1. preface
/****
* This article will try to explain something about:
* --Bubble sort.
* --Quick sort.
* --Merge sort.
* --Heap sort.
* To read this, some prerequisites is necessary:
* --a survive skill in C programming language.
* --a basic skill in CPP.
* --basic knowledge about Time complexity and Space complexity.
* --a generous mind for my fault because this article is free.
*
* Actually, their basic operating principle are easy to understand, but , unfortunately, the precise explication is big problem, at least for me. Here is the contents.
* --Analysis about Bubble sort.
* --Analysis about Quick sort.
* --Analysis about Merge sort.
* --Analysis about Heap sort.
* their source code can be find in fellowing articles. As we all know, for a programmer, the source code is also a good choice.
*/
2. Bubble sort
/****
* Bubble Sort
* This is a really simple algorithm. I learn it when I began to learn C. It just compare two value successively and swap them. The core source code is as fellowing:
*/
2.1 core code
bool BubbleSort::sort(void)
{
int i,j;
for( i=0; i<=n; i++)
for( j=0; j< n -i; j++)
{
if( this->array[ j]>this->array[ j+1])
{
this->swap( j, j + 1); //swap two values
}
}
}
2.2 Time complexity and Space complexity
/**
* For sort algorithm, it's basic operation is swap two values.So we can compute it's sentence frequency f(n):
* f(n) = n*n = n^2
* (at worst situation)
* and it's time complexity is :
* T(n) = O( f(n)) = O(n^2)
*
*
* obviously, It's space complexity is :
* S(n) = O( g(n)) = O( C) = O(1)
* because it use only constant space whatever n change.
*/
/*
* The totally example source code is here:
*
* (It maybe some fault, I will glad to your advices)
*/
3. Quick sort
/****
* Quick Sort
* This is a famous algorithm. It was developed in 1960 , but never outdated even for now. I was face a problem about sort a million of numbers. Need to say that is a
* nightmare if use bubble sort. Then I learn Quick Sort, it provide a exciting performance. I will explain this below. The principle of quicksort is "divided and process".
* In detail,
* --step1: Pick an element from the array as pivot.
* --step2: Part all elements into two areas: left area and right area.put element that's value less than pivot's value into left area,and put other elements into right area.
* --step3: Recursively do the step above to those sub-array.
*
*
*. First at all, examine it's core source code:
*/
3.1 Core Code
static void quick_sort( int array[], INDEX left, INDEX right)
{
if( right-left>=2)
{//core code
int p;
p = pivot( array, left, right); //step1 + step2
quick_sort( array, left, p-1); //step3
quick_sort( array, p+1, right);
}
else if( right -left ==1)
{//auxiliary
if( array[left] > array[right])
{
swap( array + left, array + right);
}
}
}
static int pivot( int array[], INDEX left, INDEX right)
{
//get povit, one of methods
INDEX mInd = (left + right)/2;
//divide array into two parts
int i = 0;
int LLen = 0, RLen = 0;
for( i=left; i<=right; i++ )
{
if( i==mInd)
continue;
if( array[i]< array[mInd] )
{
Arr_back[left + LLen] = array[i];
LLen++;
}
else
{
Arr_back[right - RLen] = array[i];
RLen++;
}
}
Arr_back[left + LLen] = array[mInd];
memcpy( array+left, Arr_back + left, (right-left + 1)*sizeof(int)); //use a auxiliary space
return left + LLen;
}
3.2 Time complexity
/**
* For quicksort, the basic operation is similar to swap above. So we could compute a valid sentence frequency. If there are n elements, in average situation the depth of
* recurrence is log2(n).Just as below:
*
* step1: [1.........................................................n] // n
* step2: [1......................m1] [m1.......................n] // n/2 + n/2
* step3: [1.....m2] [m2.....m1] [m1....m3] [m3......n] // n/4 + n/4 + n/4 + n/4
* .......
* stepX: [1,2][3,4]................................................
*
* and funny is that: for step N, if we want to part those arrays into sub-array, we need the number of basic operation is :
* N*(n/N)
* that's means:
* f(n) = n*log2(n)
* and
* T(n) = O( f(n)) = O( n*log2(n) )
*
*/
3.3 Space complexity
/**
* At least two points are deserve to consider.
* Point.1 : Normally, we need more auxiliary space when n increase.
* Point.2 : the recursion of function may be need more space.
*
* In my situation, the auxiliary space of Point.1 is n. For Point.2, Assume that the cost is A for ecah function call, the totally number of call is
* 2^0 + 2^1 + 2^2 + .....2^log2(n)
*
* then, the cost of point.2 is
*
* A*[1 + 2^1 + 2^2 + ....2^log2(n) ]
* =A*[1 + 2^1 + 2^2 + ....+ n]
* =A*[2*n-1] < A*2*n
*
* combine two parts:
* S(n) = O( B*n) = O(n)
*/
/*
* References
* wikipedia-Quicksort <http://en.wikipedia.org/wiki/Quicksort>
*/
4. Merge sort
/****
* Merge Sort
* The common view is that: compare with famous Quicksort and Heapsort, it is slightly worse in sort a array. but it provide a excellent performance in sort a link list,
* which is difficult to Quicksort and Heapsort. on the other side, Mergesort is a stable sort, unlike standard in-place quicksort and heapsort. It's core principle is "divide
* and conquer".
*
* conceptually, Mergesort work as fellow:
* step1: divide the array into n sublists.That means every sublist is only contain of 1 element.
* step2: repeatedly merge all sublists to create new sorted sublist untill there is only 1 sublist remaining.
*
* just like this:
* step1: [0] [1] [2] [3] [4] [5] [6] [7]
* step2: [0...1] [2...3] [4...5] [6...7]
* step3: [0............3] [4..............7]
* step4: [0..................................7]
*
* If you need more information, there will be a good place.
* http://en.wikipedia.org/wiki/Merge_sort
*
* then , examine the core source code:
*/
4.1 core code
bool MergeSort::sort(void)
{
......
int width = 1;
while( width < (this->right - this->left + 1) )
{
this->subsort( width); //sort sublists
width *= 2;
}
.....
}
bool MergeSort::subsort( int width)
{
.....
INDEX cur = this->left;
while( cur + width <= this->right)
{
//sort two sublists into a new sorted list.
this->sort2Arr( cur, width, cur + width, MIN( width, this->right-cur-width+1));
cur += 2*width;
}
memcpy( this->array, this->array_back, (this->right - this->left + 1)*sizeof(int));
.....
}
4.2 Time complexity
/**
* Time complexity
*
* Now, let me see a interesting thing before check it's Time frequency. Image this, there are two arrays ,both of them are progressive increase. they are contain of n and
* m elements respectively.
*
* [1.............n] [1..........m]
*
* How many times is necessary to merge them into a new sorted array?
*
* -- at least: MIN( n,m);
* at most: m+n;
*
* For example:
* [ 1, 2, 3] [4,5,6,7]
* and
* [1,2,3,7] [4,5,6]
*
*
* Based on the conclusions above, we could know that : at worst situation, if we want to sort n elements by the way of Mergesort, the times of compare operation is n.
*
* So, Time frequency is n*log2(n)
* and
* T(n) = O( f(n)) = O( n*log2(n) )
*
*/
4.3 Space complexity
/**
* Space complexity
* In my example, a additional array was used to auxiliary operation.
* obviously, the space complexity is :
* S(n) = O(n);
*
* but that is at worst situation. It could be optimized.
*
*/
5. Heap sort
/****
* Heap Sort
* This is another famous sort algorithm. Need to say: it's very cool. Although sometimes it is slower in practice on most machine than well-implemented quicksort, it's
*have the advantage of a more favorable worst-case O( n*log(n)) runtime. unfortunately, it is not a stable sort.
*/
/*
* Before explain heapsort, some questions are necessary to know:
* 1). How can we store a binary tree into a array ?
*
* --if we number all nodes of a tree , based on 1, you will find a rule. To a node n, it must have the fellowing relationship:
* parent : floor(n/2)
* left chil : 2*n
* right chil : 2*n + 1
*
* This feature gives us a chance to save a tree into a array.
*
* 2). What is max heapify ?
* --For a binary tree, if all of parent nodes greater than or equal to those corresponding child nodes, the root node must be the largest node in this tree. In other words,
* we can get the largest one between some nodes by arrange those number into a max binary tree. By the way, if binary tree can do that, then heap can, too.
*/
/*
* The Heapsort algorithm can be divided into two parts.
* step 1: build a max binary tree.
*
* step 2: remove the largest node( the root of the tree) ,and then update the tree repeatedly untill all of nodes has been get out.
*
*/
5.1 core code
bool HeapSort::sort(void)
{
/*
* As we all know, some of nodes haven't child node.
* For skip those nodes, we need to find the last parent node.
*
* but How can we do that?
*
* --the answer is the last child node.
*/
INDEX nInd = 0;
nInd = this->fun.GetParentInd( this->right );
/*
* Adjust nodes from bottom to top.Function MaxHeapify()
* will arrange a node and its's sublayer nodes to
* a max binary tree.
*/
while( nInd>=0)
{
// @(this->right) is the number of nodes.
this->MaxHeapify( nInd, this->right);
nInd--;
}
/*
* moving the largest one between all of nodes into a array,
* and tidy the remaining. Repeat this process untill
* we get all of nodes.
*/
nInd = this->right;
while( nInd>0 )
{
this->Swap( 0, nInd);
nInd --;
this->MaxHeapify( 0, nInd);
}
return true;
}
bool HeapSort::MaxHeapify( INDEX nInd, INDEX right)
{
INDEX max = this->GetBigNodeInd( nInd, right);
while( max!=nInd)
{
this->Swap( max, nInd);
nInd = max;
max = this->GetBigNodeInd( nInd, right);
}
return true;
}
/*
* About @MaxHeapify(), there are many problems need to solve. This article is worth to reading:
* http://shmilyaw-hotmail-com.iteye.com/blog/1775868
*
*/
5.2 Time complexity
/**
* sorry, I have no idea.
*/
5.3 Space complexity
/**
* space complexity
*
* It is simple to compute the space complexity.
* S(n) = O(1);
* because it use a constant space.
*/
/*
* The totally example source code is here:
*
* (It maybe some fault, I will glad to your advices)
*/
/**
* References:
*
* heap sort分析和总结 <http://shmilyaw-hotmail-com.iteye.com/blog/1775868>
* heapsort <http://en.wikipedia.org/wiki/Heapsort>
*
*/