CE 221 Data Structures and Algorithms Chapter 7: Sorting - - PowerPoint PPT Presentation

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Chapter 7: Sorting (Heapsort, Mergesort) CE 221 Data Structures and Algorithms

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Text: Read Weiss, § 7.5 – 7.6

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Heapsort

• Priority queues can be used sort in O(NlogN)

time. – First, build a binary heap of N elements in O(N) time – perform N deleteMin operations each taking O(logN) time, hence resulting in O(NlogN).

• Problem: needs an extra array.
• A clever solution: after each deleteMin heap

size shrinks by 1, use this space. But the result will be a decreasing sorted order. Thus we may use a max heap by changing the heap-order property

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Heapsort Example

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Max heap after buildHeap phase Max heap after first deleteMax phase

Analysis of Heapsort

• worst case analysis
• 2N comparisons-buildHeap
• N-1 deleteMax operations.
• Theorem: Average # of comparisons to heapsort a random

permutation of N distinct items is 2NlogN-O(NloglogN).

• Proof: on any input, cost sequence D: d1, d2,..., dN

for any D, distinct deleteMax sequences total # of heaps with cost less than M is at most # of heaps with cost M < N(logN-loglogN-4) is at most (N/16)N. So the average # of comparisons is at least 2M.

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 

) 2 ! ( ), ( log 2 ! log 2 log 2

2

N e N N N O N N N i

N N i

          





items N with heaps

• f

number e N N f M s comparison

• f

number d M Cost

N D N i i D

)) 4 /( ( ) ( 2 ,

1

   

 

D M D S N d d d D S 2 2 ... 2 2 1 2  

M N M i i N

N N 2 ) (log 2 ) (log

1 1





 

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Mergesort

• runs in O(NlogN), # of comparisons is nearly
• ptimal, fine example of a recursive algorithm.
• Fundamental operation is merging of 2 sorted lists.
• The basic merging algorithm takes 2 input arrays

A and B, an output array C. 3 counters, Actr, Bctr, and Cctr are initially set to the beginning of their respective arrays.

• The smaller of A[Actr] and B[Bctr] is copied to

C[Cctr ] and the appropriate counters are advanced.

• When either list is exhausted, the rest of the other

list is copied to C.

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Mergesort - merge

• The time to merge is linear, at most N-1 comparisons are

required (since each comparison adds an element to C. It should also be noted that after N-1 elements are added to array C, the last element need not be compared but it simply gets copied).

• Mergesort algorithm is then easy to describe.

If N=1 DONE else { mergesort(left half); mergesort(right half); merge(left half, right half); }

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Mergesort – Implementation - I

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Merge – Implementation - II

/* leftPos = start of left half, rightPos = start of right half Last copying may be avoided by interchanging the roles of the arrays a and tmpArray at alternate levels of recursion */

Analysis of Mergesort

• Write down recurrence relations and solve them

– T(1)=1 – T(N)=2T(N/2)+N // assumption is N=2k

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) log ( log ) ( log 1 ) 1 ( ) ( ..... .......... .......... .......... 1 1 ) 1 ( 2 ) 2 ( 1 8 / ) 8 / ( 4 / ) 4 / ( 1 4 / ) 4 / ( 2 / ) 2 / ( 1 2 / ) 2 / ( ) ( N N O N N N N T N T N N T T T N N T N N T N N T N N T N N T N N T                N N N N T N N NT N T N k use kN N T N T N N T N T N N T N T N N T N T

k k

log ) ( log ) 1 ( ) ( log , ) 2 / ( 2 ) ( 3 ) 8 / ( 8 ) ( 2 ) 4 / ( 4 ) ( ) 2 / ( 2 ) (              

Homework Assignments

• 7.11, 7.12, 7.15, 7.17, 7.43 (7.38 in 3/e),

7.53 (7.48 in 3/e)

• You are requested to study and solve the
• exercises. Note that these are for you to

practice only. You are not to deliver the results to me.

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