DAA complete notes of GIET UNIVERSITY GUNUPUR, Summaries of Design and Analysis of Algorithms

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Analysis of Algorithm | Set 5 (Amortized Analysis

Introduction)

Amortized Analysis is used for algorithms where an occasional operation is very slow, but most of the other operations are faster. In Amortized Analysis, we analyze a sequence of operations and guarantee a worst case average time which is lower than the worst case time of a particular expensive operation. The example data structures whose operations are analyzed using Amortized Analysis are Hash Tables, Disjoint Sets and Splay Trees. Let us consider an example of a simple hash table insertions. How do we decide table size? There is a trade-off between space and time, if we make hash-table size big, search time becomes fast, but space required becomes high. The solution to this trade-off problem is to use Dynamic Table (or Arrays). The idea is to increase size of table whenever it becomes full. Following are the steps to follow when table becomes full.

1. Allocate memory for a larger table of size, typically twice the old table. 2) Copy the contents of old table to new table.
2. Free the old table. If the table has space available, we simply insert new item in available space. What is the time complexity of n insertions using the above scheme? If we use simple analysis, the worst case cost of an insertion is O(n). Therefore, worst case cost of n inserts is n * O(n) which is O(n^2 ). This analysis gives an upper bound, but not a tight upper bound for n insertions as all insertions don’t take Θ(n) time.