Algorithms and Data Structures for External Memory
Algorithms and Data Structures for External Memory
Algorithms and Data Structures for External Memory
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140 Dynamic <strong>Memory</strong> Allocation<br />
For sorting, the lower bound approach of Theorem 6.1 implies that<br />
�<br />
2mi logmi =Ω(nlogn). (16.1)<br />
i<br />
We say that P is dynamically optimal <strong>for</strong> sorting if<br />
�<br />
2mi logmi = O(nlogn) (16.2)<br />
i<br />
<strong>for</strong> all possible sequences m1, m2, . . . of memory allocation. Intuitively,<br />
if P is dynamically optimal, no other algorithm can per<strong>for</strong>m more than<br />
a constant number of sorts in the worst-case <strong>for</strong> the same sequence of<br />
memory allocations.<br />
Barve <strong>and</strong> Vitter [71] define the model in generality <strong>and</strong> give dynamically<br />
optimal strategies <strong>for</strong> sorting, matrix multiplication, <strong>and</strong> buffer<br />
tree operations. Their work represents the first theoretical model of<br />
dynamic allocation <strong>and</strong> the first algorithms that can be considered<br />
dynamically optimal. Previous work was done on memory-adaptive<br />
algorithms <strong>for</strong> merge sort [279, 363] <strong>and</strong> hash join [280], but the algorithms<br />
h<strong>and</strong>le only special cases <strong>and</strong> can be made to per<strong>for</strong>m nonoptimally<br />
<strong>for</strong> certain patterns of memory allocation.
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