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

2. 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.
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3. Sorting a hand of cards
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4. Pseudocode
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5. The operation of Insertion-Sort
5, 2, 4, 6, 1, 3
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6. 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
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7. 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.
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8. 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.)
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9. 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.
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10. Analyzing insertion sort
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11. the running time
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12. The best case running time
for j = 2, 3, …, n : Linear function on n
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13. The worst case running time
for j = 2,3,…,n : quadratic function on n.
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14. 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?)
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15. In some particular cases, we shall be interested in average-case, or expect running time of an algorithm.
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16. 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.
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17. 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.
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18. 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
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19. Ch2 Getting Started

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

22. 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)
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23. Example of merge sort
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24. Analysis of merge sort
Use recurrence to analyze divide-and-conquer algorithms
Analysis of merge sort
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25. 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.
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26. Recursion tree
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27. Ch2 Getting Started

28. Ch2 Getting Started

29. Outperforms insertion sort!
Merge sort
Outperforms insertion sort!
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