applied cryptography - protocols, algorithms, and source code in c

Problems that can be solved with polynomial time

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Unformatted text preview: er: John Wiley & Sons, Inc.) Author(s): Bruce Schneier ISBN: 0471128457 Publication Date: 01/01/96 Search this book: Go! Previous Table of Contents Next ----------- Complexity of Algorithms An algorithm’s complexity is determined by the computational power needed to execute it. The computational complexity of an algorithm is often measured by two variables: T (for time complexity) and S (for space complexity, or memory requirement). Both T and S are commonly expressed as functions of n, where n is the size of the input. (There are other measures of complexity: the number of random bits, the communications bandwidth, the amount of data, and so on.) Generally, the computational complexity of an algorithm is expressed in what is called “big O” notation: the order of magnitude of the computational complexity. It’s just the term of the complexity function which grows the fastest as n gets larger; all lower-order terms are ignored. For example, if the time complexity of a given algorithm is 4n2 + 7n + 12, then the computational complexity is on the order of n2, expressed O(n2). Measuring time complexity this way is system-independent. You don’t have to know the exact timings of various instructions or the number of bits used to represent different variables or even the speed of the processor. One computer might be 50 percent faster than another and a third might have a data path twice as wide, but the order-of-magnitude complexity of an algorithm remains the same. This isn’t cheating; when you’re dealing with algorithms as complex as the ones presented here, the other stuff is negligible (is a constant factor) compared to the order-of-magnitude complexity. This notation allows you to see how the input size affects the time and space requirements. For example, if T = O(n), then doubling the input size doubles the running time of the algorithm. If T = O(2n), then adding one bit to the input size doubles the running time of the algorithm (within a constant factor). General...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.

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