Donna J. Brown

A 54 algorithm for two-dimensional packing

This paper proposes a new algorithm for a two-dimensional packing problem first studied by Baker, Coffman, and Rivest (SIAM J. Comput. 9, No. 4(1980), 846–855). In their model, a finite list of rectangles is to be packed into a rectangular bin of finite width but infinite height. The model has applications to certain scheduling and stock-cutting problems. Since the problem of finding an optimal packing is NP-hard, previous work has been directed at finding polynomial approximation algorithms for the problem, i.e., algorithms which come within a constant times the height used by an optimal packing. For the algorithm proposed in this paper, the ratio of the height obtained by the algorithm and the height used by an optimal packing is asymptotically bounded by 54. This bound is an improvement over the bound of 43 achieved by the best previous algorithm.

On-line bin packing in linear time

Abstract In this paper, we study the 1-dimensional on-line bin packing problem. A list of pieces, each of size between zero and unity are to be packed, in order of their arrival, into a minimum number of unit-capacity bins. We present a new linear-time algorithm, the Modified Harmonic Algorithm and show, by a novel use of weighting functions, that it has an asymptotic worst-case performance ratio less than 3 2 + 1 9 + 1 222 = 1.(615) ∗ . We show that for a large class of linear-time on-line algorithms including the Modified Harmonic Algorithm, the performance ratio is at least 3 2 + 1 9 = 1.61 ∗ . Then we show how to successively construct classes of improved linear-time on-line algorithms. For any algorithm in any of these classes, the performance ratio is at least 3 2 + 1 12 = 1.583 ∗ . We present an improved algorithm called Modified Harmonic-2 with performance ratio 1.612 … and present an approach to construct linear-time on-line algorithms with better performance ratios. The analysis of Modified Harmonic-2 is omitted because it is very similar to that of Modified Harmonic, but it is substantially more complicated. Our results extend to orthogonal packings in two dimensions.

Lower bounds for on-line two-dimensional packing algorithms

SummaryMany problems, such as cutting stock problems and the scheduling of tasks with a shared resource, can be viewed as two-dimensional bin packing problems. Using the two-dimensional packing model of Baker, Coffman, and Rivest, a finite list L of rectangles is to be packed into a rectangular bin of finite width but infinite height, so as to minimize the total height used. An algorithm which packs the list in the order given without looking ahead or moving pieces already packed is called an on-line algorithm. Since the problem of finding an optimal packing is NP-hard, previous work has been directed at finding approximation algorithms. Most of the approximation algorithms which have been studied are on-line except that they require the list to have been previously sorted by height or width. This paper examines lower bounds for the worst-case performance of on-line algorithms for both non-preordered lists and for lists preordered by increasing or decreasing height or width.

New Lower Bounds for Channel Width

We present here a simple yet effective technique for calculating a lower bound on the number of tracks required to solve a given channel-routing problem. The bound applies to the wiring model where horizontal wires run on one layer and vertical wires run on another layer. One of the major results is that at least \( \sqrt {{2n}} \) tracks are necessary for any dense channel routing problem with n two-terminal nets that begin and end in different columns. For example, if each net i begins in column i and ends in column i+1, at least \( \sqrt {{2n}} \) tracks are required, even though the channel “density” is only 2. This is the first technique which can give results which are significantly better than the naive channel density arguments. A modification results in the calculation of an improved bound, which we conjecture to be optimal to within a constant factor.

Polynomial-time self-reducibility: theoretical motivations and practical results ∗

Although polynomial-time complexity theory has been formulated in terms of decision problems, polynomial-time decision algorithms generally operate by attempting to construct a solution to an optimization version of the problem at hand. Thus it is that self-reducibility, the process by which a decision algorithm may be used to devise a constructive algorithm, has until now been widely considered a topic of only theoretical interest. Recent fundamental advances in graph theory, however, have made available powerful new nonconstructive tools that can be applied to guarantee membership in P. These tools are nonconstructive at two distinct levels: they neither produce the decision algorithm, establishing only the finiteness of an obstruction set, nor do they reveal whether such a decision algorithm can be of any aid in the construction of a solution. We briefly review and illustrate the use of these tools, and discuss the seemingly formidable task of finding the promised polynomial-time decision algorithms wh...

Optimal multilayer channel routing with overlap

This paper presents algorithms for multiterminal net channel routing where multiple interconnect layers are available. Major improvements are possible if wires are able to overlap, and our generalized main algorithm allows overlap, but only on everyKth (K ≥ 2) layer. Our algorithm will, for a problem with densityd onL layers,L ≥K + 3,provably use at most three tracks more than optimal: ⌈(d + 1)/⌈L/K⌉⌉ + 2 tracks, compared with the lower bound of ⌈d/⌈L/K⌉⌉. Our algorithm is simple, has few vias, tends to minimize wire length, and could be used if different layers have different grid sizes. Finally, we extend our algorithm in order to obtain improved results for adjacent (K = 1) overlap: ⌈(d + 2)/⌊2L/3⌋⌉ + 5 forL ≥ 7.

Mallard/sup TM/: asynchronous learning in two engineering courses

Mallard/sup TM/ is a World Wide Web based interactive learning environment suitable for use in a wide variety of courses. Mallard/sup TM/ provides a secure environment within which one can organize online course material and test students via interactive quizzes with instantaneous problem correction and grading. In addition, Mallard/sup TM/ provides administrative utilities for tasks such as maintaining class rosters, recording grades, and viewing up-to-the-minute tables of student progress. Mallards/sup TM/ open design allows flexibility and easy expansion. Mallard/sup TM/ is currently being used in a number of courses; we describe its use in two engineering courses at the University of Illinois at Urbana-Champaign.

Writing Web-based questions with Mallard

The World Wide Web has the potential to revolutionize the field of education. Not only can students have access to virtually unlimited information (text, audio and video), but true interactivity is possible. Using Mallard(TM), educators in virtually all fields have an interactive learning environment which makes writing questions and constructing quizzes a straightforward process. Correction/grading is handled automatically and instantaneously by intelligent grading programs. Administrative utilities allow an instructor to monitor student progress.

Mallard TM : an educational tool for digital signal processing

Mallard/sup TM/ is a World Wide Web-based interactive learning environment suitable for virtually any subject. We focus on its application as an educational tool for digital signal processing basics. Students are tested via interactive quizzes with instantaneous problem correction and grading. Intelligent grading programs not only assess the correctness of a response but also determine why an answer is incorrect. The student receives immediate feedback and can access online assistance if desired. The instructor benefits from a number of administrative utilities, the most important being automatic grading and maintenance of not only the gradebook but also detailed logfiles. Mallard/sup TM/ does this within a secure environment, using Netscapes Commerce Server. Access is restricted by password, and communication is encrypted.

Hexagonal models for channel routing

Abstract. In this paper we present new diagonal routing models based on the hexagonal grid, and investigate the potential of these models in terms of channel routing. The hexagonal grid consists of vertical columns, and positive and negative diagonal tracks with slopes