Transcript Analysis of Algorithms

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
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Abstract Data Types (ADTs)
An abstract data
type (ADT) is an
abstraction of a
data structure
An ADT specifies:
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Data stored
Operations on the
data
Error conditions
associated with
operations
Example: ADT modeling a
simple stock trading system
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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
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The Stack ADT
The Stack ADT stores
arbitrary objects
Insertions and
deletions follow the
last-in first-out scheme
Main stack operations:

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push(object): inserts an
element
object pop(): removes
and returns the last
inserted element
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Auxiliary stack
operations:

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
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
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Exceptions
Attempting the
execution of an
operation of ADT may
sometimes cause an
error condition, called
an exception
Exceptions are said to
be “thrown” by an
operation that cannot
be executed
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In the Stack ADT,
operations pop and
top cannot be
performed if the
stack is empty
Attempting the
execution of pop or
top on an empty
stack throws an
EmptyStackException
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Applications of Stacks
Direct applications

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Page-visited history in a Web browser
Undo sequence in a text editor
Chain of method calls in the Java Virtual
Machine
Indirect applications
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Auxiliary data structure for algorithms
Component of other data structures
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Method Stack in the JVM
The Java Virtual Machine (JVM)
keeps track of the chain of
active methods with a stack
When a method is called, the
JVM pushes on the stack a
frame containing
main() {
int i = 5;
foo(i);
}
foo(int j) {
int k;
 Local variables and return value
k = j+1;
 Program counter, keeping track of
bar(k);
the statement being executed
}
When a method ends, its frame
is popped from the stack and
bar(int m) {
control is passed to the method
…
on top of the stack
}
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bar
PC = 1
m=6
foo
PC = 3
j=5
k=6
main
PC = 2
i=5
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Array-based Stack
A simple way of
implementing the
Stack ADT uses an
array
We add elements
from left to right
A variable keeps
track of the index of
the top element
Algorithm size()
return t + 1
Algorithm pop()
if isEmpty() then
throw EmptyStackException
else
tt1
return S[t + 1]
…
S
0 1 2
t
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Array-based Stack (cont.)
The array storing the
stack elements may
become full
A push operation will
then throw a
FullStackException


Algorithm push(o)
if t = S.length  1 then
throw FullStackException
else
tt+1
Limitation of the arrayS[t]  o
based implementation
Not intrinsic to the
Stack ADT
…
S
0 1 2
t
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Performance and Limitations
Performance
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Let n be the number of elements in the stack
The space used is O(n)
Each operation runs in time O(1)
Limitations
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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
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Computing Spans
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We show how to use a stack 6
as an auxiliary data structure 5
in an algorithm
4
Given an an array X, the span
3
S[i] of X[i] is the maximum
2
number of consecutive
elements X[j] immediately
1
preceding X[i] and such that 0
X[j]  X[i]
Spans have applications to
financial analysis
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E.g., stock at 52-week high
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X
S
1
6
1
3
1
2
3
4
2
5
3
4
2
1
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Quadratic Algorithm
Algorithm spans1(X, n)
Input array X of n integers
Output array S of spans of X
S  new array of n integers
for i  0 to n  1 do
s1
while s  i  X[i  s]  X[i]
ss+1
S[i]  s
return S
#
n
n
n
1 + 2 + …+ (n  1)
1 + 2 + …+ (n  1)
n
1
Algorithm spans1 runs in O(n2) time
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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
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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
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6
5
4
3
2
1
0
0 1 2 3 4 5 6 7
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Linear Algorithm
Each index of the
array
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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
Algorithm spans2(X, n)
S  new array of n integers
A  new empty stack
for i  0 to n  1 do
while (A.isEmpty() 
X[top()]  X[i] ) do
A.pop()
if A.isEmpty() then
S[i]  i + 1
else
S[i]  i  A.top()
A.push(i)
return S
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#
n
1
n
n
n
n
n
n
n
1
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Growable Array-based Stack
In a push operation, when Algorithm push(o)
the array is full, instead of
if t = S.length  1 then
throwing an exception, we
A  new array of
can replace the array with
size …
a larger one
for i  0 to t do
A[i]  S[i]
How large should the new
SA
array be?
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incremental strategy:
increase the size by a
constant c
doubling strategy: double
the size
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tt+1
S[t]  o
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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 operations
We assume that we start with an empty
stack represented by an array of size 1
We call amortized time of a push operation
the average time taken by a push over the
series of operations, i.e., T(n)/n
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Incremental Strategy Analysis
We replace the array k = n/c times
The 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)/2
Since c is a constant, T(n) is O(n + k2), i.e.,
O(n2)
The amortized time of a push operation is O(n)
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Doubling Strategy Analysis
We replace the array k = log2 n
times
The 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)
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geometric series
2
4
1
1
8
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Stack Interface in Java
Java interface
corresponding to
our Stack ADT
Requires the
definition of class
EmptyStackException
Different 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;
}
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Array-based Stack in Java
public class ArrayStack
implements Stack {
// holds the stack elements
private Object S[ ];
// index to top element
private int top = -1;
// constructor
public 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;
}
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Queues
Outline and Reading
The Queue ADT (§4.2)
Implementation with a circular array
(§4.2.2)
Growable array-based queue
Queue interface in Java
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The Queue ADT
The Queue ADT stores arbitrary
objects
Insertions and deletions follow
the first-in first-out scheme
Insertions are at the rear of the
queue and removals are at the
front of the queue
Main queue operations:
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enqueue(object): inserts an
element at the end of the
queue
object dequeue(): removes and
returns the element at the front
of the queue
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Auxiliary queue
operations:
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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
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Applications of Queues
Direct applications
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Waiting lists, bureaucracy
Access to shared resources (e.g., printer)
Multiprogramming
Indirect applications
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Auxiliary data structure for algorithms
Component of other data structures
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Array-based Queue
Use an array of size N in a circular fashion
Two variables keep track of the front and rear
f index of the front element
r index immediately past the rear element
Array location r is kept empty
normal configuration
Q
0 1 2
f
r
wrapped-around configuration
Q
0 1 2
r
f
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Queue Operations
We use the
modulo operator
(remainder of
division)
Algorithm size()
return (N  f + r) mod N
Algorithm isEmpty()
return (f = r)
Q
0 1 2
f
0 1 2
r
r
Q
f
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Queue Operations (cont.)
Operation enqueue
throws an exception if
the array is full
This exception is
implementationdependent
Algorithm enqueue(o)
if size() = N  1 then
throw FullQueueException
else
Q[r]  o
r  (r + 1) mod N
Q
0 1 2
f
0 1 2
f
r
Q
r
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Queue Operations (cont.)
Operation dequeue
throws an exception
if the queue is empty
This exception is
specified in the
queue ADT
Algorithm dequeue()
if isEmpty() then
throw EmptyQueueException
else
o  Q[f]
f  (f + 1) mod N
return o
Q
0 1 2
f
r
Q
0 1 2
f
r
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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 one
Similar to what we did for an array-based
stack
The enqueue operation has amortized
running time
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
O(n) with the incremental strategy
O(1) with the doubling strategy
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Queue Interface in Java
Java interface
corresponding to
our Queue ADT
Requires the
definition of class
EmptyQueueException
No 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;
}
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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.
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Answer:
Algorithm reverseQueue(Queue Q)
Create new stack S
while (not Q.isEmpty()) do
S.push(Q.dequeue())
while (not S.isEmpty()) do
Q.enqueue(S.pop())
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