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COMP 731 - Data Struc & Algorithms - Lecture 5 - Monday, July 3, 2000
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Announcements: from now on we will meet: 
       TR 8:30-10:00
        F 11-12
Today: [ ] Heaps, lecture 2


Heaps
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Review of heaps - sort some cards, using the fast method to build a heap.

Heaps are good for implementing the Priority Queue ADT - 

ADT Priority Queue

    Data:  A set of items, S={x_1, ..., x_n} 
           each item having an associated key, key(x_1), ..., key(x_n),
           the keys being drawn from a totally ordered universe.
           Initially, there are no items in the set.

    Operations: 
        Insert (x):   S = S U {x}
        Maximum ():   If |S|==0 then error. Else
                      return an items x such that key(x) is 
                      maximal among the key(x_1), ..., key(x_n)
        Delete-Max(): Let x be an item such that key(x) is maximal 
                      among key(x_1), ..., key(x_n).
                      S = S \ {x}
                      return x

Applications: Job scheduling, where jobs have priorities.
              Event-driven simulation  
              Many algorithms use heaps; we will see some, like
                Prim's algorithm and Dijkstra's algorithm. 
 
Basic operations:   insert:     sift up      O(lg n)
                    delete_max: sift down    O(lg n)
                    make_heap:               O(1)

Building a heap with n items:
    O(n \lg n) by way of consecutive insertions.  Or ...
    [review method].  Analysis:

  
     lg  n                lg  n
      SUM     i n/2^i  = n SUM (i/2^i)
     i = 1                 i=1

 An interesting sum.  Let's us review some properties of sums
  
    1 + x + x^2 + ...  = 1/(1-x)     (for |x|<1)  (prove!)

 Differentiate both sides and multiply by x to get
 
    1 + 2x^2 + 3x^3 + ... = x / (1-x)^2   (for |x|<1)

(remember that (1/f)' = -f'/f^2)

   So, upper-bound the above by 
             n (1/2)/(1-1/2)^2 = 2n.