1. How much does your program take ?
Analysis
How much does your program take ?

2. Data Structure: Analysis
Outline
Asymptotic notation
Big O, small o
, 
How to calculate running time
Sequential statements
If statements
Loops
Recursions
Examples
4/28/2019
Data Structure: Analysis

3. Data Structure: Analysis
Asymptotic Notations
T(N) = O(f(N)) if  positive constants c and n0 such that T(N)  c f(N) when N  n0.
Upper bound
f(N)
T(N)
c=1
Running time n0
Input size (N )
4/28/2019
Data Structure: Analysis

4. Data Structure: Analysis
Asymptotic Notations
T(N) = (f(N)) if  positive constants c and n0 such that T(N)  c f(N) when N  n0.
T(N)
f(N)
lower bound
c=1
Running time n0
Input size (N )
4/28/2019
Data Structure: Analysis

5. Data Structure: Analysis
Asymptotic Notations
T(N) = (f(N)) if T(N) = O(f(N)) and T(N) = (f(N)).
cuT(N)
f(N)
upper bound
clT(N)
lower bound
Running time
nu0
nl0
Input size (N )
4/28/2019
Data Structure: Analysis

6. Data Structure: Analysis
Asymptotic Notations
T(N) = o(f(N)) if T(N) = O(f(N)) and T(N)  (f(N)).
clT(N)
cuT(N)
f(N)
lower
bound
upper bound
Running time
nu0
Input size (N )
4/28/2019
Data Structure: Analysis

7. Data Structure: Analysis
What is faster?
4/28/2019
Data Structure: Analysis

8. Data Structure: Analysis
What is faster?
4/28/2019
Data Structure: Analysis

9. Running-time Calculation
Sequential statements S1 S2
Running time: O(T(N)) = O(T1(N)+T2(N))
= O(max(T1(N), T2(N)))
O(T1(N))
O(T2(N))
4/28/2019
Data Structure: Analysis

10. Running-time Calculation
Conditional statements
if (E) S1
else S2
Running time:
O(T(N)) = O(T3(N)) + O(max(T1(N),T2(N)))
= O(max(T1(N), T2(N), T3(N)))
O(T1(N))
O(T2(N))
O(T3(N))
4/28/2019
Data Structure: Analysis

11. Running-time Calculation
Iteration statements
for (i=1; i<=n;i++) S
Running time: ni=1 T(i)
If T(i) = O(n), ni=1 T(i) = O(n2).
If T(i) = O(1), ni=1 T(i) = O(n).
T(i)
4/28/2019
Data Structure: Analysis

12. Data Structure: Analysis
Examples
1 unit
2 units
for (i=1; i<=n;i++)
sum = sum+i;
Running time: T(n) = 1 + ni=13 = 1+3n = O(n)
for (j=i; j<=n; j++)
sum = sum+a[i,j];
Running time: T(n) = 1 + ni=1 ((1+nj=i3) +2)
= 1 + ni=1 (3+3(n-i+1)) = 1 + ni=1(3n-3i+6)
= 1+3n2+6n-3 ni=1i =2.5n2+5.5n+1= O(n2)
2 units
1 unit
4/28/2019
Data Structure: Analysis

13. Running-time Calculation
Recursive call
function rec(N)
if (…) return (…)
else { rec(N'); rec(N''); return (…); }
Running time: T(N) = T(N’) +T(N’’) + f(N)
T(1) = g(N)
4/28/2019
Data Structure: Analysis

14. Running-time Calculation
Recursive call
function rec(N)
if (…) return (…)
else { rec(N-1); …; return (…); }
Running time: T(N) = T(N-1) + f(N)
T(1) = k
T(N) = k + f(2) + f(3) +…+ f(N-1) + f(N)
T(N) = O(N  f(N)).
f(N)
4/28/2019
Data Structure: Analysis

15. Running-time Calculation
Recursive call
function rec(N)
if (…) return (…)
else { rec(N/2); …; return (…); }
Running time: T(N) = T(N/2) + f(N)
T(1) = k
T(N) = T(N/2) + f(N) = T(N/4)+ f(N/2)+ f(N)
= k+f(2)+f(4)+f(8)+…+ f(N/4)+ f(N/2)+ f(N)
T(N) = O( f(N) log2N )
f(N)
4/28/2019
Data Structure: Analysis

16. Running-time Calculation
Let T(N) = aT(N/b) + cnk.
If a<bk, T (n) = O(nk) .
If a=bk, T(n) = O(nk lg n) .
If a>bk, T(n) = O(nlogbl).
4/28/2019
Data Structure: Analysis

17. Example: Binary Search
public static int Bserach(int low, high, x)
{ int mid = (low+high)/2;
if (a[mid]<x)
return(Bsearch(mid+1, high, x));
else if (a[mid]>x)
return(Bsearch(low, mid-1, x));
else return(mid); }
T(N) = T(N/2) + k
T(N) = O(log N)
4/28/2019
Data Structure: Analysis

18. Example: Max Subsequence Sum
public static int maxSum(int left, int right)
{ if (left==right)
if (a[left]>0) return(a[left]);
else return(0);
int center=(left+right)/2;
int leftBorder =leftBorderSum(left, center);
int rightBorder=rightBorderSum(center+1, right);
return(max3(maxSum(left, center),
maxSum(center+1, right),
leftBorder+rightBorder); }
T(N) = 2T(N/2) + O(N)
T(N) = O(N log N)
4/28/2019
Data Structure: Analysis