算法导论Problems 2-4: Inversions

算法导论上Problems 2-4是一道求逆序数个数的问题，原题目是这样的：

Let A[1 ‥ n] be an array of n distinct numbers. If i < j and A[i] > A[j], then the pair (i, j) is called an inversion of A.

1. List the five inversions of the array 〈2, 3, 8, 6, 1〉.

2. What array with elements from the set {1, 2, . . . , n} has the most inversions? How many does it have?

3. What is the relationship between the running time of insertion sort and the number of inversions in the input array? Justify your answer.

4. Give an algorithm that determines the number of inversions in any permutation on n elements in Θ(n lg n) worst-case time. (Hint: Modify merge sort.)

下面是根据题目提示，修改归并排序后的代码

   1: #include <iostream>

   2: #include <vector>

   3: using namespace std;

   4:  

   5: template<typename T>

   6: int mergeInversions(vector<T>& array,vector<T>& tempArray,int left,int mid,int right)

   7: {

   8:     int l = left,r = mid+1;

   9:     int i =l;

  10:     int count=0;

  11:  

  12:     while(l<=mid&&r<=right)

  13:     {

  14:         //if a inversion found

  15:         if(array[l]>array[r])

  16:         {

  17:             //In left section ,for all the elements right to array[l], counter will count.    

  18:             count+=mid-l+1;    

  19:         }

  20:  

  21:         if(array[l]<=array[r])

  22:             tempArray[i++] = array[l++];

  23:         else

  24:             tempArray[i++] = array[r++];

  25:     }

  26:  

  27:     //copy the remaining elements in L and R to tempArray. 

  28:     while(l<=mid)

  29:         tempArray[i++]= array[l++];

  30:     while (r<=right)

  31:         tempArray[i++] = array[r++];

  32:     //copy the sorted elements back to the original array.

  33:     for(i=left;i<=right;i++)

  34:         array[i] = tempArray[i];

  35:     return count;

  36: }

  37:  

  38: template<typename T>

  39: int countInversions(vector<T>& array,vector<T>& tempArray,int left,int right)

  40: {

  41:     int count = 0;

  42:     if(left < right)

  43:     {

  44:         int mid = (left + right)/2;

  45:         count += countInversions(array,tempArray,left,mid);

  46:         count += countInversions(array,tempArray,mid+1,right);

  47:         count += mergeInversions(array,tempArray,left,mid,right);

  48:     }

  49:     return count;

  50: }

  51:  

  52: template<typename T>

  53: int countInversions(vector<T>& array)

  54: {

  55:     vector<T> tempArray(array.size());//A auxiliary vector will be used in merge function.

  56:     return countInversions(array,tempArray,0,array.size()-1);

  57: }

  58:  

  59: int main()

  60: {

  61:     vector<int> array(5);

  62:     array[0]= 5;

  63:     array[1]= 4;

  64:     array[2]= 3;

  65:     array[3]= 2;

  66:     array[4]= 1;

  67:     cout<<"Number of inversions:";

  68:     cout<<countInversions<int>(array)<<endl;

  69: }