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Understanding Running Time Analysis in Algorithms: Concepts and Examples

This guide explores the importance of analyzing the running time of algorithms, detailing methods such as running and timing, and emphasizing why these options may be inadequate. The description also covers key concepts like pseudo-code, big-oh notation, and asymptotic analysis. Examples are provided to illustrate how to analyze algorithms, including finding maximums in matrices and counting matching elements in arrays. By understanding these principles, you can better assess algorithm efficiency and performance across varying input sizes.

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Understanding Running Time Analysis in Algorithms: Concepts and Examples

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  1. Algorithm/Running Time Analysis

  2. Running Time • Why do we need to analyze the running time of a program? • Option 1: Run the program and time it • Why is this option bad? • What can we do about it?

  3. Pseudo-Code • Used to specify algorithms • Part English, part code Algorithm (arrayMax(A, n)) curMax = A[0] for i=1 i<n i++ if curMax < A[i] curMax = A[i] return curMax

  4. Math Review • Summation – ∑ • Sum of n consecutive digits = n(n+1)/2

  5. Counting Operations Algorithm (arrayMax(A, n)) curMax = A[0]//1 for i=1 i<n i++ //n //1 or 2 if curMax < A[i] curMax = A[i] return curMax //1 • Best case – n+2 • Worst case – 2n + 2 • Average case – hard to analyze

  6. Asymptotic Notation • 2n + 2 • n=5 -> 12 • n=100 -> 202 • n=1,000,000 -> 2,000,002 • Running time grows proportionally to n • What happens as n gets large?

  7. Big-Oh • f(n) is O(g(n)) if there is a real constant c>0 and an integer constant n0>=1 such that f(n) <= cg(n) for every integer n>=n0 • 2n+2 is O(n) n0>=1 c=3

  8. Examples • 87n4 + 7n • 3nlogn + 12logn • 4n4 + 7n3logn

  9. Terminology • Logarithmic – O(logn) • Linear – O(n) • Linearithmic – O(nlogn) • Quadratic – O(n2) • Polynomial – O(nk) k>=1 • Exponential – O(an) a>1

  10. Example 0 n-1 0 6 4 … 2 • Find maximum number in nxn matrix • Algorithm: 12 3 … 9 … … … … n-1 5 8 … 1

  11. Example • What is the big-oh running time of this algorithm? Algorithm: Input: A, n curMax = A[0][0] for i=0 i<n i++ for j=0 j<n j++ if curMax < A[i][j] curMax = A[i][j] return curMax

  12. Another Example 0 n-1 2 4 … 6 • Determine how many elements of array 1 match elements of array 2 • Algorithm? 0 n-1 6 8 … 3

  13. Another Example 0 n-1 2 4 … 6 Algorithm: Input: A, B, n for i=0 i<n i++ for j=0 j<n j++ if A[i] == A[j] matches++ break • What is the running time of the algorithm? 0 n-1 6 8 … 3

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