Two-dimensional packing problems: A survey

From the Publisher:
Mark Allen Weiss provides a proven approach to algorithms and data structures using the exciting Java programming language as the implementation tool. With Java he highlights conceptual topics, focusing on ADTs and the analysis of algorithms for efficiency as well as performance and running time. Dr. Weiss also distinguishes this text with a logical organization of topics, his engaging writing style, and an extensive use of figures and examples showing the successive stages of an algorithm. 
Features:
Contains extensive sample code using Java 1.2, which is available over the Internet. Covers the Java Collections Library in an appendix. Includes a chapter on algorithm and design techniques that covers greedy algorithms, divide-and-conquer algorithms, dynamic programming, randomized algorithms, and backtracking. Presents current topics and new data structures such as Fibonacci heaps, skew heaps, binomial queues, skip lists, and splay trees. Offers a chapter on amortized analysis that examines the advanced data structures presented earlier in the book. Provides a chapter on advanced data structures and their implementation, covering red-black trees, top-down splay trees, treaps, k-d trees, pairing heaps, and more.

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Data Structures and Algorithm Analysis in C

This paper updates a survey [53] written about 3 years ago. All of the results mentioned there are covered here as well. However, as a major justification for this second edition we shall be presenting many new results, some of which represent important advances. As a measure of the impressive amount of research in just 3 years, the present reference list more than doubles the list in [53].

Approximation Algorithms for Bin-Packing — An Updated Survey

On genetic algorithms for the packing of polygons

A new framework for analyzing online bin packing algorithms is presented. This framework presents a unified way of explaining the performance of algorithms based on the Harmonic approach. Within this framework, it is shown that a new algorithm, Harmonic++, has asymptotic performance ratio at most 1.58889. It is also shown that the analysis of Harmonic+1 presented in Richey [1991] is incorrect; this is a fundamental logical flaw, not an error in calculation or an omitted case. The asymptotic performance ratio of Harmonic+1 is at least 1.59217. Thus, Harmonic++ provides the best upper bound for the online bin packing problem to date.

On the online bin packing problem

A 54 algorithm for two-dimensional packing

On-line bin packing in linear time

This paper presents algorithms for routing channels with L≥2 layers. For the unit vertical overlap model, we describe a two-layer channel routing algorithm that uses at most d + O(√d) tracks to route two-terminal net problems and 2d + O(√d) tracks to route multiterminal nets. We also show that d + Ω(log d) tracks are required to route two-terminal net problems in the worst case even if arbitrary vertical overlap is allowed. We generalize the algorithm to unrestricted multilayer routing and use only d/(L -1) + O(√d/L + 1)> tracks for two-terminal net problems (within O(√d/L + 1) tracks of optimal) and d/(L-2) +O(√d/L + 1) tracks for multiterminal net problems (within a factor of(L-1)/(L-2) times optimal). We demonstrate the generality of our routing strategy by showing that it can be used to duplicate some of the best previous upper bounds for other models (two-layer Manhattan routing and two and three-layer knock-knee routing of two-terminal, two-sided nets), and gives a new upper bound for rotuing with 45-degree diagonal wires.

Nearly optimal algorithms and bounds for multilayer channel routing

Many 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.

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Lower bounds for on-line two-dimensional packing algorithms

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.

Donna J. Brown

Papers

A 54 algorithm for two-dimensional packing

On-line bin packing in linear time

Nearly optimal algorithms and bounds for multilayer channel routing

Lower bounds for on-line two-dimensional packing algorithms

New Lower Bounds for Channel Width