8/19/2014. Most algorithms transform input objects into output objects The running time of an algorithm typically grows with input size

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1 1. Algorithm analysis 3. Stacks 4. Queues 5. Double Ended Queues Semester I (2014) 1 Most algorithms transform input objects into output objects The running time of an algorithm typically grows with input size Semester I (2014) 2 1

2 Big Oh Notation is used to characterize running times and space bound in terms of some parameter n. Definition: Lt Let f(n) and g(n) be function mapping nonnegative integers to real numbers We say that f(n) is O(g(n)) if there is a real constant c>0 and an integer constant n 0 1, such that f(n) < cg(n) for every integer n n 0 c g(n) running tim me n o Input size f(n) Semester I (2014) 3 Examples: 7n 2 is O( ) 20n n logn + 5 is O( ) 3 log n + log log n is O( ) is O( 1 ) 5/n is O( ) 2n 3 + 4n 2 log n is O( ) n (2i+3) is O( ) Semester I (2014) 4 2

3 Common terms used in algorithm analysis: Logarithmic O(log n) Linear O(n) Quadratic O(n 2 ) Cubic O(n 3 ) Polynomial O(n k ) (k 1) Exponential O(a n ) Any algorithm running in O(n log n) time should be considered d efficient i Semester I (2014) 5 Linear O(n) Quadratic O(n 2 ) Cubic O(n 3 ) Semester I (2014) 6 3

4 Relatives of Big Oh big Omega f(n) is Ω(g(n)) if there is a constant c > 0, and an integer constant n0 1 such that f(n) c g(n) for n n0 big Theta f(n) is Θ(g(n)) if there are constants c > 0 and c > 0 and an integer constant n0 1 such that c g(n) f(n) c g(n) for n n0 little oh f(n) is o(g(n)) if, for any constant c > 0, there is an integer constant n0 0 such that f(n) c g(n) for n n0 little omega f(n) is ω(g(n)) if, for any constant c > 0, there is an integer constant n0 0 such that f(n) c g(n) for n n0 Semester I (2014) 7 Big Oh f(n) is O(g(n)) if f(n) is asymptotically less than or equal to g(n) Big Omega f(n) is Ω(g(n)) if f(n) is asymptotically greater than or equal to g(n) Big Theta f(n) is Θ(g(n)) if f(n) is asymptotically equal to g(n) Little oh f(n) is o(g(n)) if f(n) is asymptotically strictly less than g(n) Little omega f(n) is ω(g(n)) if is asymptotically strictly greater than g(n) Semester I (2014) 8 4

5 Class Assignment 1. Algorithm A uses 10nlog n operation Algorithm B uses n 2 operation. determine the value n 0 such that A is better than B for n n 0 2. Show that log 3 n is O(n 1/3 ) 3. Show that O(max{f(n), g(n)}) = O(f(n)+g(n)) 4. Show that if p(n) is polynomial in n, then log p(n) is O(log n) 5. Show that (n+1) 5 is O(n 5 ) 6. Show that n i 1 i is O( n 2 ) Semester I (2014) 9 Definition: a function that can call itself as a subroutine Linear Recursion A function is defined so that it makes one recursive call each time it is invoked Example: calculate sum of the first n integers in A. int LinearSum(int A[],int n) { if n = 1 then return A[0]; else return LinearSum(A, n 1) + A[n 1]; } Semester I (2014) 10 5

6 2.1. Linear Recursion Test for the base case: At least one should not use recursion Recursion Each recursive step makes the progress towards a base case A = {4, 3, 6, 2, 5} return 7 + A[2] = = 13 LinearSum(A, 3) LinearSum(A, 2) return 4 + A[1] = 4+ 3 = 7 return A[0] = 4 LinearSum(A, 1) Semester I (2014) Linear Recursion Algorithm ReverseArray(A, i, n): Input: An integer array A and integers iand n Output: t The reversal of the n integers in A starting ti at index i if n>1 then Swap A[i] and A[i+n 1] Call ReverseArray(A, i+1, n 2) return Semester I (2014) 12 6

7 2.1. Linear Recursion Computing Power via Linear Recursion power(x, n) = x n 1 power( x, n) x power( x, n 1) The algorithm uses O(n) time n 0 otherwise 1 power( x, n) x. power( x,( n 1) / 2) 2 power( x, n / 2) The algorithm uses O(log n) time 2 if n 0 if n 0is odd if n 0 is even Semester I (2014) High order Recursion Binary recursion Example Algorithm BinarySum(A,i,n): Input: An integer array A and integer iand n Output: The sum of the n integer in A starting at index i If n = 1 then return A[i] return BinarySum (A,i, [n/2]) + BinarySum(A, i + [n/2], n/2]) The running time of the algorithm is O(log n) Semester I (2014) 14 7

8 2.2. High order Recursion Computing the kth Fibonacci number using binary recursion Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, Recursion relation: F 0 = 0; F 1 = 1; F n = F n 1 + F n 2 Semester I (2014) Stack Stack is an abstract data type (ADT) that uses last in first out principle Support functions: push(o): insert object o at the top of the stack Input: Object; Output: None pop(): remove from the stack and return the top on the stack; an error occurs if the stack is empty Input: None; Output: None size(): return the number of objects in the stack Input: None; Output: Integer isempty(): return a Boolean indicating if the stack is empty Input: None; Output: Boolean top(): Return the top object on the stack, without removing it; an error occurs if the stack is empty Input: None; Output: Object Note that STL stack may have different functions Semester I (2014) 16 8

9 3. Stack No. Operation Output Stack contents > top 1 Push(5) (5) 2 Push(3) (5, 3) 3 4 Pop() Push(7) 5 Pop() 6 Top() 7 Pop() 8 pop() 9 isempty() 10 Push(9) 11 Push(7) 12 Push(3) 13 Push(5) 14 Size() Semester I (2014) Stack Example for STL stack: #include <iostream> #include <stack> using namespace std; void main() { stack<int> mystack; mystack.push(3); mystack.push(4); cout << mystack.size() << \n ; cout << mystack.top() << \n ; mystack.pop(); //pop() function is void type cout << mystack.top() << \n ; cout << mystack.size() << \n ; } What shows on the output screen? Semester I (2014) 18 9

10 4. Queues Queue is a data structure which uses first in first out principle Functions: enqueue(o): insert object o at the rear of the queue Input: Object; Output: None dequeue(): remove and return from the queue the object at the front; an error occurs if the queue is empty. Input: None; Output: None size(): Return the number of the objects in the queue Input: None; Output: Integer. isempty(): return a Boolean value indicating if the queue is empty. Input: None; Output: Boolean. front(): return, but do not remove, a reference to the front element in the queue; an error occurs if the queue is empty. Input: None; Output: Object. Note that STL queue may have different functions Semester I (2014) Queues No. Operation Output Front < Queue < rear 1 enqueue(5) (5) 2 enqueue(3) (5, 3) 3 4 dequeue() enqueue(7) () 5 dequeue() 6 front() 7 dequeue() 8 dequeue() 9 isempty() 10 enqueue(9) 11 enqueue(7) 12 size() 13 enqueue(3) 14 enqueue(5) Semester I (2014) 20 10

11 4. Queues Application of Queues Direct applications waiting lists (e.g., stores, theaters, reservation centers) access to shared resources (e.g., printers, scanner) multiprogramming Indirect applications Auxiliary data structure for algorithms (e.g., scheduling) Component of other data structures (e.g., pattern) Semester I (2014) Queue Example for STL Queue: #include <iostream> #include <queue> using namespace std; void main() { queue<int> myqueue; myqueue.push(3); myqueue.push(4); myqueue.push(5); myqueue.push(6); cout << myqueue.size() << '\n'; cout << myqueue.front() << '\n'; myqueue.pop (); cout << myqueue.empty() << '\n'; cout << myqueue.size() << '\n'; getchar(); } What shows on the output screen? Semester I (2014) 22 11

12 5. Double Ended Queues Deque Deque, pronounced deck, is a queue like data structure that supports insertion and deletion at both the front and the rear of the queue. Functions of the deque (D): insertfront(o): insert a new object o at the beginning gof D Input: Object; Output: None insertback(o): insert a new object o at the end of D. Input: Object; Output: None erasefront(o): remove the first object of D; an error occurs if D is empty Input: None; Output: None eraseback(): remove the last object of D; an error occurs if D is empty Input: None; Output: None Note that STL deque may have different functions Semester I (2014) Double Ended Queues Deque Additional functions: front(): Return the first object of D; an error occurs if D is empty back(): empty. size(): Return the last object of D; an error occurs if D is Return the number of the objects in D Input: None; Output: Integer. isempty(): return a Boolean value indicating if the queue is empty. Input: None; Output: Boolean. Note that STL deque may have different functions Semester I (2014) 24 12

13 5. Double Ended Queues Deque Operation Output D insertfront(3) (3) insertfront(5) first() erasefront() insertback(7) last() erasefront() eraseback() Semester I (2014) Double Ended Queues Deque Example for STL Deque #include <iostream> #include <deque> using namespace std; void main() { deque<int> mydeque; mydeque.push_front(3); mydeque.push_back(4); mydeque.push_back(5); mydeque.push_front(6); cout << mydeque.size() << '\n'; cout << mydeque.front() << '\n'; cout << mydeque.back() << '\n'; \n; mydeque.pop_back(); cout << mydeque.empty() << '\n'; cout << mydeque.size() << '\n'; getchar(); } Semester I (2014) 26 13