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1 24 Binary counter: potential method
Potential function. Let Φ(D) = number of 1 bits in the binary counter D. ・Φ(D0) = 0.
・Φ(Di) ≥ 0 for each Di. increment 7 6 5 4 3 2 1 0 0 1 0 0
1 1
0 1
0 1
0 1
0 25 Binary counter: potential method
Potential function. Let Φ(D) = number of 1 bits in the binary counter D. ・Φ(D0) = 0.
・Φ(Di) ≥ 0 for each Di. 7 6 5 4 3 2 1 0 0 1 0 1 0 0 0 0 26 Binary counter: potential method
Potential function. Let Φ(D) = number of 1 bits in the binary counter D. ・Φ(D0) = 0.
・Φ(Di) ≥ 0 for each Di.
Theorem. Starting from the zero counter, a sequence of n INCREMENT
operations flips O(n) bits.
Pf. ・Suppose that the ith increment operation flips ti bits from 1 to 0.
operation sets one bit to 1 (unless counter resets to zero)
・The actual cost ci ≤ ti + 1.
・The amortized cost ĉi = ci + Φ(Di) – Φ(Di–1) ≤ ci + 1 – ti ≤ 2. ▪ 27 Famous potential functions
Fibonacci heaps. Φ(H) = trees(H) + 2 marks(H).
Splay trees. log2 size(x) (T ) =
xT Movetofront. Φ(L) = 2 × inversions(L, L*)...
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This document was uploaded on 02/05/2014.
 Spring '14

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