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Part-B1 Stacks
Stacks 2
Abstract Data Types (ADTs)
An abstract data type (ADT) is an abstraction of a data structureAn ADT specifies: Data stored Operations on
the data Error conditions
associated with operations
Example: ADT modeling a students record The data stored are
Student name, id No., as1, as2,as3, exam
The operations supported are
int averageAs(as1,as2,as3) Int finalMark(as1, as2,as3,
exam) ) Error conditions:
Calculate the final mark for absent student
Stacks 3
The Stack ADT (§4.2)The Stack ADT stores arbitrary objectsInsertions and deletions follow the last-in first-out schemeThink of a spring-loaded plate dispenserMain stack operations:
push(object): inserts an element
object pop(): removes and returns the last inserted element
Auxiliary stack operations:
object top(): returns the last inserted element without removing it
integer size(): returns the number of elements stored
boolean isEmpty(): indicates whether no elements are stored
Stacks 4
Stack Interface in Java
Java interface corresponding to our Stack ADTRequires the definition of class EmptyStackException
public interface Stack {
public int size();
public boolean isEmpty();
public Object top()throws EmptyStackException;
public void push(Object o);
public Object pop() throws EmptyStackException;
}
Stacks 5
ExceptionsAttempting the execution of an operation of ADT may sometimes cause an error condition, called an exceptionExceptions are said to be “thrown” by an operation that cannot be executed
In the Stack ADT, operations pop and top cannot be performed if the stack is emptyAttempting the execution of pop or top on an empty stack throws an EmptyStackException
Stacks 6
Applications of Stacks
Direct applications Undo sequence in a text editor Chain of method calls in the Java
Virtual Machine
Indirect applications Auxiliary data structure for algorithms Component of other data structures
Stacks 7
Array-based Stack (Implementation)
A simple way of implementing the Stack ADT uses an arrayWe add elements from left to rightA variable t keeps track of the index of the top element
S0 1 2 t
…
Algorithm size()return t + 1
Algorithm pop()if isEmpty() then
throw EmptyStackException else
t t 1return S[t + 1]
Stacks 8
Array-based Stack (cont.)The array storing the stack elements may become fullA push operation will then throw a FullStackException
Limitation of the array-based implementation
Not intrinsic to the Stack ADT
S0 1 2 t
…
Algorithm push(o)if t = S.length 1 then
throw FullStackException else
t t + 1S[t] o
Stacks 9
Array-based Stack (Cont.)
A Stack might be emptytop returns the element at the top of the Stack, but does not remove the top element. When the Stack is empty, an error occurs.
S0 1 2 t
…
Algorithm isEmpty() if t<0 then return true else return false
Algorithm top()if isEmpty() then
throw EmptyStackException return S[t ]
Stacks 10
Performance and Limitations for array-based Stack
Performance Let n be the number of elements in the stack The space used is O(n) Each operation runs in time O(1)
Limitations The maximum size of the stack must be
defined a priori and cannot be changed Trying to push a new element into a full stack
causes an implementation-specific exception
Stacks 11
Array-based Stack in Javapublic class ArrayStack implements Stack{// holds the stack elements
private Object S[ ];// index to top elementprivate int top = -1;// constructor
public ArrayStack(int capacity) { S = new Object[capacity]);
}
Stacks 12
Array-based Stack in Javapublic Object pop() throws EmptyStackException{
if isEmpty()throw new EmptyStackException
(“Empty stack: cannot pop”);Object temp = S[top];// facilitates garbage collection S[top] = null;top = top – 1;return temp;
} public int size() { return (top + 1);
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Array-based Stack in Java public boolean isEmpty() { return (top < 0); }public void push(Object obj) throws FullStackException { if (size() == capacity) throw new FullStackException("Stack overflow."); S[++top] = obj; } public Object top() throws EmptyStackException { if (isEmpty()) throw new EmptyStackException("Stack is empty."); return S[top]; }
Stacks 14
Parentheses Matching
An expression, i.e.,[(2+3)*x+5]*2. Each “(”, “{”, or “[” must be paired with a matching “)”, “}”, or “[” correct: ( )(( )){([( )])} correct: ((( )(( )){([( )])} incorrect: )(( )){([( )])} incorrect: ({[ ])} incorrect: (
Stacks 15
Parentheses Matching Algorithm
Algorithm ParenMatch(X,n):Input: An array X of n tokens, each of which is either a grouping symbol, avariable, an arithmetic operator, or a numberOutput: true if and only if all the grouping symbols in X matchLet S be an empty stackfor i=0 to n-1 do
if X[i] is an opening grouping symbol thenS.push(X[i])
else if X[i] is a closing grouping symbol thenif S.isEmpty() then
return false {nothing to match with} if S.pop() does not match the type of X[i] then
return false {wrong type} if S.isEmpty() then
return true {every symbol matched} else
return false {some symbols were never matched}
Stacks 16
Parentheses Matching Example 1
Input: () (() [()])
i X[i]
Operation Stack Output
0 ( Push ( (
1 ) Pop ( Test if ( and X[i] match? YES
2 ( Push ( (
3 ( Push ( ((
4 ) Pop (Test if ( and X[i] match? YES
(
5 [ Push [ ([
Stacks 17
Parentheses Matching Example 1
Input: () (() [()])
i X[i]
Operation Stack
Output
6 ( Push ( ([(
7 ) Pop (Test if ( and X[i] match? YES
([
8 ] Pop [Test if [ and X[i] match? YES
(
9 ) Pop (Test if ( and X[i] match? YES
Test if stack is Empty? YES TRUE
Stacks 18
Parentheses Matching Example 2
Input: ( () [] ]()
i X[i]
Operation Stack Output
0 ( Push ( (
1 ( Push ( ((
2 ) Pop (Test if ( and X[i] match? YES
(
3 [ Push [ ([
4 ] Pop [Test if [ and X[i] match? YES
(
5 ] Pop (Test if ( and X[i] match ? NO FASLE
Stacks 19
Computing Spans (not in book)
We show how to use a stack as an auxiliary data structure in an algorithmGiven an array X, the span S[i] of X[i] is the maximum number of consecutive elements X[j] immediately preceding X[i] and such that X[j] X[j+1] Spans have applications to financial analysis
E.g., stock at 52-week high
6 3 4 5 2
1 1 2 3 1
X
S
01234567
0 1 2 3 4
Stacks 20
Quadratic AlgorithmAlgorithm spans1(X, n)
Input array X of n integersOutput array S of spans of X #S new array of n integers nfor i 0 to n 1 do n
s 1 nwhile s i X[i s] X[i-s+1] 1 2 … (n 1)
s s 1 1 2 … (n 1)S[i] s n
return S 1X[]= 1,2,3, …, n-1, n.
Algorithm spans1 runs in O(n2) time
Stacks 21
Computing Spans with a Stack
We push the n elements in the stackWe sacn the elements in the stack in the reverse order.
We find the last element x[j] with x[j]<=x[i] and x[j-1]>x[i] and set s[i]=i-j+1.
If no such a j exists, s[i]=i+1.
Compute the s[i] for the remaining i’s as follows:
for (i=n-2; i>=0; i--) if (s[i+1]>1 & s[i]==0)
then s[i]=s[i+1]-1
01234567
0 1 2 3 4 5 6 7
Stacks 22
Example:
i= 0, 1, 2, 3, 4, 5, 6, 7 X[i]= 6, 3, 4, 1, 2, 3, 5, 4S[i]= 1, 1, 2, 1, 2, 3, 4, 1.
4 5 53 32 21 1 4 4 43 3 36 6 6 6Stack Stack Stack Stack Stack
Stacks 23
Linear AlgorithmAlgorithm spans2(X, n) #
S new array of n integers nA new empty stack 1for i 0 to n 1 doA.push(i) n
i=n-1 ; j=n-1; 1 while (i<=0) do while (A.isEmpty()
X[A.top()] X[j] ) do n
j A.pop() n if A.isEmpty() then n
S[i] i 1 1 else
S[i] i j +1 n i=j-1; n
return S 1
Each index of the array
Is pushed into the stack exactly one
Is popped from the stack at most once
The statements in the two while loops are executed at most n times Algorithm spans2 runs in O(n) time for (i=n-2; i>=0; i--)
if (s[i+1]>1 & s[i]==0) then s[i]=s[i+1]-1
Stacks 24
Summary of Stack
Understand the definition of Stack (basic)
Applications1. Parentheses (moderate)2. Computing Spans (not required)
Implementation of Stack is not required. You should be able to use stack to write
programs to solve problems
Stacks 25
Week 3: Tutorial Arrangement
Week 3 Tutorial (Feb. 6, 2006, 6:30p.m. – 7:20p.m) will be in MMW2478 (CS lab, second floor, Lift 18, on the left hand side after you go through the tunnel.)
City Uni.
Lift 18
Tunnel connecting city U and Festival Walk
Stacks 26
Remarks
Emphasize the concept of ADT.More examples about ADTDelete the Span application examples. Add Queue part (perhaps to week 3)Teach slowly.