maintain a cache of files of total size at most some specified k so as to minimize the total retrieval cost. Specifically, when a requested file is not in the cache, bring it into the cache and pay the retrieval cost, and remove other files from the cache so that the total size of files remaining in the cache is at most k . This problem generalizes previous paging and caching problems by allowing objects of arbitrary size and cost, both important attributes when caching files for world-wide-web browsers, servers, and proxies. We give a simple deterministic on-line algorithm that generalizes many well-known paging and weighted-caching strategies, including least-recently-used, first-in-first-out, flush-when-full, and the balance algorithm. On any request sequence, the total cost incurred by the algorithm is at most k/(k-h+1) times the minimum possible using a cache of size h ≤ k . For any algorithm satisfying the latter bound, we show it is also the case that for most choices of k , the retrieval cost is either insignificant or at most a constant (independent of k ) times the optimum. This helps explain why competitive ratios of many on-line paging algorithms have been typically observed to be constant in practice.

Faster parametric shortest path and minimum balance algorithms

Author(s): Young, NE; Tarjant, RE; Orlin, JB | Abstract: We use Fibonacci heaps to improve a parametric shortest path algorithm of Karp and Orlin, and we combine our algorithm and the method of Schneider and Schneiders minimum‐balance algorithm to obtain a faster minimum‐balance algorithm. For a graph with n vertices and m edges, our parametric shortest path algorithm and our minimum‐balance algorithm both run in O(nm + n2 log n) time, improved from O(nm log n) for the parametric shortest path algorithm of Karp and Orlin and O(n2m) for the minimum‐balance algorithm of Schneider and Schneider. An important application of the parametric shortest path algorithm is in finding a minimum mean cycle. Experiments on random graphs suggest that the expected time for finding a minimum mean cycle with our algorithm is O(n log n + m). Copyright

Logical-shapelets: an expressive primitive for time series classification

Time series shapelets are small, local patterns in a time series that are highly predictive of a class and are thus very useful features for building classifiers and for certain visualization and summarization tasks. While shapelets were introduced only recently, they have already seen significant adoption and extension in the community. Despite their immense potential as a data mining primitive, there are two important limitations of shapelets. First, their expressiveness is limited to simple binary presence/absence questions. Second, even though shapelets are computed offline, the time taken to compute them is significant. In this work, we address the latter problem by introducing a novel algorithm that finds shapelets in less time than current methods by an order of magnitude. Our algorithm is based on intelligent caching and reuse of computations, and the admissible pruning of the search space. Because our algorithm is so fast, it creates an opportunity to consider more expressive shapelet queries. In particular, we show for the first time an augmented shapelet representation that distinguishes the data based on conjunctions or disjunctions of shapelets. We call our novel representation Logical-Shapelets. We demonstrate the efficiency of our approach on the classic benchmark datasets used for these problems, and show several case studies where logical shapelets significantly outperform the original shapelet representation and other time series classification techniques. We demonstrate the utility of our ideas in domains as diverse as gesture recognition, robotics, and biometrics.

Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

Given an undirected graph with edge costs and a subset ofk=3 nodes calledterminals, a multiway, ork-way, cut is a subset of the edges whose removal disconnects each terminal from the others. The multiway cut problem is to find a minimum-cost multiway cut. This problem is Max-SNP hard. Recently, Calinescu et al. (Calinescu, G., H. Karloff, Y. Rabani. 2000. An improved approximation algorithm for Multiway Cut.J. Comput. System Sci.60(3) 564--574) gave a novel geometric relaxation of the problem and a rounding scheme that produced a (3/2-1/ k)-approximation algorithm.In this paper, we study their geometric relaxation. In particular, we study the worst-case ratio between the value of the relaxation and the value of the minimum multicut (the so-called integrality gap of the relaxation). Fork=3, we show the integrality gap is 12/11, giving tight upper and lower bounds. That is, we exhibit a family of graphs with integrality gaps arbitrarily close to 12/11 and give an algorithm that finds a cut of value 12/11 times the relaxation value. Our lower bound shows that this is the best possible performance guarantee for any algorithm based purely on the value of the relaxation. Our upper bound meets the lower bound and improves the factor of 7/6 shown by Calinescu et al.For allk, we show that there exists a rounding scheme with performance ratio equal to the integrality gap, and we give explicit constructions of polynomial-time rounding schemes that lead to improved upper bounds. Fork=4 and 5, our best upper bounds are based on computer-constructed rounding schemes (with computer proofs of correctness). For generalk we give an algorithm with performance ratio 1.3438-e k .Our results were discovered with the help of computational experiments that we also describe here.

Rounding algorithms for a geometric embedding of minimum multiway cut

Given an undirected graph with edge costs and a subset of k ≥ 3 nodes called terminals, a multiway, or k-way, cut is a subset of the edges whose removal disconnects each terminal from the others. The multiway cut problem is to find a minimum-cost multiway cut. T his problem is Max-SNP hard. Recently Calinescu, Karloff, and Rabani (STOC’98) gave a novel geometric relaxation of the problem and a rounding scheme that produced a (3/2 − 1/k)-approximation algorithm. In this paper, we study their geometric relaxation. In parti cular, we study the worst-case ratio between the value of the relaxation and the value of the minimum multicut (the so-called integrality gap of the relaxation). For k = 3, we show the integrality gap is 12/11, giving tight upper and lower bounds. That is, we exhibit a family of graphs with integrality gaps arbit rarily close to 12/11 and give an algorithm that finds a cut of value 12/11 times the relaxation value. Our lower bound shows that this is the best

Low-Degree Spanning Trees of Small Weight

Given