算法导论复习:第八章和第九章

定理: 在最坏情况下,任何比较排序算法都需要做nlgn次比较.

1. 计数排序

基本原理:假设n个输入元素中的每一个都是在0到k区间内的一个整数,其中k为某个整数.当k=O(n)时,排序的运行时间为n.对于每一个输入元素x,确定小于x的元素个数.

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <string.h>

#define SIZE 100

void countingSort( int *A, int *B, int k, int len );

int main( void )
{
	int		A[ SIZE ];
	int		B[ SIZE ];
	int		k = 0;
	int		i = 0;
	memset( B, 0, sizeof( int ) * SIZE );
	k = SIZE - 5;
	srand( ( unsigned int )time( NULL ) );
	for ( i = 0; i < SIZE; i++ ){
		A[ i ] = rand() % k;
	}

	printf("the array is:\n");
	for ( i = 0; i < SIZE; i++ ){
		printf("%d ", A[ i ] );
		if ( 0 == ( i + 1 ) % 10 ){
			printf("\n");
		}
	}

	countingSort( A, B, k, SIZE );

	printf("\nafter sort,the array is:\n");
	for ( i = 0; i < SIZE; i++ ){
		printf("%d ", B[ i ] );
		if ( 0 == ( i + 1 ) % 10 ){
			printf("\n");
		}
	}
	printf("\n");

	return 0;
}

void countingSort( int *A, int *B, int k, int len )
{
	int		*C = ( int * )malloc( sizeof( int ) * k );
	int		i = 0;
	memset( C, 0, sizeof( int ) * k );

	for ( i = 0; i < len; i++ ){
		C[ A[ i ] ] += 1;
	}
	for ( i = 1; i < k; i++ ){
		C[ i ] += C[ i - 1 ];
	}
	for ( i = 0; i < k; i++ ){
		C[ i ] -= 1;		//下标从0开始,否则B会存在下标越界情况
	}

	for ( i = len - 1; i >= 0; i-- ){
		B[ C[ A[ i ] ] ] = A[ i ];
		C[ A[ i ] ] -= 1;
	}
	free( C );
}

程序输出:

备注:在算法的讨论中,一般下标是从1开始的,而实际的编码却是从0开始,所以要小心下标的越界情况.

2. 基数排序

我们用基数排序对以下英文单词进行排序:COW,DOG,SEA,RUG,ROW,MOB,BOX,TAB,BAR,EAR,TAR,DIG,BIG,TEA,NOW,FOX.

#include <stdio.h>

void radixSort( char *A[], int len, int keyLen );
int main( void )
{
	int		i = 0;
	char		*A[] = { "COW", "DOG", "SEA", "RUG", "ROW", "MOB", "BOX", "TAB", "BAR", "EAR", "TAR", "DIG", "BIG", "TEA", "NOW", "FOX" };
	int		len = sizeof( A ) / sizeof( *A );
	int		keyLen = 3;		//每个单词的长度,这里不考虑长度不同的情况,只是为了更好的说明基数排序
	radixSort( A, len, keyLen );

	for ( i = 0; i < len; i++ ){
		printf("%s ", A[ i ] );
		if ( 0 == ( i + 1 ) % 5 ){
			printf("\n");
		}
	}
	printf("\n");

	return 0;
}

void radixSort( char *A[], int len, int keyLen )
{
	int		i = 0;
	int		j = 0;
	int		k = 0;
	for ( i = keyLen - 1; i >= 0; i-- ){
		//插入排序
		for ( j = 1; j < len; j++ ){
			char		keyValue = A[ j ][ i ];
			char		*key = A[ j ];
			k = j - 1;
			while ( k >= 0 && A[ k ][ i ] > keyValue ){
				A[ k + 1 ] = A[ k ];
				k -= 1;
			}
			A[ k + 1 ] = key;
		}
	}
}

程序输出:

3. 桶排序

假设有100个数据,放在10个桶中,对每个桶进行排序,然后再合并10个桶.

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <string.h>
#include <limits.h>

#define SIZE 100
#define LENBUCKET		10		

typedef struct NODE{
	int						data;
	struct NODE		*link;
}Node;

void bucketSort( int *A, int len );
void insertNode( Node *node, int value );

int main( void )
{
	int		A[ SIZE ];
	int		i = 0;
	srand( ( unsigned int )time( NULL ) );
	for ( i = 0; i < SIZE; i++ ){
		A[ i ] = rand() % SIZE;
	}

	printf("the array is:\n");
	for ( i = 0; i < SIZE; i++ ){
		printf("%3d ", A[ i ] );
		if ( 0 == ( i + 1 ) % 10 ){
			printf("\n");
		}
	}

	bucketSort( A, SIZE );

	printf("after sort, the array is:\n");
	for ( i = 0; i < SIZE; i++ ){
		printf("%3d ", A[ i ] );
		if ( 0 == ( i + 1 ) % 10 ){
			printf("\n");
		}
	}
	printf("\n");

	return 0;
}

void bucketSort( int *A, int len )
{
	Node		*B[ LENBUCKET ];
	int			i = 0;
	int			j = 0;
	for ( i = 0; i < LENBUCKET; i++ ){
		Node *node = ( Node * )malloc( sizeof( Node ) );
		node->link = NULL;
		node->data = INT_MIN;
		B[ i ] = node;
	}
	for ( i = 0; i < len; i++ ){
		int			index = A[ i ] / ( SIZE / LENBUCKET );
		insertNode( B[ index ], A[ i ] );
	}

	for ( i = 0, j = 0; j < LENBUCKET; j++ ){
		Node *node = ( Node * )malloc( sizeof( Node ) );
		node = B[ j ]->link;
		while ( NULL != node ){
			A[ i++ ] = node->data;
			node = node->link;
		}
	}
}

void insertNode( Node *node, int value )
{
	Node *tempNode = ( Node * )malloc( sizeof( Node ) );
	tempNode->data = value;
	tempNode->link = NULL;
	//为空结点
	if ( NULL == node->link ){
		node->link = tempNode;
	}
	else{
		//插入头部
		if ( value <= node->link->data ){
			tempNode->link = node->link;
			node->link = tempNode;
		}
		else{
			Node *prevNode = ( Node * )malloc( sizeof( Node ) );
			node = node->link;
			while ( NULL != node && value > node->data ){
				prevNode = node;
				node = node->link;
			}
			tempNode->link = node;
			prevNode->link = tempNode;
		}
	}
}

程序输出:

备注:1.一定要模块化,比如插入链表就应该封装成insertNode函数.

2. 对于链表的插入,初始化一个头节点专门用来标识这个链表是良好的设计方法,毕竟C语言只支持传值不支持传址(没有C++中引用).

4. 同时查找最小值和最大值

#include <stdio.h>
#include <stdlib.h>
#include <time.h>

#define SIZE	100

void searchMinMax( int *A, int *minValue, int *maxValue, int len );

int main( void )
{
	int		A[ SIZE ];
	int		i = 0;
	int		minValue = 0;
	int		maxValue = 0;
	srand( ( unsigned int )time( NULL ) );
	for ( i = 0; i < SIZE; i++ ){
		A[ i ] = rand() % SIZE;
	}

	searchMinMax( A, &minValue, &maxValue, SIZE );

	printf("the array is:\n");
	for ( i = 0; i < SIZE; i++ ){
		printf("%3d ", A[ i ] );
		if ( 0 == ( i + 1 ) % 10 ){
			printf("\n");
		}
	}

	printf("\nthe min value is: %3d\nthe max value is:%3d\n", minValue, maxValue );

	return 0;
}

void searchMinMax( int *A, int *minValue, int *maxValue, int len )
{
	int		i = 0;
	if ( 0 == len % 2 ){
		*minValue = A[ 0 ] < A[ 1 ] ? A[ 0 ] : A[ 1 ];
		*maxValue = A[ 0 ] > A[ 1 ] ? A[ 0 ] : A[ 1 ];
	}
	else{
		*minValue = *maxValue = A[ 0 ];
	}

	for ( i = 2; i < len; i += 2 ){
		if ( A[ i ] < A[ i + 1 ] ){
			if ( *minValue > A[ i ] ){
				*minValue = A[ i ];
			}
			if ( *maxValue < A[ i + 1 ] ){
				*maxValue = A[ i + 1 ];
			}
		}
		else{
			if ( *minValue > A[ i + 1 ] ){
				*minValue = A[ i + 1 ];
			}
			if ( *maxValue < A[ i ] ){
				*maxValue = A[ i ];
			}
		}
	}
}

程序输出: