AmortizedAnalysis

Di di1 2 amortized costs i ci di di1 1

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Unformatted text preview: city of array 4 5 6 43 Dynamic table: insert only Theorem. [via potential method] Starting from an empty dynamic table, any sequence of n INSERT operations takes O(n) time. Pf. Let Φ(Di) = 2 size(Di) – capacity(Di). number of elements capacity of array Case 1. [does not trigger expansion] size(Di) ≤ capacity(Di–1). ・Actual cost ci = 1. ・Φ(Di) – Φ(Di–1) = 2. ・Amortized costs ĉi = ci + Φ(Di) – Φ(Di–1) = 1 + 2 = 3. Case 2. [triggers expansion] size(Di) = 1 + capacity(Di–1). ・Actual cost ci = 1 + capacity(Di–1). ・Φ(Di) – Φ(Di–1) = 2 – capacity(Di) + capacity(Di–1) = 2 – capacity(Di–1). ・Amortized costs ĉi = ci + Φ(Di) – Φ(Di–1) = 1 + 2 = 3. ▪ 44 Dynamic table: doubling and halving Thrashing. ・Initialize table to be of fixed size, say 1. ・INSERT: if table is full, expand to a table of twice the size. ・DELETE: if table is ½-full, contract to a table of half the size. Efficient solution. ・Initialize table to be of fixed size, say 1. ・INSERT: if table is full, expand...
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