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Advanced Algorithms Analysis and Design
Lecture 6
(Continuation of 5th Lecture)
By
Engr Huma Ayub Vine
1
Algorithm Comparison (5 elements)
Quicksort
first call
2nd call
3rd call
Each time the (sub) list is partitioned exactly one value (the pivot) is placed in its correct position.
If the list is equally divided during each partitioning, it will require about lg (n) calls to quickSort to produce a sorted list.
Assignment 2
Prove that quick sort recursive case average analysis is
Submit after midterm
Algorithm Comparison (2K Elements)
9
Computation Time for 100,000 elements
250
200
150
100
50
0
Series1
COMPARISONSort Worst Average Best
Method2 2Bubble O(n ) O(n ) O(n)
Quick O(n2 ) O(n log n) O(log n)
Heap O(n log n) O(n log n) O(log n)
2 2Insertion O(n ) O(n ) O(n)2 2Selection O(n ) O(n ) O(n)
Merge O(n log n) O(n log n) O(log n)
Radix O(n2 ) O(s*n) O(n log n)2 5/3Shell O(n ) O(n ) O(n)
Counting O(n + k) O(n) O(n)
Bucket O(n2 ) O(n) O(n)
Pigeonhole O(n + s) O(n) O(n)
ALGORITHMS AND THEIR COMPLEXITY(Limits on Problem Size as Determined by Growth Rate)
Algo TimeComplexity
Maximum Problem Size (n)
1 sec 1 min 1 hour
A1 n 1000 6 x 104 3.6 x 106
A2 n log n 140 4893 2.0 x 105
A3 n2 31 244 1897
A4 3n 10 39 153
A5 2n 9 15 21
Calculating the Greatest Common DivisorFunction Euclid (m,n)
while m > 0 dot ← mm ← n mod mn ← t
return n
Euclid (14, 21) : t m n14 7 147 0 7
return = 7
Euclid (6, 15) : t m n6 3 63 0 3
return = 3Takes a time in the order of the algorithm of its arguments
Calculating the Fibonacci Sequence
ƒ0 = 0, ƒ1 = 1ƒn = ƒn-1 + ƒn-2 for n > = 2Function Fibrec (n)
if n < 2 then return n
else return Fibrec (n - 1) + Fibrec (n - 2)
f0 f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f110 1 1 2 3 5 8 13 21 34 55 89Time required to calculate fn is in the order of thevalue of fn. It is very inefficient compared to deMoivre’s formula• It is better to use Fibiter algorithm to calculate theFibonnaci Sequence
Function Fibiter (n)
{ Calculates the n-th term of the Fibonacci sequence;}
i ← 1;j ← 0
for k ← 1 to n doj ← i + j
i ← j - ireturn jFibonacci (5)
i = 1j = 0k = 1 2 3 4 5j = 1 1 2 3 5i= 0 1 1 2 3• Time required to calculate fn is in the order of n
Calculating the Fibonacci Sequence
de Moivre’s formula
fn=1 n -n[φ−(−φ) ]5
1+ 5where φ=
2(φ=1.61803)
Time required to calculate fn is value of fn is in theorder of φn
Examplef10 = f10-1 + f10-2
= f9 + f8= 34+21
= 55 ---------------------- (i)
f 10=
=
1 n − n[φ−(−φ) ]5
1 10 −10[1.61803) −(−1.61803) ]2.236
=
=
1 [2.236
1 [2.236
122.98883−0.008131]
122.9807]
= 55.0003
= 55 (ii)
Function Fibiter (n)i ← 1; j ← 0for k ← 1 to n d
j ← i+ji
← j - ireturn j
Comparison of modulo fibonacci algorithms
n 10 20 30 50 100
Fibrec 8 msec 1 sec 2 min 21 days 109 years
Fibiter 1/6 1/3 ½ msec ¾ msec 1 ½ msecmsec msec
Important Points for an Algorithm
• Correct in execution
• Execution time
• Storage needs
• Limitation of computing equipment to support
operations for desired numbers to have
precision within in limits
• Efficient methodology for the specific task
• Programming/ hardware implementation tools
• Average/worst-case performance