Exam #2 Review. Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3

Exam #2 Review

Nyhoff, A

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ll rights reserved. 0-13-140909-3

2Evolution of Reusability, Genericity

Major theme in development of programming languages Reuse code

Avoid repeatedly reinventing the wheel

Trend contributing to this Use of generic code

Can be used with different types of data

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3Function GenericityOverloading and Templates

Initially code was reusable by encapsulating it within functions

Example lines of code to swap values stored in two variables Instead of rewriting those 3 lines

Place in a functionvoid swap (int & first, int & second){ int temp = first; first = second; second = temp; }

Then call swap(x,y);

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4Template Mechanism Declare a type parameter

also called a type placeholder

Use it in the function instead of a specific type. This requires a different kind of parameter list:

void Swap(______ & first, ______ & second)

{ ________ temp = first; first = second; second = temp;}

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5Instantiating Class Templates Instantiate it by using declaration of form

ClassName<Type> object; Passes Type as an argument to the class template definition. Examples:

Stack<int> intSt;

Stack<string> stringSt;

Compiler will generate two distinct definitions of Stack two instances

one for ints and one for strings.

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6STL (Standard Template Library)

A library of class and function templates

Components:1. Containers:

• Generic "off-the-shelf" class templates for storing collections of data

2. Algorithms: • Generic "off-the-shelf" function templates for operating on

containers

3. Iterators: • Generalized "smart" pointers that allow algorithms to operate on

almost any container

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7The vector Container A type-independent pattern for an array class

capacity can expand self contained

Declaration

template <typename T>

class vector

{ . . . } ;

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8vector Operations Information about a vector's contents

v.size()

v.empty()

v.capacity()

v.reserve()

Adding, removing, accessing elements v.push_back()

v.pop_back()

v.front()

v.back()

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9Increasing Capacity of a Vector When vector v becomes full

capacity increased automatically when item added

Algorithm to increase capacity of vector<T> Allocate new array to store vector's elements

use T copy constructor to copy existing elements to new array

Store item being added in new array

Destroy old array in vector<T>

Make new array the vector<T>'s storage array

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10Iterators

Each STL container declares an iterator type

can be used to define iterator objects

Iterators are a generalization of pointers that allow a C++ program to work with different data structures (containers) in a uniform manner

To declare an iterator object

the identifier iterator must be preceded by name of container

scope operator :: Example:

vector<int>::iterator vecIter = v.begin() Would define vecIter as an iterator positioned at the first

element of v

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11IteratorsContrast use of subscript vs. use of iterator

ostream & operator<<(ostream & out, const vector<double> & v){ for (int i = 0; i < v.size(); i++) out << v[i] << " "; return out;}

for (vector<double>::iterator it = v.begin(); it != v.end(); it++) out << *it << " ";

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12Iterator Functions

Note Table 9-5

Note the capability of the last two groupings Possible to insert, erase elements of a vector anywhere in the vector

Must use iterators to do this

Note also these operations are as inefficient as for arrays due to the shifting required

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13Contrast Vectors and Arrays

Vectors Arrays

• Capacity can increase

• A self contained object• Is a class template (No specific type)• Has function members to do tasks

• Fixed size, cannot be changed during execution• Cannot "operate" on itself•Bound to specific type•Must "re-invent the wheel" for most actions

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14STL's deque Class Template

Has the same operations as vector<T> except … there is no capacity() and no reserve()

Has two new operations:

d.push_front(value);Push copy of value at front of d

d.pop_front(value);Remove value at the front of d

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15vector vs. deque

vector deque• Capacity of a vector must be increased • It must copy the objects from the old vector to the new vector • It must destroy each object in the old vector • A lot of overhead!

• With deque this copying, creating, and destroying is avoided.• Once an object is constructed, it can stay in the same memory locations as long as it exists

– If insertions and deletions take place at the ends of the deque.

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16vector vs. deque

Unlike vectors, a deque isn't stored in a single varying-sized block of memory, but rather in a collection of fixed-size blocks (typically, 4K bytes).

One of its data members is essentially an array map whose elements point to the locations of these blocks.

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17Linear SearchVector based search functiontemplate <typename t>void LinearSearch (const vector<t> &v,

const t &item, boolean &found, int &loc){

found = false; loc = 0; while(loc < n && !found){ if (found || loc == v.size()) return;

if (item == x[loc]) found = true;

else loc++; }

}

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18Binary SearchBinary search function for vector

template <typename t>void LinearSearch (const vector<t> &v,

const t &item, boolean &found, int &loc){

found = false;int first = 0; int last = v.size() - 1; while(first <= last && !found){

if (found || first > last) return;

loc = (first + last) / 2; if (item < v[loc]) last = loc + 1;

else if (item > v[loc]) first = loc + 1; else /* item == v[loc] */

found = true; }

}

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19Binary Search

Usually outperforms a linear search

Disadvantage: Requires a sequential storage

Not appropriate for linked lists (Why?)

It is possible to use a linked structure which can be searched in a binary-like manner

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20Trees

Tree terminology Root nodeRoot node

Leaf nodesLeaf nodes

• Children of the parent (3)

• Siblings to each other

• Children of the parent (3)

• Siblings to each other

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21Binary Trees

Each node has at most two children Useful in modeling processes where

a comparison or experiment has exactly two possible outcomes the test is performed repeatedly

Example multiple coin tosses encoding/decoding messages in dots and dashes such as Morse code

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22Binary Trees

Each node has at most two children Useful in modeling processes where

a comparison or experiment has exactly two possible outcomes the test is performed repeatedly

Example multiple coin tosses encoding/decoding messages in dots and dashes such as Morse code

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23Array Representation of Binary Trees

Works OK for complete trees, not for sparse trees

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24Linked Representation of Binary Trees Uses space more efficiently

Provides additional flexibility

Each node has two links one to the left child of the node

one to the right child of the node

if no child node exists for a node, the link is set to NULL

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25Binary Trees as Recursive Data Structures

A binary tree is either empty …

or

Consists of a node called the root

root has pointers to two disjoint binary (sub)trees called …

right (sub)tree

left (sub)tree

AnchorAnchor

Inductive step

Inductive step

Which is either empty … or …

Which is either empty … or … Which is either empty

… or …

Which is either empty … or …

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26ADT Binary Search Tree (BST)

Collection of Data Elements binary tree

each node x,

value in left child of x value in x in right child of x

Basic operations Construct an empty BST

Determine if BST is empty

Search BST for given item

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27ADT Binary Search Tree (BST)

Basic operations (ctd) Insert a new item in the BST

Maintain the BST property

Delete an item from the BST

Maintain the BST property

Traverse the BST

Visit each node exactly once

The inorder traversal must visit the values in the nodes in ascending order

View BST class template, Fig. 12-1View BST class template, Fig. 12-1

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28BST Traversals Note that recursive calls must be

made To left subtree To right subtree

Must use two functions Public method to send message to BST

object Private auxiliary method that can access

BinNodes and pointers within these nodes Similar solution to graphic output

Public graphic method Private graphAux method

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29BST Searches

Search begins at root If that is desired item, done

If item is less, move downleft subtree

If item searched for is greater, move down right subtree

If item is not found, we will run into an empty subtree

View search()

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30Inserting into a BST Insert function

Uses modified version of search to locate insertion location or already existing item

Pointer parent trails search pointer locptr, keeps track of parent node

Thus new node can be attached to BST in proper place

View insert() functionR

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31Recursive Deletion

Three possible cases to delete a node, x, from a BST

1. The node, x, is a leaf

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32Recursive Deletion

2. The node, x has one child

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33Recursive Deletion x has two children

Replace contents of x with inorder successorReplace contents of x with inorder successor

K

Delete node pointed to by xSucc as

described for cases 1 and 2

Delete node pointed to by xSucc as

described for cases 1 and 2

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34Problem of Lopsidedness

Trees can be totally lopsided Suppose each node has a right child only

Degenerates into a linked list

Processing time affected by

"shape" of tree

Processing time affected by

"shape" of tree

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35Hash Tables

In some situations faster search is needed Solution is to use a hash function

Value of key field given to hash function

Location in a hash table is calculated

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36Hash Functions

Simple function could be to mod the value of the key by the size of the table H(x) = x % tableSize

Note that we have traded speed for wasted space Table must be considerably larger than number of items anticipated Suggested to be 1.5-2x larger

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37Hash Functions

Observe the problem with same value returned by h(x) for different values of x Called collisions

A simple solution is linear probing Empty slots marked with -1

Linear search begins atcollision location

Continues until emptyslot found for insertion

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38Hash Functions

When retrieving a valuelinear probe until found If empty slot encountered

then value is not in table

If deletions permitted Slot can be marked so

it will not be empty and cause an invalid linear probe

Ex. -1 for unused slots, -2 for slots which used to contain data

Collision Reduction Strategies

Hash table capacity Size of table must be 1.5 to 2 times the size of the number of items

to be stored

Otherwise probability of collisions is too high

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39

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40Collision Reduction Strategies

Linear probing can result in primary clustering

Consider quadratic probing Probe sequence from location i is

i + 1, i – 1, i + 4, i – 4, i + 9, i – 9, …

Secondary clusters can still form

Double hashing Use a second hash function to determine probe sequence

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41Collision Reduction Strategies

Chaining Table is a list or vector of head nodes to linked lists

When item hashes to location, it is added to that linked list

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42Improving the Hash Function

Ideal hash function Simple to evaluate

Scatters items uniformly throughout table

Modulo arithmetic not so good for strings Possible to manipulate numeric (ASCII) value of first and last

characters of a name

43Categories of Sorting Algorithms

Selection sort Make passes through a list

On each pass reposition correctly some element (largest or smallest)

44Array 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

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45In-Class Exercise #1: Selection Sort

List of 9 elements:

90, 10, 80, 70, 20, 30, 50, 40, 60

Illustrate each pass…

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46Selection Sort Solution

Pass 0 90 10 80 70 20 30 50 40 60

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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

47Categories 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

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48Bubble Sort Algorithm

1. Initialize numCompares to n - 1

2. 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

49In-Class Exercise #2: Bubble Sort

List of 9 elements:

90, 10, 80, 70, 20, 30, 50, 40, 60

Illustrate each pass…

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50Bubble Sort Solution

Pass 0 90 10 80 70 20 30 50 40 60

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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

51Categories 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)

52Insertion 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

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53Insertion 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

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54In-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…

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55Insertion Sort Solution

Pass 0 9 3 1 5 2 key j i

1 3 9 1 5 2 3 2 1,0

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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

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56Quicksort

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

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57Quicksort

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.

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58Quicksort

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).

59In-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

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60Quicksort Solution

Pass 0

30 10 80 70 20 90 50 40 60

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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

61Heaps

62Implementing 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)

63Implementing 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]

64Basic 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


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


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

67N

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68Percolate 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

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69Heapsort

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

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70Heapsort 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]

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71Heapsort 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

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72Heapsort 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

73In-Class Exercise #4: Heapsort

For each step, want to draw the heap and array

30, 10, 80, 70, 20, 90, 40

Array?

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74Step 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)

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75Step 1 (ctd)

What is the last node that is not a leaf?

Apply percolate down

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We now have a heap!

78Step 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

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79Step 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

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80Step 2 (ctd)

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81Continue the patternN

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84Complete!N

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85Sorting 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

86Mergesort

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.

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87Merge Algorithm

1. Open File1 and File2 for input, File3 for output

2. Read first element x from File1 and first element y from File2

3. While neither eof File1 or eof File2If x < y then

a. Write x to File3b. Read a new x value from File1

Otherwisea. Write y to File3b. Read a new y from File2

End while

4. If eof File1 encountered copy rest of of File2 into File3. If eof File2 encountered, copy rest of File1 into File3

88Mergesort Algorithm

89In-Class Exercise #6

Take File1 and File2 and produce a sorted File 3

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File 1 7 9 19 33 47 51 82 99

File 2 11 18 24 49 61

File 3

90Mergesort SolutionN

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File 1 7 9 19 33 47 51 82 99

File 2 11 18 24 49 61

File 3 7 9 11

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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

91

O(n)

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92Binary Merge Sort

Given a single file

Split into two files

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93Binary 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

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94Binary Merge Sort

Split again

Merge again as before

Each time, the size of the sorted subgroups doubles

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95Binary 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

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96Natural 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

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Copy alternate groupings into two files Use the sub-groupings, not a power of 2

Look for possible larger groupings

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Merge the corresponding sub files

EOF for F2, Copy remaining groups from

F1

EOF for F2, Copy remaining groups from

F1

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Split again, alternating groups

Merge again, now two subgroups

One more split, one more merge gives sort

Documents

Exam #2 Review. Nyhoff, ADTs, Data Structures and Problem Solving with C++, Second Edition, © 2005 Pearson Education, Inc. All rights reserved. 0-13-140909-3