Algorithm Analysis

Lakshmish Ramaswamy

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

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]

Insertion Sort Algorithmpublic 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;}

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

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

Analysis• Constant cost for comparison and

assignment– 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(2 + 3 + ….+ N) = c/2*(N2-N-2)

• O(N2) algorithm

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

Bubble Sort Algorithmpublic 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]);}

}}

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

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

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

Merge Algorithmpublic 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]){A3[i+j] = A1[i]; i++; }else{A3[i+j] = A2[j]; j++; }}return(A3);}

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

Merge Sort

• Recursive algorithm• Partition array at the center• Call merge sort on each part of the array• Call merge to combine the two parts

Trace

[9, 3, 8, 12, 1, 5, 22, 18]