1. Algorithms
Lakshmish Ramaswamy

2. 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

3. 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]

4. Insertion Sort Algorithm
public static void InsertionSort(int[] Arr){
int i, j, k, currElement;
if(Arr.length == 1) return;
for (i = 1; i < (Arr.length); i++){
currElement = Arr[i];
for(j = 0; j < i && Arr[j] <= currElement; j++);
for(k = (i-1); k => j; k++)
Arr[k+1] = Arr[k];
Arr[j] = currElement; }
return;

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

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

7. Analysis Constant cost for comparison and assignment For ith iteration
Assume equal cost ‘c’ for simplicity
For ith iteration
‘t’ comparisons for some t <= (i-1)
i-t+1 assignments
Cost = c(i+1)
Total cost = c( ….+ N) = c/2*(N2-N-2)
O(N2) algorithm

8. Bubble Sort Imitates bubbles raising through water
Compares two neighboring elements
Swap if they are in the wrong order
Repeat the procedure (N-1) times

9. Bubble Sort Algorithm public static void BubbleSort(int[] Arr){
int i, j;
for(i = 0; i < (Arr.length-1); i++){
for(j = (Arr.length-1); j > i; j--){
if(Arr[j] < Arr[j-1])
swap(Arr[j], Arr[j-1]); }

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

11. Analysis Constant time for comparison and swap
Outer loop is executed (N-1) times
For a particular i the inner loop is executed N-i-1 times
# comparisons = (N-1) + (N-2) + … + 2
N(N-1)/2 – 1
O(N2) algorithm

12. Merge
Problem – Merge two sorted arrays such that the resultant array remains sorted
Logic
Keep pointers to both arrays
Compare the current pointer elements of two arrays
Place the one that is lowest and increment its pointer

13. Merge Algorithm public static int[] Merge(int[] A1, int[] A2){
int[] A3 = new int[A1.length+ A2.length];
int i, j;
while(i < A1.length && j < A2.length){
if(i => A1.length){
A3[i+j] = A2[j]; j++; }
elseif (j => A2.length){
A3[i+j] = A1[i]; i++; }
elseif(A1[i] < A2[j]){
else{ }
return(A3);

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