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Unformatted text preview: Alpha A final word about addition/subtraction chains (an idea due to Rob Miller ) Improving the 2 (log 2 n)  1 upper bound on the length of the shortest chain for generating n . Express n using redundant notation with as few 1 ’s and 1 ’s as possible. Examples: 31 = 0 1 1 1 1 1 31 = 1 0 0 0 0 1 Generate powers of two: 1, 2, 4, …, 2 log n . Add a power of two for each 1 , subtract for each 1 . How many 1 ’s and 1 ’s are necessary? Example: 1 1 1 1 1 1 1 0 0 1 1 0 1 1. Adjacent 1 ’s and 1 ’s can be eliminated, e.g., 1 1 => 0 1 1 1 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 1 How many 1 ’s and...
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This document was uploaded on 01/17/2012.
 Fall '09

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