我知道这个问题不太具体。
我想要的只是有人告诉我如何将普通的归并排序转换为就地归并排序(或具有常量额外空间开销的归并排序)。
我所能找到的(在网上)都是说“这太复杂”或“超出了本文的范围”的页面。
唯一已知的就地合并(没有任何额外空间)的方法太复杂，无法简化为实用程序。(从这里拍摄)
即使它太复杂，如何使归并排序的基本概念是什么?
当前回答
C语言中无缓冲区归并排序的一个例子。
#define SWAP(type, a, b) \
do { type t=(a);(a)=(b);(b)=t; } while (0)
static void reverse_(int* a, int* b)
{
for ( --b; a < b; a++, b-- )
SWAP(int, *a, *b);
}
static int* rotate_(int* a, int* b, int* c)
/* swap the sequence [a,b) with [b,c). */
{
if (a != b && b != c)
{
reverse_(a, b);
reverse_(b, c);
reverse_(a, c);
}
return a + (c - b);
}
static int* lower_bound_(int* a, int* b, const int key)
/* find first element not less than @p key in sorted sequence or end of
* sequence (@p b) if not found. */
{
int i;
for ( i = b-a; i != 0; i /= 2 )
{
int* mid = a + i/2;
if (*mid < key)
a = mid + 1, i--;
}
return a;
}
static int* upper_bound_(int* a, int* b, const int key)
/* find first element greater than @p key in sorted sequence or end of
* sequence (@p b) if not found. */
{
int i;
for ( i = b-a; i != 0; i /= 2 )
{
int* mid = a + i/2;
if (*mid <= key)
a = mid + 1, i--;
}
return a;
}
static void ip_merge_(int* a, int* b, int* c)
/* inplace merge. */
{
int n1 = b - a;
int n2 = c - b;
if (n1 == 0 || n2 == 0)
return;
if (n1 == 1 && n2 == 1)
{
if (*b < *a)
SWAP(int, *a, *b);
}
else
{
int* p, * q;
if (n1 <= n2)
p = upper_bound_(a, b, *(q = b+n2/2));
else
q = lower_bound_(b, c, *(p = a+n1/2));
b = rotate_(p, b, q);
ip_merge_(a, p, b);
ip_merge_(b, q, c);
}
}
void mergesort(int* v, int n)
{
if (n > 1)
{
int h = n/2;
mergesort(v, h); mergesort(v+h, n-h);
ip_merge_(v, v+h, v+n);
}
}
一个自适应归并排序的例子(优化)。
添加支持代码和修改，以在任何大小的辅助缓冲区可用时加速合并(在没有额外内存的情况下仍然可以工作)。使用前向和后向合并、环旋转、小序列合并和排序以及迭代合并排序。
#include <stdlib.h>
#include <string.h>
static int* copy_(const int* a, const int* b, int* out)
{
int count = b - a;
if (a != out)
memcpy(out, a, count*sizeof(int));
return out + count;
}
static int* copy_backward_(const int* a, const int* b, int* out)
{
int count = b - a;
if (b != out)
memmove(out - count, a, count*sizeof(int));
return out - count;
}
static int* merge_(const int* a1, const int* b1, const int* a2,
const int* b2, int* out)
{
while ( a1 != b1 && a2 != b2 )
*out++ = (*a1 <= *a2) ? *a1++ : *a2++;
return copy_(a2, b2, copy_(a1, b1, out));
}
static int* merge_backward_(const int* a1, const int* b1,
const int* a2, const int* b2, int* out)
{
while ( a1 != b1 && a2 != b2 )
*--out = (*(b1-1) > *(b2-1)) ? *--b1 : *--b2;
return copy_backward_(a1, b1, copy_backward_(a2, b2, out));
}
static unsigned int gcd_(unsigned int m, unsigned int n)
{
while ( n != 0 )
{
unsigned int t = m % n;
m = n;
n = t;
}
return m;
}
static void rotate_inner_(const int length, const int stride,
int* first, int* last)
{
int* p, * next = first, x = *first;
while ( 1 )
{
p = next;
if ((next += stride) >= last)
next -= length;
if (next == first)
break;
*p = *next;
}
*p = x;
}
static int* rotate_(int* a, int* b, int* c)
/* swap the sequence [a,b) with [b,c). */
{
if (a != b && b != c)
{
int n1 = c - a;
int n2 = b - a;
int* i = a;
int* j = a + gcd_(n1, n2);
for ( ; i != j; i++ )
rotate_inner_(n1, n2, i, c);
}
return a + (c - b);
}
static void ip_merge_small_(int* a, int* b, int* c)
/* inplace merge.
* @note faster for small sequences. */
{
while ( a != b && b != c )
if (*a <= *b)
a++;
else
{
int* p = b+1;
while ( p != c && *p < *a )
p++;
rotate_(a, b, p);
b = p;
}
}
static void ip_merge_(int* a, int* b, int* c, int* t, const int ts)
/* inplace merge.
* @note works with or without additional memory. */
{
int n1 = b - a;
int n2 = c - b;
if (n1 <= n2 && n1 <= ts)
{
merge_(t, copy_(a, b, t), b, c, a);
}
else if (n2 <= ts)
{
merge_backward_(a, b, t, copy_(b, c, t), c);
}
/* merge without buffer. */
else if (n1 + n2 < 48)
{
ip_merge_small_(a, b, c);
}
else
{
int* p, * q;
if (n1 <= n2)
p = upper_bound_(a, b, *(q = b+n2/2));
else
q = lower_bound_(b, c, *(p = a+n1/2));
b = rotate_(p, b, q);
ip_merge_(a, p, b, t, ts);
ip_merge_(b, q, c, t, ts);
}
}
static void ip_merge_chunk_(const int cs, int* a, int* b, int* t,
const int ts)
{
int* p = a + cs*2;
for ( ; p <= b; a = p, p += cs*2 )
ip_merge_(a, a+cs, p, t, ts);
if (a+cs < b)
ip_merge_(a, a+cs, b, t, ts);
}
static void smallsort_(int* a, int* b)
/* insertion sort.
* @note any stable sort with low setup cost will do. */
{
int* p, * q;
for ( p = a+1; p < b; p++ )
{
int x = *p;
for ( q = p; a < q && x < *(q-1); q-- )
*q = *(q-1);
*q = x;
}
}
static void smallsort_chunk_(const int cs, int* a, int* b)
{
int* p = a + cs;
for ( ; p <= b; a = p, p += cs )
smallsort_(a, p);
smallsort_(a, b);
}
static void mergesort_lower_(int* v, int n, int* t, const int ts)
{
int cs = 16;
smallsort_chunk_(cs, v, v+n);
for ( ; cs < n; cs *= 2 )
ip_merge_chunk_(cs, v, v+n, t, ts);
}
static void* get_buffer_(int size, int* final)
{
void* p = NULL;
while ( size != 0 && (p = malloc(size)) == NULL )
size /= 2;
*final = size;
return p;
}
void mergesort(int* v, int n)
{
/* @note buffer size may be in the range [0,(n+1)/2]. */
int request = (n+1)/2 * sizeof(int);
int actual;
int* t = (int*) get_buffer_(request, &actual);
/* @note allocation failure okay. */
int tsize = actual / sizeof(int);
mergesort_lower_(v, n, t, tsize);
free(t);
}
其他回答
这个答案有一个代码示例，实现了黄炳超和Michael a . Langston在论文《Practical in - place merge》中描述的算法。我不得不承认我不了解细节，但给定的合并步骤的复杂性是O(n)。
从实际的角度来看，有证据表明，纯就地实现在现实场景中并没有表现得更好。例如，c++标准定义了std::inplace_merge，顾名思义，这是一个就地合并操作。
假设c++库通常都得到了很好的优化，看看它是如何实现的是很有趣的:
1) libstdc++ (GCC代码库的一部分):std::inplace_merge
实现委托给__inplace_merge，通过尝试分配一个临时缓冲区来避免这个问题:
typedef _Temporary_buffer<_BidirectionalIterator, _ValueType> _TmpBuf;
_TmpBuf __buf(__first, __len1 + __len2);
if (__buf.begin() == 0)
std::__merge_without_buffer
(__first, __middle, __last, __len1, __len2, __comp);
else
std::__merge_adaptive
(__first, __middle, __last, __len1, __len2, __buf.begin(),
_DistanceType(__buf.size()), __comp);
否则，它将退回到一个实现(__merge_without_buffer)，该实现不需要额外的内存，但不再在O(n)时间内运行。
2) libc++ (Clang代码库的一部分):std::inplace_merge
看起来相似。它委托给一个函数，该函数也尝试分配一个缓冲区。它将根据是否获得了足够的元素来选择实现。常量内存回退函数称为__buffered_inplace_merge。
也许回退仍然是O(n)时间，但关键是如果有临时内存可用，他们就不使用实现。
注意，c++标准通过将所需的复杂度从O(n)降低到O(nlog n)，显式地给予实现选择这种方法的自由:
复杂性: 如果有足够的额外内存可用，则正好是N-1个比较。如果内存不足，则比较O(N log N)次。
当然，这并不能证明常数空间在O(n)时间内的合并永远不应该被使用。另一方面，如果它会更快，优化的c++库可能会切换到那种类型的实现。
这是我的C版本:
void mergesort(int *a, int len) {
int temp, listsize, xsize;
for (listsize = 1; listsize <= len; listsize*=2) {
for (int i = 0, j = listsize; (j+listsize) <= len; i += (listsize*2), j += (listsize*2)) {
merge(& a[i], listsize, listsize);
}
}
listsize /= 2;
xsize = len % listsize;
if (xsize > 1)
mergesort(& a[len-xsize], xsize);
merge(a, listsize, xsize);
}
void merge(int *a, int sizei, int sizej) {
int temp;
int ii = 0;
int ji = sizei;
int flength = sizei+sizej;
for (int f = 0; f < (flength-1); f++) {
if (sizei == 0 || sizej == 0)
break;
if (a[ii] < a[ji]) {
ii++;
sizei--;
}
else {
temp = a[ji];
for (int z = (ji-1); z >= ii; z--)
a[z+1] = a[z];
ii++;
a[f] = temp;
ji++;
sizej--;
}
}
}
C语言中无缓冲区归并排序的一个例子。
#define SWAP(type, a, b) \
do { type t=(a);(a)=(b);(b)=t; } while (0)
static void reverse_(int* a, int* b)
{
for ( --b; a < b; a++, b-- )
SWAP(int, *a, *b);
}
static int* rotate_(int* a, int* b, int* c)
/* swap the sequence [a,b) with [b,c). */
{
if (a != b && b != c)
{
reverse_(a, b);
reverse_(b, c);
reverse_(a, c);
}
return a + (c - b);
}
static int* lower_bound_(int* a, int* b, const int key)
/* find first element not less than @p key in sorted sequence or end of
* sequence (@p b) if not found. */
{
int i;
for ( i = b-a; i != 0; i /= 2 )
{
int* mid = a + i/2;
if (*mid < key)
a = mid + 1, i--;
}
return a;
}
static int* upper_bound_(int* a, int* b, const int key)
/* find first element greater than @p key in sorted sequence or end of
* sequence (@p b) if not found. */
{
int i;
for ( i = b-a; i != 0; i /= 2 )
{
int* mid = a + i/2;
if (*mid <= key)
a = mid + 1, i--;
}
return a;
}
static void ip_merge_(int* a, int* b, int* c)
/* inplace merge. */
{
int n1 = b - a;
int n2 = c - b;
if (n1 == 0 || n2 == 0)
return;
if (n1 == 1 && n2 == 1)
{
if (*b < *a)
SWAP(int, *a, *b);
}
else
{
int* p, * q;
if (n1 <= n2)
p = upper_bound_(a, b, *(q = b+n2/2));
else
q = lower_bound_(b, c, *(p = a+n1/2));
b = rotate_(p, b, q);
ip_merge_(a, p, b);
ip_merge_(b, q, c);
}
}
void mergesort(int* v, int n)
{
if (n > 1)
{
int h = n/2;
mergesort(v, h); mergesort(v+h, n-h);
ip_merge_(v, v+h, v+n);
}
}
一个自适应归并排序的例子(优化)。
添加支持代码和修改，以在任何大小的辅助缓冲区可用时加速合并(在没有额外内存的情况下仍然可以工作)。使用前向和后向合并、环旋转、小序列合并和排序以及迭代合并排序。
#include <stdlib.h>
#include <string.h>
static int* copy_(const int* a, const int* b, int* out)
{
int count = b - a;
if (a != out)
memcpy(out, a, count*sizeof(int));
return out + count;
}
static int* copy_backward_(const int* a, const int* b, int* out)
{
int count = b - a;
if (b != out)
memmove(out - count, a, count*sizeof(int));
return out - count;
}
static int* merge_(const int* a1, const int* b1, const int* a2,
const int* b2, int* out)
{
while ( a1 != b1 && a2 != b2 )
*out++ = (*a1 <= *a2) ? *a1++ : *a2++;
return copy_(a2, b2, copy_(a1, b1, out));
}
static int* merge_backward_(const int* a1, const int* b1,
const int* a2, const int* b2, int* out)
{
while ( a1 != b1 && a2 != b2 )
*--out = (*(b1-1) > *(b2-1)) ? *--b1 : *--b2;
return copy_backward_(a1, b1, copy_backward_(a2, b2, out));
}
static unsigned int gcd_(unsigned int m, unsigned int n)
{
while ( n != 0 )
{
unsigned int t = m % n;
m = n;
n = t;
}
return m;
}
static void rotate_inner_(const int length, const int stride,
int* first, int* last)
{
int* p, * next = first, x = *first;
while ( 1 )
{
p = next;
if ((next += stride) >= last)
next -= length;
if (next == first)
break;
*p = *next;
}
*p = x;
}
static int* rotate_(int* a, int* b, int* c)
/* swap the sequence [a,b) with [b,c). */
{
if (a != b && b != c)
{
int n1 = c - a;
int n2 = b - a;
int* i = a;
int* j = a + gcd_(n1, n2);
for ( ; i != j; i++ )
rotate_inner_(n1, n2, i, c);
}
return a + (c - b);
}
static void ip_merge_small_(int* a, int* b, int* c)
/* inplace merge.
* @note faster for small sequences. */
{
while ( a != b && b != c )
if (*a <= *b)
a++;
else
{
int* p = b+1;
while ( p != c && *p < *a )
p++;
rotate_(a, b, p);
b = p;
}
}
static void ip_merge_(int* a, int* b, int* c, int* t, const int ts)
/* inplace merge.
* @note works with or without additional memory. */
{
int n1 = b - a;
int n2 = c - b;
if (n1 <= n2 && n1 <= ts)
{
merge_(t, copy_(a, b, t), b, c, a);
}
else if (n2 <= ts)
{
merge_backward_(a, b, t, copy_(b, c, t), c);
}
/* merge without buffer. */
else if (n1 + n2 < 48)
{
ip_merge_small_(a, b, c);
}
else
{
int* p, * q;
if (n1 <= n2)
p = upper_bound_(a, b, *(q = b+n2/2));
else
q = lower_bound_(b, c, *(p = a+n1/2));
b = rotate_(p, b, q);
ip_merge_(a, p, b, t, ts);
ip_merge_(b, q, c, t, ts);
}
}
static void ip_merge_chunk_(const int cs, int* a, int* b, int* t,
const int ts)
{
int* p = a + cs*2;
for ( ; p <= b; a = p, p += cs*2 )
ip_merge_(a, a+cs, p, t, ts);
if (a+cs < b)
ip_merge_(a, a+cs, b, t, ts);
}
static void smallsort_(int* a, int* b)
/* insertion sort.
* @note any stable sort with low setup cost will do. */
{
int* p, * q;
for ( p = a+1; p < b; p++ )
{
int x = *p;
for ( q = p; a < q && x < *(q-1); q-- )
*q = *(q-1);
*q = x;
}
}
static void smallsort_chunk_(const int cs, int* a, int* b)
{
int* p = a + cs;
for ( ; p <= b; a = p, p += cs )
smallsort_(a, p);
smallsort_(a, b);
}
static void mergesort_lower_(int* v, int n, int* t, const int ts)
{
int cs = 16;
smallsort_chunk_(cs, v, v+n);
for ( ; cs < n; cs *= 2 )
ip_merge_chunk_(cs, v, v+n, t, ts);
}
static void* get_buffer_(int size, int* final)
{
void* p = NULL;
while ( size != 0 && (p = malloc(size)) == NULL )
size /= 2;
*final = size;
return p;
}
void mergesort(int* v, int n)
{
/* @note buffer size may be in the range [0,(n+1)/2]. */
int request = (n+1)/2 * sizeof(int);
int actual;
int* t = (int*) get_buffer_(request, &actual);
/* @note allocation failure okay. */
int tsize = actual / sizeof(int);
mergesort_lower_(v, n, t, tsize);
free(t);
}
利用c++ std::inplace_merge，就地归并排序可以实现如下:
template< class _Type >
inline void merge_sort_inplace(_Type* src, size_t l, size_t r)
{
if (r <= l) return;
size_t m = l + ( r - l ) / 2; // computes the average without overflow
merge_sort_inplace(src, l, m);
merge_sort_inplace(src, m + 1, r);
std::inplace_merge(src + l, src + m + 1, src + r + 1);
}
更多的排序算法，包括并行实现，可以在https://github.com/DragonSpit/ParallelAlgorithms repo中找到，它是开源的，是免费的。
I just tried in place merge algorithm for merge sort in JAVA by using the insertion sort algorithm, using following steps. 1) Two sorted arrays are available. 2) Compare the first values of each array; and place the smallest value into the first array. 3) Place the larger value into the second array by using insertion sort (traverse from left to right). 4) Then again compare the second value of first array and first value of second array, and do the same. But when swapping happens there is some clue on skip comparing the further items, but just swapping required.
我在这里做了一些优化;在插入排序中保持较小的比较。我发现这个解决方案的唯一缺点是它需要在第二个数组中更大的数组元素交换。
e.g)
First___Array: 3,7,8,9
第二组:1,2,4,5
然后7,8,9使第二个数组交换(向左移动一个)它的所有元素，每次将自己放在最后一个数组中。
所以这里的假设是交换物品与比较两个物品相比可以忽略不计。
https://github.com/skanagavelu/algorithams/blob/master/src/sorting/MergeSort.java
package sorting;
import java.util.Arrays;
public class MergeSort {
public static void main(String[] args) {
int[] array = { 5, 6, 10, 3, 9, 2, 12, 1, 8, 7 };
mergeSort(array, 0, array.length -1);
System.out.println(Arrays.toString(array));
int[] array1 = {4, 7, 2};
System.out.println(Arrays.toString(array1));
mergeSort(array1, 0, array1.length -1);
System.out.println(Arrays.toString(array1));
System.out.println("\n\n");
int[] array2 = {4, 7, 9};
System.out.println(Arrays.toString(array2));
mergeSort(array2, 0, array2.length -1);
System.out.println(Arrays.toString(array2));
System.out.println("\n\n");
int[] array3 = {4, 7, 5};
System.out.println(Arrays.toString(array3));
mergeSort(array3, 0, array3.length -1);
System.out.println(Arrays.toString(array3));
System.out.println("\n\n");
int[] array4 = {7, 4, 2};
System.out.println(Arrays.toString(array4));
mergeSort(array4, 0, array4.length -1);
System.out.println(Arrays.toString(array4));
System.out.println("\n\n");
int[] array5 = {7, 4, 9};
System.out.println(Arrays.toString(array5));
mergeSort(array5, 0, array5.length -1);
System.out.println(Arrays.toString(array5));
System.out.println("\n\n");
int[] array6 = {7, 4, 5};
System.out.println(Arrays.toString(array6));
mergeSort(array6, 0, array6.length -1);
System.out.println(Arrays.toString(array6));
System.out.println("\n\n");
//Handling array of size two
int[] array7 = {7, 4};
System.out.println(Arrays.toString(array7));
mergeSort(array7, 0, array7.length -1);
System.out.println(Arrays.toString(array7));
System.out.println("\n\n");
int input1[] = {1};
int input2[] = {4,2};
int input3[] = {6,2,9};
int input4[] = {6,-1,10,4,11,14,19,12,18};
System.out.println(Arrays.toString(input1));
mergeSort(input1, 0, input1.length-1);
System.out.println(Arrays.toString(input1));
System.out.println("\n\n");
System.out.println(Arrays.toString(input2));
mergeSort(input2, 0, input2.length-1);
System.out.println(Arrays.toString(input2));
System.out.println("\n\n");
System.out.println(Arrays.toString(input3));
mergeSort(input3, 0, input3.length-1);
System.out.println(Arrays.toString(input3));
System.out.println("\n\n");
System.out.println(Arrays.toString(input4));
mergeSort(input4, 0, input4.length-1);
System.out.println(Arrays.toString(input4));
System.out.println("\n\n");
}
private static void mergeSort(int[] array, int p, int r) {
//Both below mid finding is fine.
int mid = (r - p)/2 + p;
int mid1 = (r + p)/2;
if(mid != mid1) {
System.out.println(" Mid is mismatching:" + mid + "/" + mid1+ " for p:"+p+" r:"+r);
}
if(p < r) {
mergeSort(array, p, mid);
mergeSort(array, mid+1, r);
// merge(array, p, mid, r);
inPlaceMerge(array, p, mid, r);
}
}
//Regular merge
private static void merge(int[] array, int p, int mid, int r) {
int lengthOfLeftArray = mid - p + 1; // This is important to add +1.
int lengthOfRightArray = r - mid;
int[] left = new int[lengthOfLeftArray];
int[] right = new int[lengthOfRightArray];
for(int i = p, j = 0; i <= mid; ){
left[j++] = array[i++];
}
for(int i = mid + 1, j = 0; i <= r; ){
right[j++] = array[i++];
}
int i = 0, j = 0;
for(; i < left.length && j < right.length; ) {
if(left[i] < right[j]){
array[p++] = left[i++];
} else {
array[p++] = right[j++];
}
}
while(j < right.length){
array[p++] = right[j++];
}
while(i < left.length){
array[p++] = left[i++];
}
}
//InPlaceMerge no extra array
private static void inPlaceMerge(int[] array, int p, int mid, int r) {
int secondArrayStart = mid+1;
int prevPlaced = mid+1;
int q = mid+1;
while(p < mid+1 && q <= r){
boolean swapped = false;
if(array[p] > array[q]) {
swap(array, p, q);
swapped = true;
}
if(q != secondArrayStart && array[p] > array[secondArrayStart]) {
swap(array, p, secondArrayStart);
swapped = true;
}
//Check swapped value is in right place of second sorted array
if(swapped && secondArrayStart+1 <= r && array[secondArrayStart+1] < array[secondArrayStart]) {
prevPlaced = placeInOrder(array, secondArrayStart, prevPlaced);
}
p++;
if(q < r) { //q+1 <= r) {
q++;
}
}
}
private static int placeInOrder(int[] array, int secondArrayStart, int prevPlaced) {
int i = secondArrayStart;
for(; i < array.length; i++) {
//Simply swap till the prevPlaced position
if(secondArrayStart < prevPlaced) {
swap(array, secondArrayStart, secondArrayStart+1);
secondArrayStart++;
continue;
}
if(array[i] < array[secondArrayStart]) {
swap(array, i, secondArrayStart);
secondArrayStart++;
} else if(i != secondArrayStart && array[i] > array[secondArrayStart]){
break;
}
}
return secondArrayStart;
}
private static void swap(int[] array, int m, int n){
int temp = array[m];
array[m] = array[n];
array[n] = temp;
}
}
