Printing Longest Bitonic Subsequence
Last Updated :
16 Feb, 2023
The Longest Bitonic Subsequence problem is to find the longest subsequence of a given sequence such that it is first increasing and then decreasing. A sequence, sorted in increasing order is considered Bitonic with the decreasing part as empty. Similarly, decreasing order sequence is considered Bitonic with the increasing part as empty. Examples:
Input: [1, 11, 2, 10, 4, 5, 2, 1]
Output: [1, 2, 10, 4, 2, 1] OR [1, 11, 10, 5, 2, 1]
OR [1, 2, 4, 5, 2, 1]
Input: [12, 11, 40, 5, 3, 1]
Output: [12, 11, 5, 3, 1] OR [12, 40, 5, 3, 1]
Input: [80, 60, 30, 40, 20, 10]
Output: [80, 60, 30, 20, 10] OR [80, 60, 40, 20, 10]
In previous post, we have discussed about Longest Bitonic Subsequence problem. However, the post only covered code related to finding maximum sum of increasing subsequence, but not to the construction of subsequence. In this post, we will discuss how to construct Longest Bitonic Subsequence itself. Let arr[0..n-1] be the input array. We define vector LIS such that LIS[i] is itself is a vector that stores Longest Increasing Subsequence of arr[0..i] that ends with arr[i]. Therefore for an index i, LIS[i] can be recursively written as -
LIS[0] = {arr[O]}
LIS[i] = {Max(LIS[j])} + arr[i] where j < i and arr[j] < arr[i]
= arr[i], if there is no such j
We also define a vector LDS such that LDS[i] is itself is a vector that stores Longest Decreasing Subsequence of arr[i..n] that starts with arr[i]. Therefore for an index i, LDS[i] can be recursively written as -
LDS[n] = {arr[n]}
LDS[i] = arr[i] + {Max(LDS[j])} where j > i and arr[j] < arr[i]
= arr[i], if there is no such j
For example, for array [1 11 2 10 4 5 2 1],
LIS[0]: 1
LIS[1]: 1 11
LIS[2]: 1 2
LIS[3]: 1 2 10
LIS[4]: 1 2 4
LIS[5]: 1 2 4 5
LIS[6]: 1 2
LIS[7]: 1
LDS[0]: 1
LDS[1]: 11 10 5 2 1
LDS[2]: 2 1
LDS[3]: 10 5 2 1
LDS[4]: 4 2 1
LDS[5]: 5 2 1
LDS[6]: 2 1
LDS[7]: 1
Therefore, Longest Bitonic Subsequence can be
LIS[1] + LDS[1] = [1 11 10 5 2 1] OR
LIS[3] + LDS[3] = [1 2 10 5 2 1] OR
LIS[5] + LDS[5] = [1 2 4 5 2 1]
Below is the implementation of above idea –
C++
/* Dynamic Programming solution to print Longest
Bitonic Subsequence */
#include <bits/stdc++.h>
using namespace std;
// Utility function to print Longest Bitonic
// Subsequence
void print(vector<int>& arr, int size)
{
for(int i = 0; i < size; i++)
cout << arr[i] << " ";
}
// Function to construct and print Longest
// Bitonic Subsequence
void printLBS(int arr[], int n)
{
// LIS[i] stores the length of the longest
// increasing subsequence ending with arr[i]
vector<vector<int>> LIS(n);
// initialize LIS[0] to arr[0]
LIS[0].push_back(arr[0]);
// Compute LIS values from left to right
for (int i = 1; i < n; i++)
{
// for every j less than i
for (int j = 0; j < i; j++)
{
if ((arr[j] < arr[i]) &&
(LIS[j].size() > LIS[i].size()))
LIS[i] = LIS[j];
}
LIS[i].push_back(arr[i]);
}
/* LIS[i] now stores Maximum Increasing
Subsequence of arr[0..i] that ends with
arr[i] */
// LDS[i] stores the length of the longest
// decreasing subsequence starting with arr[i]
vector<vector<int>> LDS(n);
// initialize LDS[n-1] to arr[n-1]
LDS[n - 1].push_back(arr[n - 1]);
// Compute LDS values from right to left
for (int i = n - 2; i >= 0; i--)
{
// for every j greater than i
for (int j = n - 1; j > i; j--)
{
if ((arr[j] < arr[i]) &&
(LDS[j].size() > LDS[i].size()))
LDS[i] = LDS[j];
}
LDS[i].push_back(arr[i]);
}
// reverse as vector as we're inserting at end
for (int i = 0; i < n; i++)
reverse(LDS[i].begin(), LDS[i].end());
/* LDS[i] now stores Maximum Decreasing Subsequence
of arr[i..n] that starts with arr[i] */
int max = 0;
int maxIndex = -1;
for (int i = 0; i < n; i++)
{
// Find maximum value of size of LIS[i] + size
// of LDS[i] - 1
if (LIS[i].size() + LDS[i].size() - 1 > max)
{
max = LIS[i].size() + LDS[i].size() - 1;
maxIndex = i;
}
}
// print all but last element of LIS[maxIndex] vector
print(LIS[maxIndex], LIS[maxIndex].size() - 1);
// print all elements of LDS[maxIndex] vector
print(LDS[maxIndex], LDS[maxIndex].size());
}
// Driver program
int main()
{
int arr[] = { 1, 11, 2, 10, 4, 5, 2, 1 };
int n = sizeof(arr) / sizeof(arr[0]);
printLBS(arr, n);
return 0;
}
/* Dynamic Programming solution to print Longest
Bitonic Subsequence */
import java.util.*;
class GFG
{
// Utility function to print Longest Bitonic
// Subsequence
static void print(Vector<Integer> arr, int size)
{
for (int i = 0; i < size; i++)
System.out.print(arr.elementAt(i) + " ");
}
// Function to construct and print Longest
// Bitonic Subsequence
static void printLBS(int[] arr, int n)
{
// LIS[i] stores the length of the longest
// increasing subsequence ending with arr[i]
@SuppressWarnings("unchecked")
Vector<Integer>[] LIS = new Vector[n];
for (int i = 0; i < n; i++)
LIS[i] = new Vector<>();
// initialize LIS[0] to arr[0]
LIS[0].add(arr[0]);
// Compute LIS values from left to right
for (int i = 1; i < n; i++)
{
// for every j less than i
for (int j = 0; j < i; j++)
{
if ((arr[i] > arr[j]) &&
LIS[j].size() > LIS[i].size())
{
for (int k : LIS[j])
if (!LIS[i].contains(k))
LIS[i].add(k);
}
}
LIS[i].add(arr[i]);
}
/*
* LIS[i] now stores Maximum Increasing Subsequence
* of arr[0..i] that ends with arr[i]
*/
// LDS[i] stores the length of the longest
// decreasing subsequence starting with arr[i]
@SuppressWarnings("unchecked")
Vector<Integer>[] LDS = new Vector[n];
for (int i = 0; i < n; i++)
LDS[i] = new Vector<>();
// initialize LDS[n-1] to arr[n-1]
LDS[n - 1].add(arr[n - 1]);
// Compute LDS values from right to left
for (int i = n - 2; i >= 0; i--)
{
// for every j greater than i
for (int j = n - 1; j > i; j--)
{
if (arr[j] < arr[i] &&
LDS[j].size() > LDS[i].size())
for (int k : LDS[j])
if (!LDS[i].contains(k))
LDS[i].add(k);
}
LDS[i].add(arr[i]);
}
// reverse as vector as we're inserting at end
for (int i = 0; i < n; i++)
Collections.reverse(LDS[i]);
/*
* LDS[i] now stores Maximum Decreasing Subsequence
* of arr[i..n] that starts with arr[i]
*/
int max = 0;
int maxIndex = -1;
for (int i = 0; i < n; i++)
{
// Find maximum value of size of
// LIS[i] + size of LDS[i] - 1
if (LIS[i].size() + LDS[i].size() - 1 > max)
{
max = LIS[i].size() + LDS[i].size() - 1;
maxIndex = i;
}
}
// print all but last element of LIS[maxIndex] vector
print(LIS[maxIndex], LIS[maxIndex].size() - 1);
// print all elements of LDS[maxIndex] vector
print(LDS[maxIndex], LDS[maxIndex].size());
}
// Driver Code
public static void main(String[] args)
{
int[] arr = { 1, 11, 2, 10, 4, 5, 2, 1 };
int n = arr.length;
printLBS(arr, n);
}
}
// This code is contributed by
// sanjeev2552
# Dynamic Programming solution to print Longest
# Bitonic Subsequence
def _print(arr: list, size: int):
for i in range(size):
print(arr[i], end=" ")
# Function to construct and print Longest
# Bitonic Subsequence
def printLBS(arr: list, n: int):
# LIS[i] stores the length of the longest
# increasing subsequence ending with arr[i]
LIS = [0] * n
for i in range(n):
LIS[i] = []
# initialize LIS[0] to arr[0]
LIS[0].append(arr[0])
# Compute LIS values from left to right
for i in range(1, n):
# for every j less than i
for j in range(i):
if ((arr[j] < arr[i]) and (len(LIS[j]) > len(LIS[i]))):
LIS[i] = LIS[j].copy()
LIS[i].append(arr[i])
# LIS[i] now stores Maximum Increasing
# Subsequence of arr[0..i] that ends with
# arr[i]
# LDS[i] stores the length of the longest
# decreasing subsequence starting with arr[i]
LDS = [0] * n
for i in range(n):
LDS[i] = []
# initialize LDS[n-1] to arr[n-1]
LDS[n - 1].append(arr[n - 1])
# Compute LDS values from right to left
for i in range(n - 2, -1, -1):
# for every j greater than i
for j in range(n - 1, i, -1):
if ((arr[j] < arr[i]) and (len(LDS[j]) > len(LDS[i]))):
LDS[i] = LDS[j].copy()
LDS[i].append(arr[i])
# reverse as vector as we're inserting at end
for i in range(n):
LDS[i] = list(reversed(LDS[i]))
# LDS[i] now stores Maximum Decreasing Subsequence
# of arr[i..n] that starts with arr[i]
max = 0
maxIndex = -1
for i in range(n):
# Find maximum value of size of LIS[i] + size
# of LDS[i] - 1
if (len(LIS[i]) + len(LDS[i]) - 1 > max):
max = len(LIS[i]) + len(LDS[i]) - 1
maxIndex = i
# print all but last element of LIS[maxIndex] vector
_print(LIS[maxIndex], len(LIS[maxIndex]) - 1)
# print all elements of LDS[maxIndex] vector
_print(LDS[maxIndex], len(LDS[maxIndex]))
# Driver Code
if __name__ == "__main__":
arr = [1, 11, 2, 10, 4, 5, 2, 1]
n = len(arr)
printLBS(arr, n)
# This code is contributed by
# sanjeev2552
/* Dynamic Programming solution to print longest
Bitonic Subsequence */
using System;
using System.Linq;
using System.Collections.Generic;
class GFG
{
// Utility function to print longest Bitonic
// Subsequence
static void print(List<int> arr, int size)
{
for (int i = 0; i < size; i++)
Console.Write(arr[i] + " ");
}
// Function to construct and print longest
// Bitonic Subsequence
static void printLBS(int[] arr, int n)
{
// LIS[i] stores the length of the longest
// increasing subsequence ending with arr[i]
List<int>[] LIS = new List<int>[n];
for (int i = 0; i < n; i++)
LIS[i] = new List<int>();
// initialize LIS[0] to arr[0]
LIS[0].Add(arr[0]);
// Compute LIS values from left to right
for (int i = 1; i < n; i++)
{
// for every j less than i
for (int j = 0; j < i; j++)
{
if ((arr[i] > arr[j]) &&
LIS[j].Count > LIS[i].Count)
{
foreach (int k in LIS[j])
if (!LIS[i].Contains(k))
LIS[i].Add(k);
}
}
LIS[i].Add(arr[i]);
}
/*
* LIS[i] now stores Maximum Increasing Subsequence
* of arr[0..i] that ends with arr[i]
*/
// LDS[i] stores the length of the longest
// decreasing subsequence starting with arr[i]
List<int>[] LDS = new List<int>[n];
for (int i = 0; i < n; i++)
LDS[i] = new List<int>();
// initialize LDS[n-1] to arr[n-1]
LDS[n - 1].Add(arr[n - 1]);
// Compute LDS values from right to left
for (int i = n - 2; i >= 0; i--)
{
// for every j greater than i
for (int j = n - 1; j > i; j--)
{
if (arr[j] < arr[i] &&
LDS[j].Count > LDS[i].Count)
foreach (int k in LDS[j])
if (!LDS[i].Contains(k))
LDS[i].Add(k);
}
LDS[i].Add(arr[i]);
}
// reverse as vector as we're inserting at end
for (int i = 0; i < n; i++)
LDS[i].Reverse();
/*
* LDS[i] now stores Maximum Decreasing Subsequence
* of arr[i..n] that starts with arr[i]
*/
int max = 0;
int maxIndex = -1;
for (int i = 0; i < n; i++)
{
// Find maximum value of size of
// LIS[i] + size of LDS[i] - 1
if (LIS[i].Count + LDS[i].Count - 1 > max)
{
max = LIS[i].Count + LDS[i].Count - 1;
maxIndex = i;
}
}
// print all but last element of LIS[maxIndex] vector
print(LIS[maxIndex], LIS[maxIndex].Count - 1);
// print all elements of LDS[maxIndex] vector
print(LDS[maxIndex], LDS[maxIndex].Count);
}
// Driver Code
public static void Main(String[] args)
{
int[] arr = { 1, 11, 2, 10, 4, 5, 2, 1 };
int n = arr.Length;
printLBS(arr, n);
}
}
// This code is contributed by PrinciRaj1992
// Function to print the longest bitonic subsequence
function _print(arr, size) {
for (let i = 0; i<size; i++) {
process.stdout.write(arr[i]+' ');
}
}
// Function to construct and print the longest bitonic subsequence
function printLBS(arr, n) {
// LIS[i] stores the length of the longest increasing subsequence ending with arr[i]
let LIS = new Array(n);
for (let i = 0; i < n; i++) {
LIS[i] = [];
}
// initialize LIS[0] to arr[0]
LIS[0].push(arr[0]);
// Compute LIS values from left to right
for (let i = 1; i < n; i++) {
// for every j less than i
for (let j = 0; j < i; j++) {
if (arr[j] < arr[i] && LIS[j].length > LIS[i].length) {
LIS[i] = LIS[j].slice();
}
}
LIS[i].push(arr[i]);
}
// LIS[i] now stores the Maximum Increasing Subsequence of arr[0..i] that ends with arr[i]
// LDS[i] stores the length of the longest decreasing subsequence starting with arr[i]
let LDS = new Array(n);
for (let i = 0; i < n; i++) {
LDS[i] = [];
}
// initialize LDS[n-1] to arr[n-1]
LDS[n - 1].push(arr[n - 1]);
// Compute LDS values from right to left
for (let i = n - 2; i >= 0; i--) {
// for every j greater than i
for (let j = n - 1; j > i; j--) {
if (arr[j] < arr[i] && LDS[j].length > LDS[i].length) {
LDS[i] = LDS[j].slice();
}
}
LDS[i].push(arr[i]);
}
// reverse the LDS vector as we're inserting at the end
for (let i = 0; i < n; i++) {
LDS[i].reverse();
}
// LDS[i] now stores the Maximum Decreasing Subsequence of arr[i..n] that starts with arr[i]
let max = 0;
let maxIndex = -1;
for (let i = 0; i < n; i++) {
// Find maximum value of size of LIS[i] + size of LDS[i] - 1
if (LIS[i].length + LDS[i].length - 1 > max) {
max = LIS[i].length + LDS[i].length - 1;
maxIndex = i;
}
}
// print all but
// print all but last element of LIS[maxIndex] array
_print(LIS[maxIndex].slice(0, -1), LIS[maxIndex].length - 1);
// print all elements of LDS[maxIndex] array
_print(LDS[maxIndex], LDS[maxIndex].length);
}
// Driver program
const arr = [1, 11, 2, 10, 4, 5, 2, 1];
const n = arr.length;
printLBS(arr, n);
Output:
1 11 10 5 2 1
Time complexity of above Dynamic Programming solution is O(n2). Auxiliary space used by the program is O(n2).
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