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Source: https://en.wikipedia/wiki/Insertion_sort#/media/File:Insertion_sort.gif

Source: https://en.wikipedia.org/wiki/Insertion_sort#/media/File:Insertion_sort.gif. Source: http://math.hws.edu/eck/cs124/javanotes3/c8/fig4.gif. Insertion Sort. for (int i = 1; i < n; i++) {// insert a[i] into a[0:i-1] int t = a[i]; int j; for (j = i - 1; j >= 0 && t < a[j]; j--)

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Source: https://en.wikipedia/wiki/Insertion_sort#/media/File:Insertion_sort.gif

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  1. Source: https://en.wikipedia.org/wiki/Insertion_sort#/media/File:Insertion_sort.gif

  2. Source: http://math.hws.edu/eck/cs124/javanotes3/c8/fig4.gif

  3. Insertion Sort for (int i = 1; i < n; i++) {// insert a[i] into a[0:i-1] int t = a[i]; int j; for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a[j + 1] = t; }

  4. Complexity • Space/Memory • Time • Count a particular operation • Count number of steps • Asymptotic complexity • What could be more important than performance? • Maintainability, robustness, extensibility, programmer time, etc.

  5. Comparison Count for (int i = 1; i < n; i++) {// insert a[i] into a[0:i-1] int t = a[i]; int j; for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a[j + 1] = t; }

  6. Comparison Count • Pick an instance characteristic … n • For insertion sort, pick n = number of elements to be sorted. • Determine count as a function of this instance characteristic.

  7. Comparison Count for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; How many comparisons are made?

  8. Comparison Count for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; number of compares depends on a[]s and t as well as on i

  9. Comparison Count • Worst-case count = maximum count • Best-case count = minimum count • Average count

  10. Worst-Case Comparison Count for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a = [1, 2, 3, 4] and t = 0 => 4 compares a = [1,2,3,…,i] and t = 0 => i compares

  11. Worst-Case Comparison Count for (int i = 1; i < n; i++) for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; total compares = 1 + 2 + 3 + … + (n-1) = (n-1)n/2

  12. Step Count A step is an amount of computing that does not depend on the instance characteristic n 10 adds, 100 subtracts, 1000 multiplies can all be counted as a single step 10n  O(n), 100n  O(n), etc. n adds cannot be counted as 1 step

  13. Step Count Steps per execution for (int i = 1; i < n; i++)1 {// insert a[i] into a[0:i-1] 0 int t = a[i]; 1 int j; 0 for (j = i - 1; j >= 0 && t < a[j]; j--) 1 a[j + 1] = a[j]; 1 a[j + 1] = t; 1 } 0 for (int i = 1; i < n; i++) {// insert a[i] into a[0:i-1] int t = a[i]; int j; for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a[j + 1] = t; }

  14. Step Count s/e isn’t always 0 or 1 x = sum(a[], n); where n is the instance characteristic has a s/e count of n (sum() is Program 1.30).

  15. Step Count s/e steps for (int i = 1; i < n; i++)1 {// insert a[i] into a[0:i-1] 0 int t = a[i]; 1 int j; 0 for (j = i - 1; j >= 0 && t < a[j]; j--) 1 a[j + 1] = a[j]; 1 a[j + 1] = t; 1 } 0 for (int i = 1; i < n; i++) {// insert a[i] into a[0:i-1] int t = a[i]; int j; for (j = i - 1; j >= 0 && t < a[j]; j--) a[j + 1] = a[j]; a[j + 1] = t; } i+ 1 i

  16. Step Count step count for for (int i = 1; i < n; i++) is n i = 1 i = 2 … 1+2+3+…+n = n(n-1)/2

  17. Space Complexity • O(n) total • O(1) additional space Source: https://en.wikipedia.org/wiki/Insertion_sort#/media/File:Insertion-sort-example-300px.gif

  18. Asymptotic Complexity of Insertion Sort • O(n2) • What does this mean?

  19. Complexity of Insertion Sort • Time or number of operations does not exceed c.n2 on any input of size n (n suitably large)(c constant). • Actually, the worst-case time is Q(n2) and the best-case is Q(n) • So, the worst-case time is expected to quadruple each time n is doubled

  20. Complexity of Insertion Sort • Is O(n2) too much time? • Is the algorithm practical?

  21. Practical Complexities 109 instructions/second

  22. Impractical Complexities 109 instructions/second Age of the universe: 13.8 * 10^9 years!*

  23. Faster Computer Vs Better Algorithm Algorithmic improvement more useful than hardware improvement. E.g. 2n to n3

  24. To-do • Read: • https://en.wikipedia.org/wiki/Big_O_notation • Do: • Create GitHub (or similar) account • Insert 0 into 1 2 3 4 5 – Count Steps • Insert 10 into 1 2 3 4 5 – Count Steps • Implement (ungraded): • Insertion Sort (C++) • Sort a sorted sequence/reversely sorted sequence (n=1000,1000000), what’s the difference in runtime?

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