Ch 2: Getting Started Ming-Te Chi Algorithms Ch 2: Getting Started Ming-Te Chi Ch2 Getting Started

2.1 Insertion sort Example: Sorting problem Input: A sequence of n numbers Output: A permutation of the input sequence such that . The number that we wish to sort are known as the keys. Ch2 Getting Started

Sorting a hand of cards Ch2 Getting Started

Pseudocode Ch2 Getting Started

The operation of Insertion-Sort 5, 2, 4, 6, 1, 3 Ch2 Getting Started

Sorted in place The numbers are rearranged within the array A, with at most a constant number of them sorted outside the array at any time Ch2 Getting Started

Loop invariant Initialization: Maintenance: Termination: It is true prior to the first iteration of the loop Maintenance: If it is true before an iteration of loop, it remains true before next iteration. Termination: When the loop terminates, the invariant gives us a useful property that show that the alogorithm is correct. Ch2 Getting Started

2.2 Analyzing algorithms Analyzing an algorithm has come to mean predicting the resources that the algorithm requires. Resources: memory, communication bandwidth, logic gate(hardware), and time. Assumption(Model):RAM(Random Access Model). single processor (We shall have occasion to investigate models for parallel computers and digital hardware.) Ch2 Getting Started

2.2 Analyzing algorithms The best notion for input size depends on the problem being studied. The running time of an algorithm on a particular input is the number of primitive operations or “steps” executed. It is convenient to define the notion of step so that it is as machine-independent as possible. Ch2 Getting Started

Analyzing insertion sort Ch2 Getting Started

the running time Ch2 Getting Started

The best case running time for j = 2, 3, …, n : Linear function on n Ch2 Getting Started

The worst case running time for j = 2,3,…,n : quadratic function on n. Ch2 Getting Started

Worst-case and average-case analysis Usually, we concentrate on finding only on the worst-case running time Reason: It is an upper bound on the running time. The worst case occurs fair often for some algorithm. The average case is often as bad as the worst case. For example, the insertion sort. Again, quadratic function. (Why?) Ch2 Getting Started

In some particular cases, we shall be interested in average-case, or expect running time of an algorithm. Ch2 Getting Started

Order of growth The leading term of formula (ex: ) It is the rate of growth, or order of growth, of the running time that really interests us. (More on this subject in a later lecture.) Probabilistic analysis. Ch2 Getting Started

2.3 Designing algorithms There are many ways to design algorithms: Incremental approach: insertion sort Divide-and-conquer: merge sort recursive: Divide the problem into a number of subproblems. Conquer the subproblems by solving them recursively. Combine the solutions to the subproblems into the solution for the original problem. Ch2 Getting Started

Merge sort Key operation of Merge sort Merge two sorted subsequences Use an auxiliary procedure Merge(A, p, q, r) A: an array p, q, r: indices with p <= q < r Ch2 Getting Started

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Example of Merge(A,p,q,r) Ch2 Getting Started

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MERGE-SORT(A,p,r) 1 if p < r 2 then q = (p+r)/2 3 MERGE-SORT(A,p,q) 4 MERGE-SORT(A,q+1,r) 5 MERGE(A,p,q,r) Ch2 Getting Started

Example of merge sort Ch2 Getting Started

Analysis of merge sort Use recurrence to analyze divide-and-conquer algorithms   Analysis of merge sort Ch2 Getting Started

Analysis of merge sort where the constant c represents the time require to solve problems of size 1 as well as the time per array element of the divide and combine steps. Ch2 Getting Started

Recursion tree Ch2 Getting Started

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Outperforms insertion sort! Merge sort Outperforms insertion sort! Ch2 Getting Started