The.Algorithm.Design.Manual.Springer-Verlag.1998

Logarithms
justified in ignoring the base of the logarithm when analyzing algorithms.
When performing binary search in a telephone book, how important is it that each query split the
book exactly in half? Not much. Suppose we did such a sloppy job of picking our queries such
that each time we split the book 1/3 to 2/3 instead of 1/2 to 1/2. For the Manhattan telephone
book, we now use queries in the worst case, not a significant change from
. The power of binary search comes from its logarithmic complexity, not the
base of the log.
● Logarithms cut any function down to size: The growth rate of the logarithm of any polynomial
function is . This follows because
The power of binary search on a wide range of problems is a consequence of this observation. For
example, note that doing a binary search on a sorted array of things requires only twice as many
comparisons as a binary search on n things. Logarithms efficiently cut any function down to size.
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Algorithms
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