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31 Theorem. Starting from the17.2 An counter, a sequence of n0 Ito 16 by a sequence of 16 I NCREMENT
zero 8bit binary counter as its value goes from NCREMENT
Figure
operations.
operations flips O(n k) bits. Pf. At most k bits Bits that ﬂip to achieve the next value are shaded. The running cost for ﬂipping bits is
shown at the right. Notice that the total cost is always less than twice the total number of I NCREMENT
operations.
flipped per increment. ▪
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operations on an initially zero counter causes AŒ1 to ﬂip bn=2c times. Similarly,
bit AŒ2 ﬂips only every fourth time, or bn=4c times in a sequence of n I NCREMENT i Aggregate method (brute force)
Aggregate method. Sum up sequence of operations, weighted by their cost. Counter
value
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 Spring '14

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