几天来,我一直在努力想出遵循下面伪代码的代码,以计算未排序的排列数列表的反转次数。我需要算法在 O(nlogn) 时间内运行,但我只能在 O(n^2logn) 时间内想到一个解决方案。
更具体地说,我想知道如何通过不使用嵌套的 for 循环来加快第二步。我知道还有其他有效的算法(即合并排序)可以工作,但我需要遵循伪代码的步骤。
Instance: An array A[1] . . . A[n], a permutation of n numbers 1, . . . , n
Question: Calculate vector B[j] = |{A[i] : j > i and A[i] > A[j]}| (the same as
B[j] = |{i : j > i and A[i] > A[j]}|) B[j] is the number of element
larger than A[j] to the left of A[j] in t the array A. In other words,
the sum of the elements in B is equal to the number of inversions in
the permutation A[1] . . . A[n].
(1) Initialize B[i] to 0.
(2) For each even A[j] find elements with indices smaller than j that are by one larger
than A[j]: increase B[j] by the number of such elements;
(3) Divide each A[i] by 2 (in the integer sense);
(4) Stop when all A[i] are 0.
以下是我到目前为止提出的代码:
long long numInversions = 0;
// number of elements that are zero in the array
unsigned int zeros = 0;
do {
// solution will need to replace this nested
// for loop so it is O(n) not O(n^2)
for (int i = 0; i < permNumber; i++){
// checks if element is even
if(array[i] % 2 == 0){
for (int j = i; j >= 0; j--){
if (array[j] == array[i] + 1){
numInversions++;
}
}
}
}
// resets value of zeros for each pass
zeros = 0;
for (int k = 0; k < permNumber; k++){
array[k] = array[k] / 2;
if (array[k] == 0)
zeros++;
}
} while(zeros != permNumber);
注意:该算法应返回列表中的反转数,一个标量。伪代码要求一个数组,但最后将数组的元素相加以计算反转计数。
Example: Consider a permutation (2, 3, 6, 1, 3, 5) with six inversions. The
above algorithm works as follows:
2 4 6 1 3 5 (no pairs) ÷2
1 2 3 0 1 2 1 = 0: one '1' to left, 2: one 3 to left ÷2
0 1 1 0 0 1 1 = 0: two '1's to left, 0: two '1's to left ÷2
0 0 0 0 0 0 total: 6 pairs