给定长度N的arr [] ，任务是找到最长的反向双音子序列的长度。如果子序列先减小然后增大，则称为反向双音。
例子：
Input: arr[] = {10, 11, 2, 1, 1, 5, 2, 4}
Output: 5
Explanation: The longest subsequence which first decreases than increases is {10, 2, 1, 1, 2, 4}
Input: arr[] = {0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15}
Output: 7
Explanation: The longest subsequence which first decreases than increases is {12, 10, 6, 1, 9, 11, 15}
方法：
此问题是标准最长增加子序列(LIS)问题的变体。使用动态编程构造两个数组lis []和lds []来分别存储最长的递增和递减子序列，直到该数组的第i个索引为止。最后，返回lds [i] + lis [i] – 1的最大值，其中i的范围为0到N-1。
下面是上述方法的实现：
C++
// C++ program to find the
// longest Reverse bitonic
// Subsequence
#include
using namespace std;
// Function to return the length
// of the Longest Reverse Bitonic
// Subsequence in the array
int ReverseBitonic(int arr[], int N)
{
int i, j;
// Allocate memory for LIS[] and
// initialize LIS values as 1 for
// all indexes
int lds[N];
for (i = 0; i < N; i++) {
lds[i] = 1;
}
// Compute LIS values from left
// to right for every index
for (i = 1; i < N; i++) {
for (j = 0; j < i; j++) {
if (arr[i] < arr[j]
&& lds[i] < lds[j] + 1) {
lds[i] = lds[j] + 1;
}
}
}
// Initialize LDS for
// all indexes as 1
int lis[N];
for (i = 0; i < N; i++) {
lis[i] = 1;
}
// Compute LDS values for every
// index from right to left
for (i = N - 2; i >= 0; i--) {
for (j = N - 1; j > i; j--) {
if (arr[i] < arr[j]
&& lis[i] < lis[j] + 1) {
lis[i] = lis[j] + 1;
}
}
}
// Find the maximum value of
// lis[i] + lds[i] - 1
// in the array
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 the maximum
return max;
}
// Driver Program
int main()
{
int arr[] = { 0, 8, 4, 12, 2, 10, 6,
14, 1, 9, 5, 13, 3, 11,
7, 15 };
int N = sizeof(arr) / sizeof(arr[0]);
printf("Length of LBS is %d\n",
ReverseBitonic(arr, N));
return 0;
} Java
// Java program to find the
// longest Reverse bitonic
// Subsequence
import java.io.*;
class GFG{
// Function to return the length
// of the Longest Reverse Bitonic
// Subsequence in the array
static int ReverseBitonic(int arr[], int N)
{
int i, j;
// Allocate memory for LIS[] and
// initialize LIS values as 1 for
// all indexes
int[] lds = new int[N];
for(i = 0; i < N; i++)
{
lds[i] = 1;
}
// Compute LIS values from left
// to right for every index
for(i = 1; i < N; i++)
{
for(j = 0; j < i; j++)
{
if (arr[i] < arr[j] &&
lds[i] < lds[j] + 1)
{
lds[i] = lds[j] + 1;
}
}
}
// Initialize LDS for
// all indexes as 1
int[] lis = new int[N];
for(i = 0; i < N; i++)
{
lis[i] = 1;
}
// Compute LDS values for every
// index from right to left
for(i = N - 2; i >= 0; i--)
{
for(j = N - 1; j > i; j--)
{
if (arr[i] < arr[j] &&
lis[i] < lis[j] + 1)
{
lis[i] = lis[j] + 1;
}
}
}
// Find the maximum value of
// lis[i] + lds[i] - 1
// in the array
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 the maximum
return max;
}
// Driver code
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 " +
ReverseBitonic(arr, N));
}
}
// This code is contributed by jana_sayantanPython3
# Python3 program to find the
# longest Reverse bitonic
# Subsequence
# Function to return the length
# of the Longest Reverse Bitonic
# Subsequence in the array
def ReverseBitonic(arr):
N = len(arr)
# Allocate memory for LIS[] and
# initialize LIS values as 1
# for all indexes
lds = [1 for i in range(N + 1)]
# Compute LIS values from left to right
for i in range(1, N):
for j in range(0 , i):
if ((arr[i] < arr[j]) and
(lds[i] < lds[j] + 1)):
lds[i] = lds[j] + 1
# Allocate memory for LDS and
# initialize LDS values for
# all indexes
lis = [1 for i in range(N + 1)]
# Compute LDS values from right to left
# Loop from n-2 downto 0
for i in reversed(range(N - 1)):
# Loop from n-1 downto i-1
for j in reversed(range(i - 1, N)):
if (arr[i] < arr[j] and
lis[i] < lis[j] + 1):
lis[i] = lis[j] + 1
# Return the maximum value of
# (lis[i] + lds[i] - 1)
maximum = lis[0] + lds[0] - 1
for i in range(1, N):
maximum = max((lis[i] +
lds[i] - 1), maximum)
return maximum
# Driver code
arr = [ 0, 8, 4, 12,
2, 10, 6, 14,
1, 9, 5, 13,
3, 11, 7, 15 ]
print("Length of LBS is", ReverseBitonic(arr))
# This code is contributed by grand_masterC#
// C# program to find the
// longest Reverse bitonic
// Subsequence
using System;
class GFG{
// Function to return the length
// of the Longest Reverse Bitonic
// Subsequence in the array
static int ReverseBitonic(int[] arr, int N)
{
int i, j;
// Allocate memory for LIS[] and
// initialize LIS values as 1 for
// all indexes
int[] lds = new int[N];
for(i = 0; i < N; i++)
{
lds[i] = 1;
}
// Compute LIS values from left
// to right for every index
for(i = 1; i < N; i++)
{
for(j = 0; j < i; j++)
{
if (arr[i] < arr[j] &&
lds[i] < lds[j] + 1)
{
lds[i] = lds[j] + 1;
}
}
}
// Initialize LDS for
// all indexes as 1
int[] lis = new int[N];
for(i = 0; i < N; i++)
{
lis[i] = 1;
}
// Compute LDS values for every
// index from right to left
for(i = N - 2; i >= 0; i--)
{
for(j = N - 1; j > i; j--)
{
if (arr[i] < arr[j] &&
lis[i] < lis[j] + 1)
{
lis[i] = lis[j] + 1;
}
}
}
// Find the maximum value of
// lis[i] + lds[i] - 1
// in the array
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 the maximum
return max;
}
// Driver code
public static void Main ()
{
int[] arr = new int[] { 0, 8, 4, 12,
2, 10, 6, 14,
1, 9, 5, 13,
3, 11, 7, 15 };
int N = arr.Length;
Console.WriteLine("Length of LBS is " +
ReverseBitonic(arr, N));
}
}
// This code is contributed by sanjoy_62Javascript
输出：
Length of LBS is 7时间复杂度： O(N 2 )
辅助空间： O(N)
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