寻找最长双音素子序列的 Java 程序
原文:https://www . geesforgeks . org/Java-program-to-find-最长的双音素子序列/
给定一个包含 n 个正整数的数组 arr[0 … n-1]，arr[]的一个子序列如果先递增后递减，则称为 Bitonic。编写一个函数，以数组为参数，返回最长的双音素子序列的长度。 按递增顺序排序的序列被认为是双音素的，递减部分为空。类似地，递减顺序被认为是 Bitonic，递增部分为空。 示例:
Input arr[] = {1, 11, 2, 10, 4, 5, 2, 1};
Output: 6 (A Longest Bitonic Subsequence of length 6 is 1, 2, 10, 4, 2, 1)
Input arr[] = {12, 11, 40, 5, 3, 1}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 12, 11, 5, 3, 1)
Input arr[] = {80, 60, 30, 40, 20, 10}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 80, 60, 30, 20, 10)
来源:微软面试问题
解 这个问题是标准最长递增子序列(LIS)问题的变种。假设输入数组是长度为 n 的 arr[]，我们需要使用 LIS 问题的动态规划解来构造两个数组 lis[]和 lds[]。lis[i]存储以 arr[i]结尾的最长递增子序列的长度。lds[i]存储从 arr[i]开始的最长递减子序列的长度。最后，我们需要返回 lis[I]+LDS[I]–1 的最大值，其中 I 是从 0 到 n-1。 以下是上述动态规划解决方案的实现。
Java 语言(一种计算机语言，尤用于创建网站)
/* Dynamic Programming implementation in Java for longest bitonic
subsequence problem */
import java.util.*;
import java.lang.*;
import java.io.*;
class LBS
{
/* lbs() returns the length of the Longest Bitonic Subsequence in
arr[] of size n. The function mainly creates two temporary arrays
lis[] and lds[] and returns the maximum lis[i] + lds[i] - 1.
lis[i] ==> Longest Increasing subsequence ending with arr[i]
lds[i] ==> Longest decreasing subsequence starting with arr[i]
*/
static int lbs( int arr[], int n )
{
int i, j;
/* Allocate memory for LIS[] and initialize LIS values as 1 for
all indexes */
int[] lis = new int[n];
for (i = 0; i < n; i++)
lis[i] = 1;
/* Compute LIS values from left to right */
for (i = 1; i < n; i++)
for (j = 0; j < i; j++)
if (arr[i] > arr[j] && lis[i] < lis[j] + 1)
lis[i] = lis[j] + 1;
/* Allocate memory for lds and initialize LDS values for
all indexes */
int[] lds = new int [n];
for (i = 0; i < n; i++)
lds[i] = 1;
/* Compute LDS values from right to left */
for (i = n-2; i >= 0; i--)
for (j = n-1; j > i; j--)
if (arr[i] > arr[j] && lds[i] < lds[j] + 1)
lds[i] = lds[j] + 1;
/* Return the maximum value of lis[i] + lds[i] - 1*/
int max = lis[0] + lds[0] - 1;
for (i = 1; i < n; i++)
if (lis[i] + lds[i] - 1 > max)
max = lis[i] + lds[i] - 1;
return max;
}
public static void main (String[] args)
{
int arr[] = {0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5,
13, 3, 11, 7, 15};
int n = arr.length;
System.out.println("Length of LBS is "+ lbs( arr, n ));
}
}
输出:
Length of LBS is 7
时间复杂度:O(n^2) 辅助空间:O(n)
详情请参考最长双音素子序列| DP-15 的完整文章！
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