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Algorithm Analysis Lectures 3 & 4

Algorithm Analysis Lectures 3 & 4. Resources Data Structures & Algorithms Analysis in C++ (MAW): Chap. 2 Introduction to Algorithms (Cormen, Leiserson, & Rivest): Chap.1 Algorithms Theory & Practice (Brassard & Bratley): Chap. 1. Algorithms .

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Algorithm Analysis Lectures 3 & 4

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  1. Algorithm AnalysisLectures 3 & 4 Resources Data Structures & Algorithms Analysis in C++ (MAW): Chap. 2 Introduction to Algorithms (Cormen, Leiserson, & Rivest): Chap.1 Algorithms Theory & Practice (Brassard & Bratley): Chap. 1

  2. Algorithms • An algorithm is a well-defined computational procedure that takes some value or a set of values, as input and produces some value, or a set of values as output. • Or, an algorithm is a well-specified set of instructions to be solve a problem.

  3. Efficiency of Algorithms • Empirical • Programming competing algorithms and trying them on different instances • Theoretical • Determining mathematically the quantity of resources (execution time, memory space, etc) needed by each algorithm

  4. Analyzing Algorithms • Predicting the resources that the algorithm requires: • Computational running time • Memory usage • Communication bandwidth • The running time of an algorithm • Number of primitive operations on a particular input size • Depends on • Input size (e.g. 60 elements vs. 70000) • The input itself ( partially sorted input for a sorting algorithm)

  5. Order of Growth • The order (rate) of growth of a running time • Ignore machine dependant constants • Look at growth of T(n) asn •  notation • Drop low-order terms • Ignore leading constants • E.g. • 3n3 + 90n2 – 2n +5 =  (n3)

  6. Mathematical Background

  7. Mathematical Background • Definitions: • T(N) = O(f(N)) iff c and n0  T(N)  c.f(N) when N  n0 • T(N) = (g(N)) iff c and n0  T(N)  c.g(N) when N  n0 • T(N) = (h(N)) iffT(N) = O(h(N)) and T(N) = (h(N))

  8. Mathematical Background • Definitions: • T(N) = o(f(N)) iff c and n0  T(N)  c.f(N) when N  n0 • T(N) = (g(N)) iff c and n0  T(N)  c.g(N) when N  n0

  9. Mathematical Background • Rules: • If T1(N) = O(f(N)) and T2(N) = O(g(N)) then a) T1(N) + T2(N) = max( O(f(N)),O(g(N)) b) T1(N) * T2(N) = O(f(N) * g(N)) • If T(N) is a polynomial of degree k, then T(N) = (Nk) • Logk N = O(N) for any constant k.

  10. More … • 3n3 + 90n2 – 2n +5 = O(n3 ) • 2n2 + 3n +1000000 = (n2) • 2n = o(n2)( set membership) • 3n2 = O(n2) tighter (n2) • n log n = O(n2) • True or false: • n2 = O(n3 ) • n3 = O(n2) • 2n+1= O(2n) • (n+1)! = O(n!)

  11. Ranking by Order of Growth 1 n n log n n2 nk (3/2)n 2n (n)! (n+1)!

  12. Running time calculations • Rule 1 – For Loops The running time of a for loop is at most the running time of the statement inside the for loop (including tests) times the number of iterations • Rule 2 – Nested Loops Analyze these inside out. The total running time of a statement inside a group of nested loops is the running time of the statement multiplied by the product of the sizes of all the loops

  13. Running time calculations: Examples Example 1: sum = 0; for (i=1; i <=n; i++) sum += n; Example 2: sum = 0; for (j=1; j<=n; j++) for (i=1; i<=j; i++) sum++; for (k=0; k<n; k++) A[k] = k;

  14. How to rank?

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