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aa:ammortized-analysis [2017/01/11 16:14]
pdmatei 		aa:ammortized-analysis [2019/11/26 13:25] (current)
pdmatei
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 For the FIFO, we estimate: For the FIFO, we estimate:
   * $math[\hat{c}_{enq} = 3] since each inserted element in $math[l] **may** be subsequently removed from $math[l] and inserted in $math[r]. We charge extra to **amortise** for this potential cost.   * $math[\hat{c}_{enq} = 3] since each inserted element in $math[l] **may** be subsequently removed from $math[l] and inserted in $math[r]. We charge extra to **amortise** for this potential cost.
-  * $mat[\hat{c}_{deq} = 1] which represents the deletion from $math[r].+  * $math[\hat{c}_{deq} = 1] which represents the deletion from $math[r].
  
 To validate this estimation, we verify the **golden rule**. To validate this estimation, we verify the **golden rule**.
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 ===== The potential method ===== ===== The potential method =====
 +
 +We fix $math[\Phi(L_0) = 0] and $math[\Phi(L_i) = 2 * elems(L_i) - size(L_i)].
 +
 +We observe that $math[elems(L_i) \geq size(L_i)/​2] since we can never have fewer elements that half the capacity of the array. Thus, the golden rule of the potential method:
 +
 +$math[\forall S: \Phi(F_n) - \Phi(F_0) \geq 0]
 +
 +is immediately verified.
 +
 +To compute the ammortised cost, we observe two cases:
 +  * the operation does not trigger doubling: $math[size(L_i) = size(L_{i-1})]. Hence $math[\hat{c_i} = 1 + 2*elems(L_i) - size(L_i) -2*elems(L_{i-1}) + size(L_{i-1}) = 1 + 2 = 3]
 +  * the operation triggers doubling: $math[size(L_i) = 2*size(L_{i-1})]. Hence $math[\hat{c_i} = 1 + size(L_{i-1}) + 2*elems(L_i) - size(L_i) -2*elems(L_{i-1}) + size(L_{i-1}) = 1 + 2 = 3]
 +
 +Incidentally,​ we have identified precisely the same ammortised cost as in the previous method.