1. 2018, Fall Pusan National University Ki-Joune Li
Basic Concepts
2018, Fall
Pusan National University
Ki-Joune Li

2. Data Structures ? Refrigerator Problem Data Structures
If we have only one item in my refrigerator, no problem.
If we have several items in a refrigerator, it does matter.
Some Organization or Structures
Data Structures
How to place data in memory

3. What is good data structure?
What is good placement in refrigerator ?
Easy to cook (and place)
What is good data structure ?
Easy (and efficient) to use
It depends on what you want to do with the data structures
Two different aspects
Functions and
Internal Implementations

4. Two viewpoints and Abstract Data Types
Example: Suppose we have 1,000,000 integer values, then what the difference between
Array and
Stack ?
ADT
We need a clear separation
Interface for each function
Implementation 
Data Structure +
Algorithm
getElement(i)
putElement(i,v)
number()

5. Abstract Data Type Object-Oriented Programming
Abstraction (or Encapsulation)
Hiding the internal details
Implementation
Internal mechanism and process
Only provide Interfaces
Abstract Data Type
Hiding the internal structures once it has been implemented
Provide only the interface to the users

6. Algorithms Algorithm A sequence of instructions with specifications of
Input
Output : at least one output
Definiteness : Clear instructions
Finiteness
Effectiveness
Abstract description of a program
Can be easily converted to a program

7. Performance Analysis What is a good algorithm ?
Correctness
Good documentation and readable code
Proper structure
Effective
How to measure the effectiveness of an algorithm ?
Space complexity
Amount of memory it needs to run
Time complexity
Amount of time (mostly CPU time) it needs to run

8. Space Complexity Notation
Space complexity, f (n) : function of input size n
How should the constants be determined ?
Is it meaningful to count the number of bytes ?
int sumAB(int a, int b) { int sum; sum=a+b; return sum; }
int sumArray(int n, int *a)
{ int sum=0; for(int i=0;i<n;i++) sum += a[i]; return sum; }
{ if(n<=0) return 0; return a[n-1]+sumArrary(n-1,a);
f (n) = c1 c1 : constant
f (n) = c2 a[n] ?
f (n) = c3 n why ?

9. Is it really meaningful to determine these constants ?
Time Complexity
Notation
Time complexity, f (n) : function of input size n
int sumAB(int a, int b) { int sum; sum=a+b; return sum; }
int sumArray(int n, int *a)
{ int sum=0; for(int i=0;i<n;i++) sum += a[i]; return sum; }
{ if(n<=0) return 0; return a[n-1]+sumArrary(n-1,a);
f (n) = 2 + 
f (n) = 3 n 
f (n) = (2+  ) n
Is it really meaningful to determine these constants ?

10. Asymptotic Notation Exact time (or step count) to run
Depends on machines and implementation (program)
NOT good measure
More general (but inexact) notation : Asymptotic Notation
Big-O O(n) : Upper Bound
Omega-O (n) : Lower Bound
Theta-O (n) : More precise than Big-O and Omega-O
Big-O notation is the most popular one.

11. Big-O notation Definition ( f of n is big-O of g of n) Example
f (n)  O(g (n))  there exist c and n0 (c, n0 >0) such that f (n)  c·g(n), for all n ( n0)
Example
3n + 2 = O(n), 3n + 2 = O(n2)
Time complexity of the following algorithm f (n)= O(n)
int sumArray(int n, int *a)
{ if(n<=0) return 0; return a[n-1]+sumArrary(n-1,a); }

12. Big-O notation : Some Properties
Classification
O(1): constant, O(n): Linear, O(n2): Quadratic, O(2n): exponential
Polynomial function
If f (n) = amnm + am-1nm-1 + … + a1n + a0, then f (n)  O(nm), where ai > 0
Big-O is determined by the highest order
Only the term of the highest order is of our concern
Big-O : useful for determining the upper bound of time complexity
When only an upper bound is known, Big-O notation is useful
In most cases, not easy to find the exact f (n)
Big-O notation is the most used.

13. Omega-O notation Definition ( f of n is Omega-O of g of n) Example
f (n)  (g (n))  there exist c and n0 (c, n0 >0) such that f (n)  c·g(n), for all n ( n0)
Example
3n + 2 = (n), 3n + 2 = (1), 3n2 + 2 = (n),
Time complexity of the following algorithm f (n)= (n)
If f (n) = amnm + am-1nm-1+…+ a1n+a0, then f (n)  (nm)
Omega-O is determined by the highest order
Omega-O notation : useful to describe the lower bound
int sumArray(int n, int *a)
{ if(n<=0) return 0; return a[n-1]+sumArrary(n-1,a); }

14. Theta-O notation Definition ( f of n is theta-O of g of n) Example
f (n)  (g (n))  there exist c1, c2 and n0 (c1, c2, and n0 >0) such that
c1·g(n)  f (n)  c2·g(n), for all n ( n0)
Example
3n + 2 = (n), 3n + 2  (1), 3n2 + 2  (n),
Time complexity of the following algorithm f (n)= (n)
If f (n) = amnm + am-1nm-1+…+ a1n+a0, then f (n)  (nm)
Theta-O is determined by the highest order
Theta-O
Possible Only if f (n)=(g(n)), and f(n)=(g(n))
Lower bound and Upper bound is the same
very exact but not easy to find such a g(n)
int sumArray(int n, int *a)
{ if(n<=0) return 0; return a[n-1]+sumArrary(n-1,a); }

15. Complexity Analysis Worst-Case Analysis Average Analysis
Time complexity for the worst case
Example
Linear Search : f (n) = n
Average Analysis
Average time complexity
f (n) = p1f1(n) + p2f2(n) + … pkfk(n),
where pi is the probability for the i -th case.
Not easy to find pi
In most cases, only worst-case analysis
Why not Best-Case Analysis ?

16. Example : Worst-Case Time complexity of Binary Search
Big-O : O(n2), O(log n)
Omega O : (1), (log n)
Theta O : (log n)
int BinarySearch(int v, int *a, int lower, int upper)
// search m among a sorted array a[1ower], a[lower+1], … a[upper]
{ if(l<=u) {
int m=(lower+upper)/2;
if(v>a[m]), return BinarySearch(v,a,m+1,upper);
else if(v<a[m]) return BinarySearch(v,a,lower,m-1);
else /* v==a[m] */ return m; }
return -1; // not found

17. Comparison : O(1), O(log n), O(n), O(n2), O(2n)
Graph
In general
Algorithm of O(1) is almost impossible
Algorithms of O(log n) is excellent
Algorithms of O(n) is very good
Algorithms of O(n2 ) is not bad
Algorithms of O(n3 ) is acceptable
Algorithms of O(2n ) is not useful