📜  最长递增子序列的C / C++程序

📅  最后修改于: 2021-05-28 04:16:42             🧑  作者: Mango

最长增长子序列(LIS)问题是找到给定序列的最长子序列的长度,以使子序列的所有元素都按升序排序。例如,{10、22、9、33、21、50、41、60、80}的LIS长度为6,LIS为{10、22、33、50、60、80}。
最长增长子序列

更多示例:

Input  : arr[] = {3, 10, 2, 1, 20}
Output : Length of LIS = 3
The longest increasing subsequence is 3, 10, 20

Input  : arr[] = {3, 2}
Output : Length of LIS = 1
The longest increasing subsequences are {3} and {2}

Input : arr[] = {50, 3, 10, 7, 40, 80}
Output : Length of LIS = 4
The longest increasing subsequence is {3, 7, 40, 80}

最佳子结构:
令arr [0..n-1]为输入数组,令L(i)为以索引i结尾的LIS的长度,以使arr [i]为LIS的最后一个元素。
然后,L(i)可以递归写为:
L(i)= 1 + max(L(j))其中0 如果不存在这样的j,则L(i)= 1。
为了找到给定数组的LIS,我们需要返回max(L(i)),其中0 因此,我们看到LIS问题满足最佳子结构属性,因为可以使用子问题的解决方案来解决主要问题。

以下是LIS问题的简单递归实现。它遵循上面讨论的递归结构。

/* A Naive C/C++ recursive implementation of LIS problem */
#include 
#include 
  
/* To make use of recursive calls, this function must return
   two things:
   1) Length of LIS ending with element arr[n-1]. We use
      max_ending_here for this purpose
   2) Overall maximum as the LIS may end with an element
      before arr[n-1] max_ref is used this purpose.
   The value of LIS of full array of size n is stored in
   *max_ref which is our final result */
int _lis(int arr[], int n, int* max_ref)
{
    /* Base case */
    if (n == 1)
        return 1;
  
    // 'max_ending_here' is length of LIS ending with arr[n-1]
    int res, max_ending_here = 1;
  
    /* Recursively get all LIS ending with arr[0], arr[1] ...
       arr[n-2]. If   arr[i-1] is smaller than arr[n-1], and
       max ending with arr[n-1] needs to be updated, then
       update it */
    for (int i = 1; i < n; i++) {
        res = _lis(arr, i, max_ref);
        if (arr[i - 1] < arr[n - 1] && res + 1 > max_ending_here)
            max_ending_here = res + 1;
    }
  
    // Compare max_ending_here with the overall max. And
    // update the overall max if needed
    if (*max_ref < max_ending_here)
        *max_ref = max_ending_here;
  
    // Return length of LIS ending with arr[n-1]
    return max_ending_here;
}
  
// The wrapper function for _lis()
int lis(int arr[], int n)
{
    // The max variable holds the result
    int max = 1;
  
    // The function _lis() stores its result in max
    _lis(arr, n, &max);
  
    // returns max
    return max;
}
  
/* Driver program to test above function */
int main()
{
    int arr[] = { 10, 22, 9, 33, 21, 50, 41, 60 };
    int n = sizeof(arr) / sizeof(arr[0]);
    printf("Length of lis is %d\n",
           lis(arr, n));
    return 0;
}
输出:
Length of lis is 5

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