排序算法：归并算法

归并算法理解起来还是比较简单的，基本原理是将两个已排序的数列归并成一个排序的数列。那么要将一个无序的数列利用归并算法排序，首先生成短的有序序列，利用归并算法，逐渐合成长的有序序列。最直接的归并方法为：对于两个有序序列p1, p2，从p2中逐个选择元素插入到p1中，这种方法简单，但是效率不高，如果采用二分法查找并且分段插入，将能提高归并效率。

分段归并算法的原理是：对于有序序列p1和p2，建立三元结构（b, e, p），b为begin，代表p2序列中要插入到p1中的片段头标识; e为end，代表p2序列要插入到p1中的片段的尾标识，p为point，代表p2要插入到p1中的位置。遍历p2序列，找到所有的三元结构，然后利用三元结构将p2序列归并到p1序列。

可以通过一个例子来说明如何使用三元结构：

有序序列：p1: 10, 18, 30, 41; p2: 3, 6, 12, 52

从p2序列第一个元素开始：p1: 10, 18, 30, 41; p2:3, 6, 12, 52 —— (0, e, p)，begin = 0

可以看到p2第一个元素的插入点为0：p1: 10, 18, 30, 41; p2: 3, 6, 12, 52 —— (0, e, 0)，point = 0, 因为p2(0) = 3小于p1(0) = 10

然后通过快速查找算法搜索10在p2中的位置：p1: 10, 18, 30, 41; p2: 3,6, 12, 52 —— (0, 2, 0)，end = 2，这样就找到了一个三元体；

类似的有：p1: 10,18, 30, 41; p2: 3, 6, 12,52 —— (2, 3, 1)

                    p1: 10, 18, 30, 41; p2: 3, 6, 12, 52 —— (3, 4, 4)

利用生成的所有三元体结构，将p2数据插入到p1中得到：3, 6, 10,12, 18, 30, 41, 52

C++代码实现：

#include <iostream>#include <vector>#define LOG2(x) ((log(x)/log(2)))using namespace std;template<typename T>void MergeSort(vector<T> &vec);int main(){int att[] = { 10, 4, 23, 46, 20, 5, 3, 88, 8, 44, 53, 25, 86, 32, 16, 11};vector<int> vec(&att[0], &att[sizeof(att)/sizeof(int)]);MergeSort(vec);return 0;}template<typename T>void MergeSort(vector<T> &vec){int VSize = vec.size();if (VSize <= 1)return;double Kf = LOG2(VSize);int K = ((int)floor(Kf) == (int)ceil(Kf)) ? (int)floor(Kf) : (int)ceil(Kf);T maxV = vec[0];for (int vIdx = 0; vIdx < VSize; vIdx++){if ((0 == (vIdx % 2)) && (vIdx + 1 < VSize) && (vec[vIdx] > vec[vIdx + 1])){vec[vIdx] ^= vec[vIdx + 1];vec[vIdx + 1] ^= vec[vIdx];vec[vIdx] ^= vec[vIdx + 1];}maxV = (maxV > vec[vIdx]) ? maxV : vec[vIdx];}// insert max value at the end of vector to make up of vector length to pow(2, K)if ((int)pow(2, K) > VSize)vec.insert(vec.end(), pow(2, K) - VSize, maxV);int newVSize = vec.size();int Km = 2;typedef struct{int begin;int end;int point;} TripleStruct;for (int kIdx = 2; kIdx <= K; kIdx++){for (int kmIdx = 0; kmIdx < newVSize/(Km*2); kmIdx++){vector<T> vecTMP1(&vec[kmIdx * Km * 2], &vec[kmIdx * Km * 2 + Km]);vector<T> vecTMP2(vec.begin() + kmIdx * Km * 2 + Km, vec.begin() + kmIdx * Km * 2 + Km * 2);vector<TripleStruct> tripleList;if (vecTMP2[0] > vecTMP1[Km - 1])continue;for (int pIdx = 0; pIdx < Km; pIdx++){int left = pIdx;int right = Km - 1;if (vecTMP2[Km - 1] < vecTMP1[0])  // vecTMP2 the whole element locates at the front of vecTMP1{TripleStruct triple = { 0, Km, 0 };tripleList.push_back(triple);break;}if (vecTMP2[pIdx] > vecTMP1[Km - 1]){TripleStruct triple = { pIdx, Km, Km };tripleList.push_back(triple);break;}// get insert pointTripleStruct triple;triple.begin = pIdx;if (vecTMP2[pIdx] < vecTMP1[left])triple.point = left;else{while (left != right){int avg = (left + right) / 2;if (vecTMP2[pIdx] > vecTMP1[avg])left = avg;elseright = avg;if (left + 1 == right)left = right;}triple.point = left;}// get insert endint l_left = pIdx + 1;int l_right = Km - 1;if (pIdx == Km - 1){triple.end = Km;}else if (vecTMP1[left] < vecTMP2[l_left])triple.end = l_left;else{while (l_left != l_right){int avg = (l_left + l_right) / 2;if (vecTMP1[left] < vecTMP2[avg])l_right = avg;elsel_left = avg;if (l_left + 1 == l_right)l_left = l_right;}triple.end = l_left;}pIdx = l_left - 1;tripleList.push_back(triple);}// Process Insertingint insertLength = 0;int tripleSize = tripleList.size();for (int tIdx = 0; tIdx < tripleSize; tIdx++){vecTMP1.insert(vecTMP1.begin() + insertLength + tripleList[tIdx].point, vecTMP2.begin() + tripleList[tIdx].begin, vecTMP2.begin() + tripleList[tIdx].end);insertLength += tripleList[tIdx].end - tripleList[tIdx].begin;}memcpy((char*)&vec[kmIdx * Km * 2], (char*)&vecTMP1[0], sizeof(T)*vecTMP1.size());}Km *= 2;}if ((int)pow(2, K) > VSize){vector<T> tmpvec(&vec[0], &vec[VSize]);vec.assign(tmpvec.begin(), tmpvec.end());}for (int vIdx = 0; vIdx < VSize; vIdx++){cout << "indx " << vIdx << " value " << vec[vIdx] << endl;}return;}