Space-time tradeoff

In computer science, a space-time or time-memory tradeoff is a situation where the memory use can be reduced at the cost of slower program execution, or vice versa, the computation time can be reduced at the cost of increased memory use. As the relative costs of CPU cycles, RAM space, and hard drive space change — hard drive space has for some time been getting cheaper at a much faster rate than other components of computers^{[citation needed]} — the appropriate choices for space-time tradeoffs have changed radically. Often, by exploiting a space-time tradeoff, a program can be made to run much faster.

The most common situation is an algorithm involving a lookup table: an implementation can include the entire table, which reduces computing time, but increases the amount of memory needed, or it can compute table entries as needed, increasing computing time, but reducing memory requirements.

A space-time tradeoff can be applied to the problem of data storage. If data is stored uncompressed, it takes more space but less time than if the data were stored compressed (since compressing the data reduces the amount of space it takes, but it takes time to run the compression algorithm). Depending on the particular instance of the problem, either way is practical. Another example is displaying mathematical formulae on primarily text-based websites, such as Wikipedia. Storing only the LaTeX source and rendering it as an image every time the page is requested would be trading time for space - more time used, but less space. Rendering the image when the page is changed and storing the rendered images would be trading space for time - more space used, but less time. Note that there are also rare instances where it is possible to directly work with compressed data, such as in the case of compressed bitmap indices, where it is faster to work with compression than without compression.

Larger code size can be traded for higher program speed when applying loop unwinding. This technique makes the code longer for each iteration of a loop, but saves the computation time required for jumping back to the beginning of the loop at the end of each iteration.

Algorithms that make use of space-time tradeoffs to achieve better running times include the baby-step giant-step algorithm for calculating discrete logarithms.

In the field of cryptography, where the adversary is trying to do better than the exponential time required for a brute force attack, the advantages of a space-time tradeoff can readily be seen. Rainbow tables use partially precomputed values in the hash space of a cryptographic hash function to crack passwords in minutes instead of weeks. Decreasing the size of the rainbow table increases the time required to iterate over the hash space. The meet-in-the-middle attack uses a space-time tradeoff to find the cryptographic key in only 2^{n + 1} encryptions (and O(2ⁿ) space) versus the expected 2²ⁿ encryptions (but only O(1) space) of the naive attack.

Dynamic programming is another example where the time complexity of a problem can be reduced significantly by using more memory.

See also

• Blum's speedup theorem
• Savitch's theorem

External links

• Philippe Oechslin: Making a Faster Cryptanalytic Time-Memory Trade-Off.
• Once Upon a Time-Memory Tradeoff.

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