# Chapter12

Amortized analysis

1. Dynamic table

We increase the table size by 2 times when the table is full
+ The initial table size is 1
+ Insert the value when the table has free slot which takes time: $O(1)$
+ Insert the value when the table has been full
+ Increase the table size by 2 times
+ Copy original elements from old table to new table
+ Insert value into new table.
+ Let $c_i$ represents the $i-th$ insertion:
+ $c_i=i$ if $i-1$ is an exact power of 2.
+ 1 otherwise
+ Total time cost：$$\sum_{i=1}^{n}c_i \le n+\sum_{j=0}^{\lfloor lg(n-1) \rfloor}2^j \le n+2^{\lfloor lg(n-1) \rfloor} \le 3n = \Theta(n)$$
+ the average cost of each dynamic-table operation is $$\Theta(n)/n=\Theta(1)$$

1. amortization arguments
• aggerate method
• accounting method
• potential method