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Adapted from instructor resource slidesNyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson
Education, Inc. All rights reserved. 0-13-140909-3
1
Mergesort and Review
Chapter 13
6/15/15
2
Today• Any questions on project?• Exams
– Review questions– Easy question/fix (I added your points wrong)
come see me in class. If you need a re-grade, follow instructions in syllabus. Come see me in office hours, or make an appointment
• Review Thursday• Break• More (new) Sorting
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
3
Categories of Sorting Algorithms
• Selection sort– Make passes through a list– On each pass reposition correctly some element
(largest or smallest)
4
Array Based Selection Sort Pseudo-Code//x[0] is reserved
For i = 1 to n-1 do the following:
//Find the smallest element in the sublist x[i]…x[n]
Set smallPos = i and smallest = x[smallPos]
For j = i + 1 to n-1 do the following:
If x[j] < smallest: //smaller element found
Set smallPos = j and smallest = x[smallPos]
End for
//No interchange smallest with x[i], first element of this sublist.
Set x[smallPos] = x[i] and x[i] = smallest
End for
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
5
In-Class Exercise #1: Selection Sort
• List of 9 elements:
90, 10, 80, 70, 20, 30, 50, 40, 60
Illustrate each pass…
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
6
Selection Sort Solution
Pass 0 90 10 80 70 20 30 50 40 60
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
1 10 90 80 70 20 30 50 40 60
2 10 20 80 70 90 30 50 40 60
3 10 20 30 70 90 80 50 40 60
4 10 20 30 40 90 80 50 70 60
5 10 20 30 40 50 80 90 70 60
6 10 20 30 40 50 60 90 70 80
7 10 20 30 40 50 60 70 90 80
8 10 20 30 40 50 60 70 80 90
7
Categories of Sorting Algorithms
• Exchange sort– Systematically interchange pairs of elements
which are out of order– Bubble sort does this
Out of order, exchange In order, do not exchange
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
8
Bubble Sort Algorithm
1. Initialize numCompares to n - 12. While numCompares != 0, do following
a. Set last = 1 // location of last element in a swap
b. For i = 1 to numPairsif xi > xi + 1
Swap xi and xi + 1 and set last = i
c. Set numCompares = last – 1End while
9
In-Class Exercise #2: Bubble Sort
• List of 9 elements:
90, 10, 80, 70, 20, 30, 50, 40, 60
Illustrate each pass…
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
10
Bubble Sort Solution
Pass 0 90 10 80 70 20 30 50 40 60
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
1 10 80 70 20 30 50 40 60 90
2 10 70 20 30 50 40 60 80 90
3 10 20 30 50 40 60 70 80 90
4 10 20 30 40 50 60 70 80 90
5 10 20 30 40 50 60 70 80 90
11
Categories of Sorting Algorithms
• Insertion sort– Repeatedly insert a new element into an already
sorted list
– Note this works well with a linked list implementation
All these have computing time O(n2)
All these have computing time O(n2)
12
Insertion Sort Pseduo Code (Instructor’s Recommendation)
for j = 2 to A.length
key = A[j]
//Insert A[j] into the sorted sequence A[1..j-1]
i = j-1
while i > 0 and A[i] > key
A[i+1] = A[i]
i = i-1
A[i+1] = key
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
13
Insertion Sort Example
Pass 0 5 2 4 6 1 3
1 2 5 4 6 1 3
2 2 4 5 6 1 3
3 2 4 5 6 1 3
4 1 2 4 5 6 3
5 1 2 3 4 5 6
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
14
In-Class Exercise #3: Insertion Sort
• List of 5 elements:
9, 3, 1, 5, 2
Illustrate each pass, along with algorithm values of key, j and i…
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
15
Insertion Sort Solution
Pass 0 9 3 1 5 2 key j i
1 3 9 1 5 2 3 2 1,0
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
2 1 3 9 5 2 1 3 2,1,0
3 1 3 5 9 2 5 4 3,2
4 1 2 3 5 9 2 5 4,3,2,1
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
16
Quicksort• A more efficient exchange sorting scheme than
bubble sort – A typical exchange involves elements that are far
apart – Fewer interchanges are required to correctly position
an element.• Quicksort uses a divide-and-conquer strategy
– A recursive approach – The original problem partitioned into simpler sub-
problems,– Each sub problem considered independently.
• Subdivision continues until sub problems obtained are simple enough to be solved directly
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
17
Quicksort
• Choose some element called a pivot • Perform a sequence of exchanges so that
– All elements that are less than this pivot are to its left and – All elements that are greater than the pivot are to its
right.
• Divides the (sub)list into two smaller sub lists, • Each of which may then be sorted independently in
the same way.
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
18
Quicksort
If the list has 0 or 1 elements, return. // the list is sorted
Else do:Pick an element in the list to use as the pivot.
Split the remaining elements into two disjoint groups:SmallerThanPivot = {all elements < pivot}LargerThanPivot = {all elements > pivot}
Return the list rearranged as: Quicksort(SmallerThanPivot),
pivot, Quicksort(LargerThanPivot).
19
In-Class Exercise #4: Quicksort
• List of 9 elements– 30,10, 80, 70, 20, 90, 50, 40, 60
• Pivot is the first element • Illustrate each pass• Clearly denote each sublist
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
20
Quicksort Solution
Pass 0
30 10 80 70 20 90 50 40 60
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
1 20 10 30 70 80 90 50 40 60
2 10 20 30 50 60 40 70 90 80
3 10 20 30 40 50 60 70 80 90
TO DO: How does this change if you choose the pivot as the median?
A heap is a binary tree with properties:
1. It is complete• Each level of tree completely filled• Except possibly bottom level (nodes in left most
positions)
2. The key in any node dominates the keys of its children
– Min-heap: Node dominates by containing a smaller key than its children
– Max-heap: Node dominates by containing a larger key than its children
21
Heaps
22
Implementing a Heap
• Use an array or vector• Number the nodes from top to bottom
– Number nodes on each row from left to right• Store data in ith node in ith location of array
(vector)
23
Implementing a Heap
• In an array implementation children of ith node are at myArray[2*i] and
myArray[2*i+1]• Parent of the ith node is at
myArray[i/2]
24
Basic Heap Operations
• Construct an empty heap• Check if the heap is empty• Insert an item• Retrieve the largest/smallest element• Remove the largest/smallest element
25
Basic Heap Operations
• Insert an item– Place new item at end of array– “Bubble” it up to the correct place
– Interchange with parent so long as it is greater/less than its parent
26
Basic Heap Operations
• Delete max/min item– Max/Min item is the root, swap with last node in
tree– Delete last element– Bubble the top element down until heap property
satisfied• Interchange with larger of two children
27Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
28
Percolate Down Algorithm
1. Set c = 2 * r2. While r <= n do following
a. If c < n and myArray[c] < myArray[c + 1]Increment c by 1
b. If myArray[r] < myArray[c]i. Swap myArray[r] and myArray[c]ii. set r = ciii. Set c = 2 * c
else Terminate repetitionEnd while
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
29
Heapsort
• Given a list of numbers in an array– Stored in a complete binary tree
• Convert to a heap– Begin at last node not a leaf– Apply percolated down to this subtree– Continue
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
30
Heapsort Algorithm
1. Consider x as a complete binary tree, use heapify to convert this tree to a heap
2. for i = n down to 2:a. Interchange x[1] and x[i] (puts largest element at end)b. Apply percolate_down to convert binary tree corresponding to sublist in x[1] .. x[i-1]
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
31
Heapsort• Now swap element 1 (root of tree) with last
element
– This puts largest element in correct location• Use percolate down on remaining sublist
– Converts from semi-heap to heap
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
32
Heapsort• Now swap element 1 (root of tree) with last
element
– This puts largest element in correct location• Use percolate down on remaining sublist
– Converts from semi-heap to heap
33
In-Class Exercise #4: Heapsort
• For each step, want to draw the heap and array
• 30, 10, 80, 70, 20, 90, 40
Array?
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
1 2 3 4 5 6 7
30 10 80 70 20 90 40
30
10 80
70 20 90 40
34
Step 1: Convert to a heap
• Begin at the last node that is not a leaf, apply the percolate down procedure to convert to a heap the subtree rooted at this node, move to the preceding node and percolat down in that subtree and so on, working our way up the tree, until we reach the root of the given tree. (HEAPIFY)
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
35
Step 1 (ctd)
• What is the last node that is not a leaf?
• Apply percolate down
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
80
80
90 40
90
80 401 2 3 4 5 6 7
30 10 90 70 20 80 40
36
Step 1 (ctd)
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
10
70 20
70
10 20
1 2 3 4 5 6 7
30 70 90 10 20 80 40
37
Step 1(ctd)
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
30
70 90
10 20 80 40
90
70 80
10 20 30 40
1 2 3 4 5 6 7
90 70 80 10 20 30 40
We now have a heap!
38
Step 2: Sort and Swap
• The largest element is now at the root• Correctly position the largest element by
swapping it with the element at the end of the list and go back and sort the remaining 6 elements
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
1 2 3 4 5 6 7
90 70 80 10 20 30 40
1 2 3 4 5 6 7
40 70 80 10 20 30 90
39
Step 2 (ctd)
• This is not a heap. However, since only the root changed, it is a semi-heap
• Use percolate down to convert to a heap
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
40
70 80
10 20 30
40
Step 2 (ctd)
80
70 40
10 20 30
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
1 2 3 4 5 6 7
80 70 40 10 20 30 90
1. Swap
30
70 40
10 20 80
2. Prune
1 2 3 4 5 6 7
30 70 40 10 20 80 90
41
Continue the pattern
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
70
30 40
10 20
1 2 3 4 5 6 7
70 30 40 10 20 80 90
1 2 3 4 5 6 7
20 30 40 10 70 80 90
20
30 40
10 70
42
Continue the pattern
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
40
30 20
10
1 2 3 4 5 6 7
40 30 20 10 70 80 90
10
30 20
40
1 2 3 4 5 6 7
10 30 20 40 70 80 90
43
Continue the pattern
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
30
10 20
1 2 3 4 5 6 7
30 10 20 40 70 80 90
20
10 30
1 2 3 4 5 6 7
20 10 30 40 70 80 90
44
Complete!
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
20
10
1 2 3 4 5 6 7
20 10 30 40 70 80 9010
20
1 2 3 4 5 6 7
10 20 30 40 70 80 90
45
Sorting Facts• Sorting schemes are either …
– internal -- designed for data items stored in main memory
– external -- designed for data items stored in secondary memory. (Disk Drive)
• Previous sorting schemes were all internal sorting algorithms:– required direct access to list elements
• not possible for sequential files– made many passes through the list
• not practical for files
46
Mergesort
• Mergesort can be used both as an internal and an external sort.
• A divide and conquer algorithm• Basic operation in mergesort is merging,
– combining two lists that have previously been sorted
– resulting list is also sorted.
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
47
Merge Algorithm
1. Open File1 and File2 for input, File3 for output2. Read first element x from File1 and
first element y from File23. While neither eof File1 or eof File2
If x < y thena. Write x to File3b. Read a new x value from File1
Otherwisea. Write y to File3b. Read a new y from File2
End while4. If eof File1 encountered copy rest of of File2 into File3.
If eof File2 encountered, copy rest of File1 into File3
49
In-Class Exercise #6
• Take File1 and File2 and produce a sorted File 3
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
File 1 7 9 19 33 47 51 82 99
File 2 11 18 24 49 61
File 3
50
Mergesort Solution
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
File 1 7 9 19 33 47 51 82 99
File 2 11 18 24 49 61
File 3 7 9 11 18 19 24 33 47 49 51 61 82 99
Fun Facts
• Most of the time spent in merging– Combining two sorted lists of size n/2– What is the runtime of merge()?
• Does not sort in-place– Requires extra memory to do the merging– Then copied back into the original memory
• Good for external sorting– Disks are slow– Writing in long streams is more efficient
51
O(n)
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
52
Binary Merge Sort
• Given a single file
• Split into two files
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
53
Binary Merge Sort
• Merge first one-element "subfile" of F1 with first one-element subfile of F2– Gives a sorted two-element subfile of F
• Continue with rest of one-element subfiles
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
54
Binary Merge Sort
• Split again• Merge again as before
• Each time, the size of the sorted subgroups doubles
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
55
Binary Merge Sort
• Last splitting gives two files each in order
• Last merging yields a single file, entirely in order
Note we always are limited to subfiles of some power of 2
Note we always are limited to subfiles of some power of 2
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
56
Natural Merge Sort
• Allows sorted subfiles of other sizes– Number of phases can be reduced when file
contains longer "runs" of ordered elements• Consider file to be sorted, note in order
groups
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
57
Natural Merge Sort
• Copy alternate groupings into two files– Use the sub-groupings, not a power of 2
• Look for possible larger groupings
Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3
58
Natural Merge Sort
• Merge the corresponding sub files
EOF for F2, Copy remaining groups from F1
EOF for F2, Copy remaining groups from F1