SDSU CS 660: Combinatorial Algorithms
Introduction Part A

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San Diego State University -- This page last updated Septemper 3, 1995

Contents of Introduction Part A Lecture

  1. References
  2. Mathematical Analysis of Algorithms
    1. Model of Computing
    2. Asymptotic Notation


Introduction To Algorithms, Corman, Leiserson,Rivest, Chapters 1-4

Mathematical Analysis of Algorithms

Model of Computing

If analysis of algorithms is the answer, what is the question?

Given two or more algorithms for the same task, which is better?
Under which condition is bubble sort better than insertion sort?

What computing resources does an algorithm require?
How long will it take bubble sort to sort a list of N items?

Goal of mathematical analysis is a function of the resources required of an algorithm

On what computer?
What is a Computer?
Random-access machine (RAM)
Single processor
Instructions executed sequentially
Each operation requires the same amount of time

Single cost vs. Lg(N) cost
Time required for basic operation?
3 + 6
Insertion Sort
A[0] = - infinity
for K = 2 to N do
J = K;
Key = A[J];
while Key < A[J-1] do
A[J] = A[J-1];
J = J - 1;
end while;
A[J] = Key;
end for;
Resources required by the algorithm as a function of the input size

Worst-case Analysis
Complexity of an algorithm based on worst input of each size

Average-case Analysis
Complexity of an algorithm averaged over all inputs of each size
Insertion Sort
		Comparisons	Element moves

	worst case	(N+1)N/2 - 1	(N-1)N/2

	average case	(N+1)N/4 - 1/2	(N-1)N/4

Asymptotic Notation

Asymptotically tight bound

Asymptotic upper bounds
Common Myths and Errors
instead of:
or even that there is an n such that

Let f(n) = 2n + 10, and g(n) = n then
f(n) = O(g(n)) but f(n) > g(n)

Bubble vs. Insertion Sort
		Worst case	Average case

	Bubble sort		

	Insertion Sort		

Bubble Sort
		Comparisons	Element moves

	worst case	(N-1)N/2	3(N-1)N/2

	average case	(N-1)N/2	3(N-1)N/4

	best case	(N-1)N/2	0

Insertion Sort
		Comparisons	Element moves

	worst case	(N+1)N/2 - 1	(N-1)N/2

	average case	(N+1)N/4 - 1/2	(N-1)N/4

	best case	N - 1	0
Bubble vs. Insertion SortTiming Results
Worst Case
	N	Bubble	Insertion
	100	1	1
	200	5	3
	400	19	11
	800	79	42
	1600	317	166

Average Case
	N	Bubble	Insertion
	100	1	0
	200	3	1
	400	14	5
	800	56	21
	1600	228	84

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