1. تصميم وتحليل الخوارزميات عال311 Chapter 3 Growth of Functions

2. How fast will your program run?
The running time of any program will depend on:
The algorithm
The input
Your implementation of the algorithm in a programming language
The compiler you use
The OS on your computer
Your computer hardware
other things: temperature outside; other programs on your computer; …

3. Complexity
Complexity is the number of steps required to solve a problem
The goal is
to find the best algorithm to solve the problem with a less number of steps

4. Measures of Algorithm Complexity
Let T(n) denote the number of operations required by an algorithm to solve a given problem.
Often T(n) depends on the input n, there are 3 cases:
Worst-case complexity,
Best-case complexity,
Average-case complexity of an algorithm

5. Measures of Algorithm Complexity
Worst-Case Running Time: the longest time for any input size of n
provides an upper bound on running time for any input
Best-Case Running Time: the shortest time for any input size of n
provides lower bound on running time for any input
Average-Case Behavior: the expected performance averaged over all possible inputs
it is generally better than worst case behavior, but sometimes it’s roughly as bad as worst case

6. Example: Sequential Search
Sequential Search Algorithm
Step Count
// Searches for x in array A of n items returns
// index of found item, or n+1 if not found
Seq_Search( A[n]: array, x: item){
i = 1
while ((i  n) & (A[i] <> x)){
i = i +1 }
return i
_________________________________
Total time 1
n + 1 n
____________
=2n+3
= 2(n+1)

7. Example: Sequential Search
worst-case running time (Sequential Search)
when x is not in the original array A
in this case, while loop needs 2(n + 1) comparisons + c other operations
So, T(n) = 2(n + 1) + c  Linear complexity
best-case running time (Sequential Search)
when x is found in A[1]
in this case, while loop needs 2 comparisons + c other operations
So, T(n) = 2 + c  Constant complexity

8. Big-O notation (Upper Bound – Worst Case)
n, n + 1, n + 80, 40n, n + lg n is O(n)
n n is O(n1.1)
n n is O(n2)
3n2 + 6n + lg n is O(n2)
O(1)  O(lg n)  O((lg n)3)  O(n)  O(n2)  O(n3)  O(nlg n)  O(2sqrt(n))  O(2n)  O(n!)  O(nn)
Constant  Logarithmic  Linear  Quadratic  Cubic  Exponential  Factorial

9. Some Common Name for Complexity
Constant time
O(lg n)
Logarithmic time
O(lg2 n)
Log-squared time
O(n)
Linear time
O(n2)
Quadratic time
O(n3)
Cubic time
O(ni) for some i
Polynomial time
O(2n)
Exponential time

10. (Mathematics) Exponents
x0 = 1 x1 = x x-1 = 1/x
xa . xb = xa+b
xa / xb = xa-b
(xa)b = (xb)a = xab
xn + xn = 2xn  x2n
2n + 2n = 2.2n = 2n+1

11. Summation

12. Factorials n! (“n factorial”) is defined for integers n  0 as n! =
n!  nn for n  2
Examples: 5! = 1 * 2 * 3 *4 *5 = 120
2! = 1*2 = 2
0! = 1