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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



                 week 2a               2
Abstract Data Types (ADTs)
 An abstract data          Example: ADT modeling a
 type (ADT) is an          simple stock trading system
 abstraction of a              The data stored are buy/sell
 data structure                 orders
 An ADT specifies:             The operations supported are
     Data stored                    order buy(stock, shares, price)
     Operations on the              order sell(stock, shares, price)
      data                           void cancel(order)
     Error conditions         Error conditions:
      associated with                Buy/sell a nonexistent stock
      operations
                                     Cancel a nonexistent order

                          week 2a                                  3
The Stack ADT
  The Stack ADT stores             Auxiliary stack
  arbitrary objects                operations:
  Insertions and                      object top(): returns the
                                       last inserted element
  deletions follow the                 without removing it
  last-in first-out scheme            integer size(): returns the
  Main stack operations:               number of elements
                                       stored
     push(object): inserts an
                                       boolean isEmpty():
      element                      

                                       indicates whether no
     object pop(): removes            elements are stored
      and returns the last
      inserted element

                         week 2a                              4
Exceptions
  Attempting the               In the Stack ADT,
  execution of an              operations pop and
  operation of ADT may         top cannot be
  sometimes cause an           performed if the
  error condition, called      stack is empty
  an exception                 Attempting the
  Exceptions are said to       execution of pop or
  be “thrown” by an            top on an empty
  operation that cannot        stack throws an
  be executed                  EmptyStackException
                     week 2a                   5
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                     6
Method Stack in the JVM
 The Java Virtual Machine (JVM) main() {
 keeps track of the chain of            int i = 5;
                                                     bar
 active methods with a stack            foo(i);       PC = 1
 When a method is called, the           }             m=6
 JVM pushes on the stack a            foo(int j) {
 frame containing                                    foo
                                        int k;         PC = 3
   Local variables and return value

   Program counter, keeping track of
                                        k = j+1;       j=5
     the statement being executed       bar(k);        k=6
 When a method ends, its frame          }
 is popped from the stack and                        main
                                      bar(int m) {    PC = 2
 control is passed to the method
                                        …             i=5
 on top of the stack
                                        }
                          week 2a                           7
Array-based Stack
  A simple way of           Algorithm size()
  implementing the           return t + 1
  Stack ADT uses an
  array                     Algorithm pop()
  We add elements            if isEmpty() then
  from left to right            throw EmptyStackException
  A variable keeps            else
  track of the index of         tt1
  the top element               return S[t + 1]

                                    …
   S
       0 1 2                                   t
                          week 2a                      8
Array-based Stack (cont.)
 The array storing the
 stack elements may
                               Algorithm push(o)
 become full
                                if t = S.length  1 then
 A push operation will
                                   throw FullStackException
 then throw a
 FullStackException              else
     Limitation of the array-     tt+1
      based implementation         S[t]  o
     Not intrinsic to the
      Stack ADT


                                       …
      S
          0 1 2                                           t
                             week 2a                          9
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                      10
Computing Spans
                                7
 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
 S[i] of X[i] is the maximum    3
 number of consecutive          2
 elements X[j] immediately      1
 preceding X[i] and such that 0
 X[j]  X[i]
                                           0       1   2   3       4
 Spans have applications to
 financial analysis
                                       X       6   3   4   5   2
     E.g., stock at 52-week high
                                       S       1   1   2   3   1
                             week 2a                           11
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                n
  for i  0 to n  1 do                      n
    s1                                      n
    while 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                        12
Computing Spans with a Stack
 We keep in a stack the                 7
 indices of the elements
                                        6
 visible when “looking
 back”                                  5
                                        4
 We scan the array from
 left to right                          3
    Let i be the current index         2
    We pop indices from the            1
     stack until we find index j        0
     such that X[i]  X[j]
                                            0 1 2 3 4 5 6 7
    We set S[i]  i  j
    We push i onto the stack

                              week 2a                    13
Linear Algorithm
  Each index of the        Algorithm spans2(X, n)              #
  array                      S  new array of n integers       n
     Is pushed into the     A  new empty stack               1
      stack exactly one
                              for i  0 to n  1 do            n
      Is popped from
  
      the stack at most
                                while (A.isEmpty() 
      once                              X[top()]  X[i] ) do   n
  The statements in                A.pop()                     n
  the while-loop are            if A.isEmpty() then            n
  executed at most                 S[i]  i + 1                n
  n times                       else
                                   S[i]  i  A.top()          n
  Algorithm spans2
                                A.push(i)                      n
  runs in O(n) time
                             return S                          1
                           week 2a                             14
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
 How large should the new            A[i]  S[i]
 array be?                           SA
    incremental strategy:        tt+1
     increase the size by a       S[t]  o
     constant c
    doubling strategy: double
     the size
                        week 2a                    15
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

                     week 2a                     16
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)

                    week 2a                  17
Doubling Strategy Analysis
  We replace the array k = log2 n
  times
                                    geometric series
  The total time T(n) of a series
  of n push operations is                2
  proportional to                                    4
    n + 1 + 2 + 4 + 8 + …+ 2k =      1       1
        n + 2k + 1 1 = 2n 1
                                                 8
  T(n) is O(n)
  The amortized time of a push
  operation is O(1)

                    week 2a                          18
Stack Interface in Java
                         public interface Stack {
 Java interface
 corresponding to            public int size();
 our Stack ADT               public boolean isEmpty();
 Requires the
 definition of class         public Object top()
                                 throws EmptyStackException;
 EmptyStackException
 Different from the          public void push(Object o);
 built-in Java class         public Object pop()
 java.util.Stack                  throws EmptyStackException;
                         }

                       week 2a                             19
Array-based Stack in Java
public class ArrayStack                     public Object pop()
     implements Stack {                           throws EmptyStackException {
  // holds the stack elements                 if isEmpty()
  private Object S[ ];                            throw new EmptyStackException
                                                    (“Empty stack: cannot pop”);
  // index to top element                     Object temp = S[top];
  private int top = -1;                       // facilitates garbage collection
  // constructor                              S[top] = null;
  public ArrayStack(int capacity) {           top = top – 1;
      S = new Object[capacity]);              return temp;
   }                                        }


                                      week 2a                              20
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



                  week 2a                22
The Queue ADT
 The Queue ADT stores arbitrary           Auxiliary queue
 objects                                  operations:
 Insertions and deletions follow             object front(): returns the
 the first-in first-out scheme                element at the front without
 Insertions are at the rear of the            removing it
 queue and removals are at the               integer size(): returns the
 front of the queue                           number of elements stored
                                             boolean isEmpty(): indicates
 Main queue operations:
                                              whether no elements are
     enqueue(object): inserts an             stored
      element at the end of the
      queue                               Exceptions
     object dequeue(): removes and          Attempting the execution of
      returns the element at the front        dequeue or front on an
      of the queue                            empty queue throws an
                                              EmptyQueueException
                                week 2a                             23
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                      24
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
                          week 2a                  25
Queue Operations
  We use the          Algorithm size()
  modulo operator      return (N  f + r) mod N
  (remainder of
                      Algorithm isEmpty()
  division)            return (f = r)


   Q
       0 1 2   f                    r
   Q
       0 1 2   r              f

                    week 2a                       26
Queue Operations (cont.)
 Operation enqueue           Algorithm enqueue(o)
 throws an exception if       if size() = N  1 then
 the array is full               throw FullQueueException
 This exception is             else
 implementation-                 Q[r]  o
 dependent                       r  (r + 1) mod N

    Q
        0 1 2    f                       r
    Q
        0 1 2    f                           r

                          week 2a                     27
Queue Operations (cont.)
 Operation dequeue         Algorithm dequeue()
 throws an exception        if isEmpty() then
 if the queue is empty         throw EmptyQueueException
 This exception is           else
 specified in the              o  Q[f]
 queue ADT                     f  (f + 1) mod N
                               return o

   Q
       0 1 2    f                       r
   Q
       0 1 2        f                   r
                         week 2a                    28
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
     O(n) with the incremental strategy
     O(1) with the doubling strategy

                        week 2a            29
Queue Interface in Java
                         public interface Queue {
 Java interface
 corresponding to            public int size();
 our Queue ADT               public boolean isEmpty();
 Requires the                public Object front()
 definition of class             throws EmptyQueueException;
 EmptyQueueException         public void enqueue(Object o);
 No corresponding            public Object dequeue()
 built-in Java class              throws EmptyQueueException;
                         }

                       week 2a                                30
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.




                 week 2a               31
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())


                week 2a           32

				
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