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Algorithm Analysis
Lakshmish Ramaswamy
Formal Definitions
• Big-Oh: T(N) is O(F(N)) if there exists
positive constants N0 and c such that T(N)
<= cF(N) for all N => N0
• Big-Omega: T(N) is Ω(F(N)) if there exists
positive constants N0 and c such that T(N)
>= cF(N) for all N => N0
• Big-Theta: T(N) is Θ(F(N)) if and only if
T(N) is both O(F(N)) and Ω(F(N))
• Little-Oh: T(N) is o(F(N)) if and only if
T(N) is O(F(N)) and T(N) is not Θ(F(N))
Meaning of Notations
Logarithm
• Definition – For any B, N > 0 logBN = k if Bk
=N
• Base has no effect on Big-Oh
• For any B > 1 logBN = O(log N)
– logB(N) = log2 (N)/log2 (B)
• Log functions are characterized by slow
growth
– log 10 = 2.3; log 1000 = 6.9; log 100000 = 11.5
Examples of Logarithm
• Bits required to represent N (in binary)
– Upper limit of log2 N
• Repeated doubling
– Starting from X = 1, how many times should X
be doubled before it is at least as large as N?
• Repeated halving
– Starting from X = N, how many times should X
be halved before it is lesser than or equal to
1?
Sequential Search
• Problem – In an array of numbers, search
whether a given number is present
public static boolen sequentialsearch(int[]
A, num)
for (i = 0; i < A.length; i++){
if (A[i] == num) return(1);
}
return(0)
• O(N) algorithm
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?
Illustration
2
7
5
7
11
14
19
25
31
25
• Depending upon the relationship part of
the array can be safely eliminated from
future comparisons
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
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);
}
Trace
[2, 4, 6, 9, 13, 16, 20, 24, 26, 29, 30, 32, 36,
38, 41, 50]
given = 36
given = 4
given =3
Analysis
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•
•
•
•
At most two comparisons at each iteration
Constant time per comparison
Repeated halving
log2N comparisons
O(log N) algorithm
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
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
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]);
}
}
Trace
[ 9, 3, 8, 12, 1, 5, 22, 18, 14, 2]
Analysis
•
•
•
•
•
•
•
Constant time for comparison
(N-1) comparisons when i = 0
(N-2) comparisons when i = 1
1 comparison when i = (N-2)
Total comparisons = 1+2+…+(N-2) + (N-1)
Total comparisons = (N-1)N/2
O(N2) algorithm