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This material is provided primarily for review and reference. You might wish to return to the relevant sections when you encounter unfamiliar notation or mathematical techniques in later chapters. 7 on estimating might be unfamiliar to many readers. Estimating is not a mathematical technique, but rather a general engineering skill. It is enormously useful to computer scientists doing design work, because any proposed solution whose estimated resource requirements fall well outside the problem’s resource constraints can be discarded immediately.

Log(nm) = log n + log m. 2. log(n/m) = log n − log m. 3. log(nr ) = r log n. 4. loga n = logb n/ logb a. 4 Property (3) is simply an extension of property (1). Property (4) tells us that, for variable n and any two integer constants a and b, loga n and logb n differ by the constant factor logb a, regardless of the value of n. Most runtime analyses in this book are of a type that ignores constant factors in costs. Property (4) says that such analyses need not be concerned with the base of the logarithm, because this can change the total cost only by a constant factor.

4) 34 Chap. 2 Mathematical Preliminaries n ai = i=0 an+1 − 1 for a = 1. 6) i=1 and n 2i = 2n+1 − 1. 7, log n 2i = 2log n+1 − 1 = 2n − 1. 8) n+2 . 9) i=0 Finally, n i=1 i 2i = 2− The sum of reciprocals from 1 to n, called the Harmonic Series and written Hn , has a value between loge n and loge n + 1. 5772... 3). Unfortunately, induction does not help us derive a closed-form solution. It only confirms when a proposed closed-form solution is correct. 1. The running time for a recursive algorithm is most easily expressed by a recursive expression because the total time for the recursive algorithm includes the time to run the recursive call(s).

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A Practical Introduction to Data Structures and Algorithm Analysis Third Edition (Java Version) by Clifford A. Shaffer

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