Computer Science 101 A Survey of Computer Science

1. Computer Science 101A Survey of Computer Science Efficiency of Divide and Conquer

2. Background - Logarithms (Base 2) • Definition. The logarithm to base 2 of n is that number k such that 2k=n. • Example: 25=32 so Lg(32)=5. • Another way to think of this is that Lg(n) is the number of times we must divide n by 2 until we get 1. • Note: Lg(n) is usually a fraction, but the closest integer will work for us.

3. Base 2 Logarithms - Table • n Lg(n) n Lg(n)1 0 1024 102 1 2048 114 2 4096 128 3 8192 1316 4 1,048,576 2032 564 6128 7256 8512 9

4. Quicksort - Rough Analysis • For simplification, assume that we always get even splits when we partition. • When we partition the entire list, each element is compared with the pivot - approximately n comparisons.

5. Quicksort - Rough Analysis (cont.) • Each of the “halves” is partitioned, each taking about n/2 comparisons, thus about n more comparisons. • Each of the “fourths” is partitioned,each taking about n/4 comparisons - n more.

6. Quicksort - Rough Analysis (cont.) • How many “levels” of “about n comparisons” do we get? • Roughly, we keep splitting until the pieces are about size 1.

7. Quicksort - Rough Analysis (cont.) • How many times must we divide n by 2 before we get 1? • Lg(n) times, of course. • Thus Comparisons  n Lg(n) • Quicksort is O(n Lg(n)) and T(n)  KnLg(n) (Linearithmic)

8. Growth rates

9. Binary Search Algorithm • Note: For searching sorted list • Given: n, N1,N2,…Nn (sorted) and target T • Want: Position where T is located or message indicating that T is not in list

10. Binary Search - The algorithm Set Found to falseSet B to 1Set E to nWhile not Found and B ≤ E Set M to (B+E)/2 (whole part) If N(M)=T then Print M Set Found to true else If T<NM then Set E to M-1 else Set B to M+1end-of-loopIf not Found then Print “Target not in list”

11. 40 42 50 63 70 85 M=(B+E)/2 4 8 20 26 31 39 40 42 50 63 70 85 Target 50 4 8 20 26 31 39 40 42 50 63 70 85 B=1 4 8 20 26 31 39 40 42 50 63 70 85 E=n 4 8 20 26 31 39 40 42 50 63 70 85 M=(B+E)/2 4 8 20 26 31 39 40 42 50 63 70 85 T>NM B=M+1 Binary Search Example Output location 9

12. 4 8 20 26 31 39 40 42 50 63 70 85 Target 31 4 8 20 26 31 39 40 42 50 63 70 85 B=1,E=n 4 8 20 26 31 39 40 42 50 63 70 85 T<NM E=M-1 4 8 20 26 31 39 40 42 50 63 70 85 M=(B+E)/2 4 8 20 26 31 M=(B+E)/2 4 8 20 26 31 T>NM B=M+1 Binary Search Example

13. 26 31 M=(B+E)/2 26 31 T>NM B=M+1 31 M=(B+E)/2 Binary Search Example (cont.) Output location 5

14. Efficiency - Binary Search • Note with 1000 elements, Sequential Search would have a worst case number of comparisons of 1000. • After 1 comparison, Binary Search would be left with at most 500 elements. • After 2 comparisons 250 at worst • After 3 comparisons 125 at worst • After 4 comparisons 62 at worst • After 5 comparisons 31 at worst • After 6 comparisons 15 at worst • After 7 comparisons 7 at worst • After 8 comparisons 3 at worst • After 9 comparisons 1 at worst

15. Efficiency - Binary Search (cont) • Each time we make a comparison, we cut the list in half. • How many times must we do this to end up with 1 element? • Lg(n), of course. • Binary search is O(Lg(n)) • T(n)  K Lg(n)

16. Growth rates

17. If it takes that long for 100, we'll dang well be here all night!