Abstract. We consider problems of packing an arbitrary collection of rectangular pieces into an open-ended, rectangular bin so as to minimize the height achieved by any piece. This problem has numerous applications in operations research and studies of computer operation. We devise efficient approximation algorithms, study their limitations, and derive worst-case bounds on the performance of the packings they produce.Key words, two-dimensional packing, bin packing, resource constrained scheduling 1. Introduction. Efficiently packing sets of rectangular figures into a given rectangular area is a problem with widespread application in operations research. Thus, one is inclined to attribute the scarcity of results on this problem, and others of its type, to inherent difficulty rather than to lack of importance. Motivated by the intractability of these problems, we define and analyze certain approximation algorithms. These algorithms are natural in the sense that they would probably be among the first to occur to anyone wishing to design simple, fast procedures for determining easily computed packings. The analysis of these algorithms leads to bounds on the performance of approximate packings relative to the best achievable.
The problem of finding sections of code that either are identical or are related by the systematic renaming of variables or constants can be modeled in terms of parameterized strings ( p-strings) and parameterized matches ( p-matches). P-strings are strings over two alphabets, one of which represents parameters. Two p-strings are a parameterized match ( p-match) if one p-string is obtained by renaming the parameters of the other by a one-to-one function. In this paper, we investigate parameterized pattern matching via parameterized suffix trees (p-suffix trees). We give two algorithms for constructing p-suffix trees: one (eager) that runs in linear time for fixed alphabets, and another that uses auxiliary data structures and runs in O(n log(n)) time for variable alphabets, where n is input length. We show that using a p-suffix tree for a pattern p-string P, it is possible to search for all p-matches of P within a text p-string T in space linear in |P| and time linear in |T | for fixed alphabets, or O( |T| log(min( |P|, _)) time and O( |P| ) space for variable alphabets, where _ is the sum of the alphabet sizes. The simpler p-suffix tree construction algorithm eager has been implemented, and experiments show it to be practical. Since it runs faster than predicted by the above worst-case bound, we reanalyze the algorithm and show that eager runs in time O(min(t |S | +m(t, S) | t>0) log _)), where for an input p-string S, m(t, S ) is the number of maximal p-matches of length at least t that occur within S, and _ is the sum of the alphabet sizes. Experiments with the author's program dup (B. Baker, in``Comput. Sci. Statist., '' Vol. 24, 1992) for finding all maximal p-matches within a p-string have found m(t, S ) to be less than |S| in practice unless t is small. ]
Approximation Algorithms for NP.Complete Problems on Planar Graphs (Preliminary Version> nodes! (Djidjev's recent improvements [D) in the planar separator theorem bounds reduce this number to about 22)00.)Both the Lipton-Tarjan approach and our approach have tradeoffs between running time and convergence rate. However, for the same running time the convergence rate of our approximation scheme is much better. For example, if we n is the number of nodes and c is some constant, we get polynomial time approximation schemes, i.e. algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum Hmatching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k -outerplanar graphs also enlarges the class of planar graphs for which the problems are known to be solvable.
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