CSC300: Ungraded HW [2/2] Previous pageContents

2.4.2
  Criticize the following idea: To implement find the maximum in constant
  time, why not use a stack or a queue, but keep track of the maximum value
  inserted so far, then return that value for find the maximum?

  The proposal is to keep all the values in a queue/stack and to separately 
  keep the max value inserted so far.  So if you were to implement this, 
  you would have fields something like the following (without generics):

  class CrappyPQ {
    Stack s;
    Comparable maxSoFar;

    void insert (Comparable x) {
       s.push (x);
       if  (x.compareTo(maxSoFar) > 0) maxSoFar = x;

     
  The problem is that the CrappyPQ must support the delete operation, so 
  when you delete the max it will take you linear time to find the new max
  in the stack or queue.

2.4.4
  Is an array that is sorted in decreasing order a max-oriented heap?

  Yes.
  
2.4.5
  Give the heap that results when the keys E A S Y Q U E S T I O N are inserted
  in that order into an initially empty max-oriented heap.

2.4.9*
  Draw all of the different heaps that can be made from the five 
  keys A B C D E, then draw all of the different heaps that can be 
  made from the five keys A A A B B.

  MAX HEAPS:
  ------------------------------------------------------------------
  |     E    |     E    |     E    |     E    |     E    |     E   | 
  |   D   C  |   D   C  |   D   B  |   D   B  |   D   A  |   D   A | 
  |  B A     |  A B     |  A C     |  C A     |  B C     |  C B    | 
  ------------------------------------------------------------------
  |     E    |     E    |                                          |
  |   C   D  |   C   D  |                                          |
  |  B A     |  A B     |                                          |
  ------------------------------------------------------------------
  |     B    |     B    |                                          |
  |   B   A  |   A   B  |                                          |
  |  A A     |  A A     |                                          |
  ------------------------------------------------------------------
  MIN HEAPS:
  ------------------------------------------------------------------
  |     A    |     A    |     A    |     A    |     A    |     A   | 
  |   B   C  |   B   C  |   B   D  |   B   D  |   B   E  |   B   E | 
  |  D E     |  E D     |  E C     |  C E     |  D C     |  C D    | 
  ------------------------------------------------------------------
  |     A    |     A    |                                          |
  |   C   B  |   C   B  |                                          |
  |  D E     |  E D     |                                          |
  ------------------------------------------------------------------
  |     A    |     A    |     A    |                               |
  |   A   A  |   A   B  |   A   B  |                               |
  |  B B     |  A B     |  B A     |                               |
  ------------------------------------------------------------------

2.4.11
  Suppose that your application will have a huge number of insert operations, 
  but only a few remove the maximum operations. Which priority-queue 
  implementation do you think would be most effective: heap, unordered 
  array, or ordered array?
  (Also say why.  Don't confuse find-the-max and remove-the-max)

  unordered array, because cost of insert is constant
  or heap, because cost of insert is logarithmic
  
2.4.12
  Suppose that your application will have a huge number of find the maximum 
  operations, but a relatively small number of insert and remove the maximum 
  operations. Which priority-queue implementation do you think would be most
  effective: heap, unordered array, or ordered array?
  (Also say why.  Don't confuse find-the-max and remove-the-max)
  
  heap or ordered array, because cost of find-max is constant

2.4.15*
  Design a linear-time certification algorithm to check whether an array pq[]
  is a min-oriented heap.
  (The method MinPQ.isMinHeap does this recursively.  What would
  an iterative solution be?  You should make sure you understand
  this and can do it without looking at MinPQ.isMinHeap.)
  
2.4.27*
  Find the minimum. Add a min() method to MaxPQ. Your implementation should use
  constant time and constant extra space.
  (Argue that your solution is constant time/space.)

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