U. Glaeser

Barve and Vitter [37] discuss the design and analysis of EM algorithms that adapt gracefully to changing
memory allocations. In their model, without loss of generality, an algorithm (or program) P is allocated
internal memory in phases: During the ith phase, P is allocated m i blocks of internal memory, and this
memory remains allocated to P until P completes 2m i I/O operations, at which point the next phase
begins. The process continues until P finishes execution. We say that P is dynamically optimal, no other
algorithm can perform more than a constant number of sorts in the worst-case for the same sequence
of memory allocations.
Barve and Vitter [37] define a precise model and give dynamically optimal strategies for sorting, matrix
multiplication, and buffer tree operations. Previous work was done on memory-adaptive algorithms for
merge sort [162,216] and hash join [163], but the algorithms handle only special cases and can be made
to perform nonoptimally for certain patterns of memory allocation.
32.16 Conclusions
In this chapter, several useful paradigms for the design and implementation of efficient external memory
(EM) algorithms and data structures were described. The problem domains we have considered include
sorting, permuting, FFT, scientific computing, computational geometry, graphs, databases, geographic
information systems, and text and string processing. Interesting challenges remain in virtually all these
problem domains. One difficult problem is to prove lower bounds for permuting and sorting without an
item indivisibility assumption. Another promising area is the design and analysis of EM algorithms for
efficient use of multiple disks. Optimal bounds have not yet been determined for several basic EM graph
problems like topological sorting, shortest paths, breadth-first and depth-first search, and connected components.
There is an intriguing connection between problems that have good I/O speedups and problems
that have fast and work-efficient parallel algorithms. Several problems remain open in the dynamic and
kinetic settings, such as range searching, ray shooting, point location, and finding nearest neighbors.
A continuing goal is to develop optimal EM algorithms and to translate theoretical gains into observable
improvements in practice. For some of the problems that can be solved optimally up to a constant
factor, the constant overhead is too large for the algorithm to be of practical use, and simpler approaches
are needed. In practice, algorithms cannot assume a static internal memory allocation; they must adapt
in a robust way when the memory allocation changes.
Many interesting challenges and opportunities in algorithm design and analysis arise from new architectures
being developed, such as networks of workstations and hierarchical storage devices. Active (or
intelligent) disks, in which disk drives have some processing capability and can filter information sent
to the host, have recently been proposed to further reduce the I/O bottleneck, especially in large database
applications [3,167]. MEMS-based nonvolatile storage has the potential to serve as an intermediate level
in the memory hierarchy between DRAM and disks. It could ultimately provide better latency and
bandwidth than disks, at less cost per bit than DRAM [178,198].
Acknowledgments
The author thanks the TPIE group and his colleagues for helpful comments and suggestions. This work
was supported in part by Army Research Office MURI grant DAAH04-96-1-0013 and by National Science
Foundation Research Grants CCR-9522047, EIA-9870734, CCR-9877133, and CCR-0082986.
References
1. D. J. Abel. A B + -tree structure for large quadtrees. Computer Vision, Graphics, and Image Processing,
27(1), 19–31, July 1984.
2. J. Abello, A. Buchsbaum, and J. Westbrook. A functional approach to external graph algorithms. In
Proceedings of the European Symposium on Algorithms, volume 1461 of Lecture Notes in Computer
Science, 332–343, Venice, Italy, Aug. 1998. Springer-Verlag.
© 2002 by CRC Press LLC