Lecture No.04. Data Structures
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1 Lecture No.04 Data Structures Josephus Problem #include "CList.cpp" void main(int argc, char *argv[]) { CList list; int i, N=10, M=3; for(i=1; i <= N; i++ ) list.add(i); } list.start(); while( list.length() > 1 ) { for(i=1; i <= M; i++ ) list.next(); cout << "remove: " << list.get() << endl; list.remove(); } cout << "leader is: " << list.get() << endl; 1
2 Josephus Problem Using a circularly-linked list made the solution trivial. Josephus Problem Using a circularly-linked list made the solution trivial. The solution would have been more difficult if an array had been used.
3 Josephus Problem Using a circularly-linked list made the solution trivial. The solution would have been more difficult if an array had been used. This illustrates the fact that the choice of the appropriate data structures can significantly simplify an algorithm. It can make the algorithm much faster and efficient. Josephus Problem Using a circularly-linked list made the solution trivial. The solution would have been more difficult if an array had been used. This illustrates the fact that the choice of the appropriate data structures can significantly simplify an algorithm. It can make the algorithm much faster and efficient. Later we will see how some elegant data structures lie at the heart of major algorithms. 3
4 Josephus Problem Using a circularly-linked list made the solution trivial. The solution would have been more difficult if an array had been used. This illustrates the fact that the choice of the appropriate data structures can significantly simplify an algorithm. It can make the algorithm much faster and efficient. Later we will see how some elegant data structures lie at the heart of major algorithms. An entire CS course Design and Analysis of Algorithms is devoted to this ic. Abstract Data Type We have looked at four different implementations of the List data structures: Using arrays Singly linked list Doubly linked list Circularly linked list. 4
5 Abstract Data Type We have looked at four different implementations of the List data structures: Using arrays Singly linked list Doubly linked list Circularly linked list. The interface to the List stayed the same, i.e., add(), get(), next(), start(), remove() etc. Abstract Data Type We have looked at four different implementations of the List data structures: Using arrays Singly linked list Doubly linked list Circularly linked list. The interface to the List stayed the same, i.e., add(), get(), next(), start(), remove() etc. The list is thus an abstract data type; we use it without being concerned with how it is implemented.
6 Abstract Data Type What we care about is the methods that are available for use with the List ADT. Abstract Data Type What we care about is the methods that are available for use with the List ADT. We will follow this theme when we develop other ADT. 6
7 Abstract Data Type What we care about is the methods that are available for use with the List ADT. We will follow this theme when we develop other ADT. We will publish the interface and keep the freedom to change the implementation of ADT without effecting users of the ADT. Abstract Data Type What we care about is the methods that are available for use with the List ADT. We will follow this theme when we develop other ADT. We will publish the interface and keep the freedom to change the implementation of ADT without effecting users of the ADT. The C++ classes provide us the ability to create such ADTs. 7
8 Stacks Stacks in real life: stack of books, stack of plates Add new items at the Remove an item at the Stack data structure similar to real life: collection of elements arranged in a linear order. Can only access element at the Stack Operations Push(X) insert X as the element of the stack Pop() remove the element of the stack and return it. Top() return the element without removing it from the stack. 8
9 Stack Operations push() push() push(7) push(1) pop() push(1) 1 pop() 7 pop() pop() Stack Operation The last element to go into the stack is the first to come out: LIFO Last In First Out. What happens if we call pop() and there is no element? Have IsEmpty() boolean function that returns true if stack is empty, false otherwise. Throw StackEmpty exception: advanced C++ concept. 9
10 Stack Implementation: Array Worst case for insertion and deletion from an array when insert and delete from the beginning: shift elements to the left. Best case for insert and delete is at the end of the array no need to shift any elements. Implement push() and pop() by inserting and deleting at the end of an array. Stack using an Array = 3 10
11 Stack using an Array In case of an array, it is possible that the array may fill-up if we push enough elements. Have a boolean function IsFull() which returns true is stack (array) is full, false otherwise. We would call this function before calling push(x). Stack Operations with Array int pop() { return A[current--]; } void push(int x) { } A[++current] = x; 11
12 Stack Operations with Array int () { return A[current]; } int IsEmpty() { return ( current == -1 ); } int IsFull() { return ( current == size-1); } A quick examination shows that all five operations take constant time. Stack Using Linked List We can avoid the size limitation of a stack implemented with an array by using a linked list to hold the stack elements. As with array, however, we need to decide where to insert elements in the list and where to delete them so that push and pop will run the fastest. 1
13 Stack Using Linked List We can avoid the size limitation of a stack implemented with an array by using a linked list to hold the stack elements. As with array, however, we need to decide where to insert elements in the list and where to delete them so that push and pop will run the fastest. Stack Using Linked List For a singly-linked list, insert at start or end takes constant time using the head and current pointers respectively. Removing an element at the start is constant time but removal at the end required traversing the list to the node one before the last. Make sense to place stack elements at the start of the list because insert and removal are constant time. 13
14 Stack Using Linked List No need for the current pointer; head is enough. 1 7 head 1 7 Stack Operation: List int pop() { int x = head->get(); Node* p = head; head = head->getnext(); delete p; return x; } head
15 Stack Operation: List void push(int x) { Node* newnode = new Node(); newnode->set(x); newnode->setnext(head); head = newnode; } head 9 7 newnode 9 7 push(9) Stack Operation: List int () { return head->get(); } int IsEmpty() { return ( head == NULL ); } All four operations take constant time. 1
16 Stack: Array or List Since both implementations support stack operations in constant time, any reason to choose one over the other? Allocating and deallocating memory for list nodes does take more time than preallocated array. List uses only as much memory as required by the nodes; array requires allocation ahead of time. List pointers (head, next) require extra memory. Array has an upper limit; List is limited by dynamic memory allocation. Use of Stack Example of use: prefix, infix, postfix expressions. Consider the expression A+B: we think of applying the operator + to the operands A and B. + is termed a binary operator: it takes two operands. Writing the sum as A+B is called the infix form of the expression. 16
17 Prefix, Infix, Postfix Two other ways of writing the expression are + A B prefix A B + postfix The prefixes pre and post refer to the position of the operator with respect to the two operands. Prefix, Infix, Postfix Consider the infix expression A + B * C We know that multiplication is done before addition. The expression is interpreted as A + ( B * C ) Multiplication has precedence over addition. 17
18 Prefix, Infix, Postfix Conversion to postfix A + ( B * C ) infix form Prefix, Infix, Postfix Conversion to postfix A + ( B * C ) A + ( B C * ) infix form convert multiplication 18
19 Prefix, Infix, Postfix Conversion to postfix A + ( B * C ) A + ( B C * ) A ( B C * ) + infix form convert multiplication convert addition Prefix, Infix, Postfix Conversion to postfix A + ( B * C ) A + ( B C * ) A ( B C * ) + A B C * + infix form convert multiplication convert addition postfix form 19
20 Prefix, Infix, Postfix Conversion to postfix (A + B ) * C infix form Prefix, Infix, Postfix Conversion to postfix (A + B ) * C infix form ( A B + ) * C convert addition 0
21 Prefix, Infix, Postfix Conversion to postfix (A + B ) * C infix form ( A B + ) * C convert addition ( A B + ) C * convert multiplication Prefix, Infix, Postfix Conversion to postfix (A + B ) * C infix form ( A B + ) * C convert addition ( A B + ) C * convert multiplication A B + C * postfix form 1
22 Precedence of Operators The five binary operators are: addition, subtraction, multiplication, division and exponentiation. The order of precedence is (highest to lowest) Exponentiation Multiplication/division *, / Addition/subtraction +, - Precedence of Operators For operators of same precedence, the left-to-right rule applies: A+B+C means (A+B)+C. For exponentiation, the right-to-left rule applies A B C means A ( B C )
23 Infix to Postfix Infix Postfix A + B A B (A + B)*(C D ) A B + C D * A B * C D + E/F A B C*D E F/+ 3
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