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Stacks. Outline and Reading. The Stack ADT ( §4.1 ) Applications of Stacks Array-based implementation ( §4. 1.2) Growable array-based stack Think of a spring-loaded plate dispenser. An abstract data type (ADT) is an abstraction of a data structure An ADT specifies: Data stored - PowerPoint PPT Presentation
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week 2a 2
Outline and Reading
The Stack ADT (§4.1)Applications of Stacks Array-based implementation (§4.1.2)Growable array-based stackThink of a spring-loaded plate
dispenser
week 2a 3
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 simple stock trading system The data stored are buy/sell
orders The operations supported are
order buy(stock, shares, price) order sell(stock, shares, price) void cancel(order)
Error conditions: Buy/sell a nonexistent stock Cancel a nonexistent order
week 2a 4
The Stack ADTThe Stack ADT stores arbitrary objectsInsertions and deletions follow the last-in first-out schemeMain 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
week 2a 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
week 2a 6
Applications of Stacks
Direct applications Page-visited history in a Web browser 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
week 2a 7
Method Stack in the JVMThe Java Virtual Machine (JVM) keeps track of the chain of active methods with a stackWhen a method is called, the JVM pushes on the stack a frame containing
Local variables and return value Program counter, keeping track
of the statement being executed
When a method ends, its frame is popped from the stack and control is passed to the method on top of the stack
main() {int i = 5;foo(i);}
foo(int j) {int k;k = j+1;bar(k);}
bar(int m) {…}
bar PC = 1 m = 6
foo PC = 3 j = 5 k = 6
main PC = 2 i = 5
week 2a 8
Array-based StackA simple way of implementing the Stack ADT uses an arrayWe add elements from left to rightA variable 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]
week 2a 9
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
week 2a 10
Performance and Limitations
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
week 2a 11
Computing SpansWe show how to use a stack as an auxiliary data structure in an algorithmGiven an 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[i] 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
week 2a 12
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] 1 2 … (n 1)
s s 1 1 2 … (n 1)S[i] s n
return S 1
Algorithm spans1 runs in O(n2) time
week 2a 13
Computing Spans with a Stack
We keep in a stack the indices of the elements visible when “looking back”We scan the array from left to right
Let i be the current index
We pop indices from the stack until we find index j such that X[i] X[j]
We set S[i] i j We push i onto the stack
01234567
0 1 2 3 4 5 6 7
week 2a 14
Linear AlgorithmAlgorithm spans2(X, n) #
S new array of n integers nA new empty stack 1
for i 0 to n 1 do nwhile (A.isEmpty()
X[top()] X[i] ) do n A.pop() n
if A.isEmpty() then nS[i] i 1 n
else S[i] i A.top() n
A.push(i) nreturn 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 while-loop are executed at most n times Algorithm spans2 runs in O(n) time
week 2a 15
Growable Array-based Stack
In a push operation, when the array is full, instead of throwing an exception, we can replace the array with a larger oneHow large should the new array be? incremental strategy:
increase the size by a constant c
doubling strategy: double the size
Algorithm push(o)if t = S.length 1 then
A new array ofsize …
for i 0 to t do A[i] S[i] S A
t t + 1S[t] o
week 2a 16
Comparison of the Strategies
We compare the incremental strategy and the doubling strategy by analyzing the total time T(n) needed to perform a series of n push operationsWe assume that we start with an empty stack represented by an array of size 1We call amortized time of a push operation the average time taken by a push over the series of operations, i.e., T(n)/n
week 2a 17
Incremental Strategy Analysis
We replace the array k = n/c timesThe total time T(n) of a series of n push operations is proportional to
n + c + 2c + 3c + 4c + … + kc =n + c(1 + 2 + 3 + … + k) =
n + ck(k + 1)/2Since c is a constant, T(n) is O(n + k2), i.e., O(n2)The amortized time of a push operation is O(n)
week 2a 18
Doubling Strategy AnalysisWe replace the array k = log2 n timesThe total time T(n) of a series of n push operations is proportional to
n + 1 + 2 + 4 + 8 + …+ 2k =n 2k + 1 1 = 2n 1
T(n) is O(n)The amortized time of a push operation is O(1)
geometric series
1
2
14
8
week 2a 19
Stack Interface in Java
Java interface corresponding to our Stack ADTRequires the definition of class EmptyStackExceptionDifferent from the built-in Java class java.util.Stack
public interface Stack {
public int size();
public boolean isEmpty();
public Object top()throws EmptyStackException;
public void push(Object o);
public Object pop() throws EmptyStackException;
}
week 2a 20
Array-based Stack in Javapublic class ArrayStack
implements Stack {
// holds the stack elements private Object S[ ];
// index to top elementprivate int top = -1;
// constructorpublic ArrayStack(int capacity) {
S = new Object[capacity]); }
public 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;
}
week 2a 22
Outline and Reading
The Queue ADT (§4.2)Implementation with a circular array (§4.2.2)Growable array-based queueQueue interface in Java
week 2a 23
The Queue ADTThe Queue ADT stores arbitrary objectsInsertions and deletions follow the first-in first-out schemeInsertions are at the rear of the queue and removals are at the front of the queueMain queue operations:
enqueue(object): inserts an element at the end of the queue
object dequeue(): removes and returns the element at the front of the queue
Auxiliary queue operations:
object front(): returns the element at the front without removing it
integer size(): returns the number of elements stored
boolean isEmpty(): indicates whether no elements are stored
Exceptions Attempting the execution
of dequeue or front on an empty queue throws an EmptyQueueException
week 2a 24
Applications of Queues
Direct applications Waiting lists, bureaucracy Access to shared resources (e.g.,
printer) Multiprogramming
Indirect applications Auxiliary data structure for algorithms Component of other data structures
week 2a 25
Array-based QueueUse an array of size N in a circular fashionTwo variables keep track of the front and rearf index of the front elementr index immediately past the rear element
Array location r is kept empty
Q0 1 2 rf
normal configuration
Q0 1 2 fr
wrapped-around configuration
week 2a 26
Queue OperationsWe use the modulo operator (remainder of division)
Algorithm size()return (N f + r) mod N
Algorithm isEmpty()return (f r)
Q0 1 2 rf
Q0 1 2 fr
week 2a 27
Queue Operations (cont.)Algorithm enqueue(o)
if size() = N 1 thenthrow FullQueueException
else Q[r] or (r + 1) mod N
Operation enqueue throws an exception if the array is fullThis exception is implementation-dependent
Q0 1 2 rf
Q0 1 2 f r
week 2a 28
Queue Operations (cont.)Operation dequeue throws an exception if the queue is emptyThis exception is specified in the queue ADT
Algorithm dequeue()if isEmpty() then
throw EmptyQueueException else
o Q[f]f (f + 1) mod Nreturn o
Q0 1 2 rf
Q0 1 2 rf
week 2a 29
Growable Array-based Queue
In an enqueue operation, when the array is full, instead of throwing an exception, we can replace the array with a larger oneSimilar to what we did for an array-based stackThe enqueue operation has amortized running time O(n) with the incremental strategy O(1) with the doubling strategy
week 2a 30
Queue Interface in Java
Java interface corresponding to our Queue ADTRequires the definition of class EmptyQueueExceptionNo corresponding built-in Java class
public interface Queue {
public int size();
public boolean isEmpty();
public Object front()throws EmptyQueueException;
public void enqueue(Object o);
public Object dequeue() throws EmptyQueueException;
}
week 2a 31
Creativity:
Describe in pseudo-code a linear-time algorithm for reversing a queue Q. To access the queue, you are only allowed to use the methods of queue ADT.