1. Algorithms
Lakshmish Ramaswamy

2. Binary Search Algorithm
Problem – In an sorted array of numbers, search whether a given number is present
Can the fact that the array is sorted be used to reduce comparisons?

3. Illustration 2 5 7 11 14 19 25 31 7 25
Depending upon the relationship part of the array can be safely eliminated from future comparisons

4. Logic of Binary Search Compare given number to center of the array
If found terminate
Eliminate one half of the array
Continue until no more comparisons left

5. Binary Search Algorithm
public static int (int[] Arr, int given){
int high = Arr.length-1;
int low = 0;
int mid;
while(low <= high){
mid = (low+high)/2;
if(Arr[mid] < given) low = mid+1;
else if(Arr[mid] > given) high = mid-1;
else return (mid); }
return(-1);

6. Trace
[2, 4, 6, 9, 13, 16, 20, 24, 26, 29, 30, 32, 36, 38, 41, 50]
given = 36
given = 4
given =3

7. Analysis At most two comparisons at each iteration
Constant time per comparison
Repeated halving
log2N comparisons
O(log N) algorithm

8. Sorting Algorithms
Problem – Given an array, rearrange the elements such that they are in increasing (or decreasing order)
Fundamental operation with wide range of applications
Several algorithms
Selection sort
Insertion sort
Bubble sort
Merge sort
Quick sort
Radix sort

9. Selection Sort
Logic – In ith iteration select the ith smallest element and place it ith location
How to select the ith minimum element
Repeated use of minimum element algorithm

10. Selection Sort Algorithm
public static void SelectionSort(int[] Arr){
for(int i = 0; i < Arr.length-1; i++){
minElement = Arr[i];
minIndex = i;
for(j = (i+1); j < Arr.length; j++)
if(Arr[j] < minElement){
minElement = Arr[j];
minIndex = j; }
swap(Arr[i], Arr[minIndex]);

11. Trace
[ 9, 3, 8, 12, 1, 5, 22, 18, 14, 2]

12. Analysis Constant time for comparison (N-1) comparisons when i = 0
1 comparison when i = (N-2)
Total comparisons = 1+2+…+(N-2) + (N-1)
Total comparisons = (N-1)N/2
O(N2) algorithm

13. Insertion Sort Conceptual Logic Create a duplicate array
Insert elements from original array into duplicate array (ith element in iteration i)
Maintain sortedness of duplicate array at all times
May need to move existing elements to accommodate the incoming element

14. Trace
[9, 3, 8, 12, 1, 5, 22, 18, 14, 2]
[9]
[3, 9]
[3, 8, 9]
[3, 8, 9, 12]
[1, 3, 8, 9, 12] …
[1, 2, 3, 5, 8, 9, 12, 14, 18, 22]