Abstract
DYNAMIC STORAGE ALLOCATION is the problem of packing given axis-aligned rectangles into a horizontal strip of minimum height by sliding the rectangles vertically but not horizontally. Where L=LOAD is the maximum sum of heights of rectangles that intersect any vertical line and OPT is the minimum height of the enclosing strip, it is obvious that OPT≥LOAD; previous work showed that OPT≤ 3• LOAD. We continue the study of the relationship between OPT and LOAD, proving that OPT=L+O((hmax/L)1/7)L, where hmax is the maximum job height. Conversely, we prove that for any e>0, there exists a c>0 such that for all sufficiently large integers hmax, there is a DYNAMIC STORAGE ALLOCATION instance with maximum job height hmax, maximum load at most L, and OPT≥ L+c(hmax/L)1/2+eL, for infinitely many integers L. En route, we construct several new polynomial-time approximation algorithms for DYNAMIC STORAGE ALLOCATION.
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