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aa:ammortized-analysis [2017/01/11 16:11] pdmatei created |
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**. | ||
| Line 122: | Line 122: | ||
| where $math[size(l_i)] is the size of the left list after the $math[ith] operation. | where $math[size(l_i)] is the size of the left list after the $math[ith] operation. | ||
| - | $math[\hat{c}_{deq} = 1 + 2*size(l_i) - 2*size(l_{i-1}) = 1 + 0 = 1] | + | For dequeue, we consider two cases: |
| + | * no normalization takes place: $math[\hat{c}_{deq} = 1 + 2*size(l_i) - 2*size(l_{i-1}) = 1 + 0 = 1] | ||
| + | * normalization takes place: $math[\hat{c}_{deq} = 1 + 2*size(l_{i-1}) + 0 - 2*size(l_{i-1}) = 1] | ||
| note that during dequeueing, the size of the left list does not change. | note that during dequeueing, the size of the left list does not change. | ||