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AmortizedAnalysis

# Invariant each bit that is set to 1 has one credit

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Unformatted text preview: crement 7 6 5 4 3 2 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 19 Binary counter: accounting method Credits. One credit pays for a bit flip. Invariant. Each bit that is set to 1 has one credit. Accounting. ・Flip bit j from 0 to 1: charge two credits (use one and save one in bit j). ・Flip bit j from 1 to 0: pay for it with saved credit in bit j. 7 6 5 4 3 2 1 0 0 1 0 1 0 0 0 0 20 Binary counter: accounting method Credits. One credit pays for a bit flip. Invariant. Each bit that is set to 1 has one credit. Accounting. ・Flip bit j from 0 to 1: charge two credits (use one and save one in bit j). ・Flip bit j from 1 to 0: pay for it with saved credit in bit j. Theorem. Starting from the zero counter, a sequence of n INCREMENT operations flips O(n) bits. Pf. The algorithm maintains the invariant that any bit that is currently set to 1 has one credit ⇒ number of credits in each bit ≥ 0. ▪ 21 Potential method (physicist's method) Potential function. Φ(Di) maps each data structure Di to a real number s.t.: ・Φ(D0) = 0. ・Φ(Di) ≥ 0 for eac...
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