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Basic Concepts 2015, Fall Pusan National University Ki-Joune Li Slide 2 STEMPNU Refrigerator Problem 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 2 Data Structures ? Slide 3 STEMPNU 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 Slide 4 STEMPNU Example: Suppose we have 1,000,000 integer values, then what the difference between Array and Stack ? ADT 4 Two viewpoints and Abstract Data Types Implementation Data Structure + Algorithm Implementation Data Structure + Algorithm getElement(i) putElement(i,v) number() Interface for each function We need a clear separation Slide 5 STEMPNU 5 Abstract Data Type Object-Oriented Programming Object ? 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 Slide 6 STEMPNU 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 Slide 7 STEMPNU 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 Slide 8 STEMPNU 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 STEMPNU 11 Big-O notation Definition ( f of n is big-O of g of n) f (n) O(g (n)) there exist c and n 0 (c, n 0 >0) such that f (n) cg(n), for all n ( n 0 ) Example 3n + 2 = O(n), 3n + 2 = O(n 2 ) Time complexity of the following algorithm f (n)= O(n) int sumArray(int n, int *a) { if(n STEMPNU 12 Big-O notation : Some Properties Classification O(1): constant, O(n): Linear, O(n 2 ): Quadratic, O(2 n ): exponential Polynomial function If f (n) = a m n m + a m-1 n m-1 + + a 1 n + a 0, then f (n) O(n m ), where a i > 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. Slide 13 STEMPNU 13 Omega-O notation Definition ( f of n is Omega-O of g of n) f (n) (g (n)) there exist c and n 0 (c, n 0 >0) such that f (n) cg(n), for all n ( n 0 ) Example 3n + 2 = (n), 3n + 2 = (1), 3n 2 + 2 = (n), Time complexity of the following algorithm f (n)= (n) If f (n) = a m n m + a m-1 n m-1 ++ a 1 n+a 0, then f (n) (n m ) 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 STEMPNU 14 Theta-O notation Definition ( f of n is theta-O of g of n) f (n) (g (n)) there exist c 1, c 2 and n 0 (c 1, c 2, and n 0 >0) such that c 1 g(n) f (n) c 2 g(n), for all n ( n 0 ) Example 3n + 2 = (n), 3n + 2 (1), 3n 2 + 2 (n), Time complexity of the following algorithm f (n)= (n) If f (n) = a m n m + a m-1 n m-1 ++ a 1 n+a 0, then f (n) (n m ) 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 STEMPNU 16 Example : Worst-Case Time complexity of Binary Search Binary Search Big-O : O(n 2 ), 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(la[m]), return BinarySearch(v,a,m+1,upper); else if(v