1. Data Structures Types of Data Structures Algorithms
Algorithms Running times

2. Algorithm is a specified set of instructions for solving a problem

3. Various Criteria for Goodness of Algorithm
Correctness
Performance
Running Times (minimum)
Memory requirement (minimum)
Generality (maximum)
Clarity (maximum)

4. Example: Insertion sort
Input: 5, 2, 4, 6, 1, 3
Output: 1, 2, 3, 4, 5, 6

5. Example: Insertion sort
Input: A= {5, 2, 4, 6, 1, 3}
Output: A={1, 2, 3, 4, 5, 6} i j
sorted
key
n=length[A]=6

6. Insertion Sort InsertionSort(A) 5 2 4 6 1 3 for j2 to length[A] do
keyA[j]
ij-1
while i>0 and A[i]>key do
A[i+1]A[i]
ii-1
A[i+1]key

7. Insertion Sort –Running Time
Cost Times
C1 n
C n-1
C n-1 C4 C5 C6
C n-1
InsertionSort(A)
for j2 to length[A] do
keyA[j]
ij-1
while i>0 and A[i]>key do
A[i+1]A[i]
ii-1
A[i+1]key

8. Insertion Sort –Running Time
Cost Times
C1 n
C n-1
C n-1 C4 C5 C6
C n-1
T(n)=C1n+ C2(n-1)+C3(n-1)+
+ C7(n-1)+

9. Running Time Algorithm depends on:
Input size (n)
Input itself (e.g. partially sorted)
Speed of primitive operations(constants) (will be ignored in future)

10. Running Time Algorithm Kinds of Analysis:
Worst Case: T(n)=max time of any input size n (Upper Bound)
Average Case: T(n)=average time of any input size n
Best Case: (Lower bound)

11. Insertion Sort The best case: A is already sorted
tj=1 for any j
T(n)=C1n+ C2(n-1)+C3(n-1)+ C7(n-1)+ C4(n-1)
= (C1 + C2 +C3 + C4+ C7 )n-(C2+ C3+ C4+ C7)
= an+b
T(n) is linear function of n

12. Insertion Sort The worst case: A is sorted in reverse order (A[j] is compared with all elements in A[1…j-1])
tj=j for each j
T(n)=C1n+ C2(n-1)+C3(n-1)+ C7(n-1)+
= (C1 + C2 +C3 + C7 )n-(C2+ C3+ C7)+ C4 (n(n+1)/2-1)
+(C5+ C6 )n(n-1)/2 = A n2+B n+C
T(n) is quadratic function of n

13. Insertion Sort The average case:
tj=j/2 for each j
T(n) is still quadratic function of n

14. Insertion Sort-The worst case and Average case Asymptotic Analysis (large n) Notations
Example: T(n)=3n2+5n-2=
(n2)
For large n average case is not much better than the worst
(g(n))
( will be defined formally later)