Given a string s, the task is to find the minimum number of cuts needed for palindrome partitioning of the given string. A partitioning of the string is a palindrome partitioning if every sub-string of the partition is a palindrome.
Examples:
Input: s = "geek"
Output: 2
Explanation: We need to make minimum 2 cuts, i.e., "g | ee | k".
Input: s= "aaaa"
Output: 0
Explanation: The string is already a palindrome.
Input: s = "ababbbabbababa"
Output: 3
Explanation: We need to make minimum 3 cuts, i.e., "aba | bb | babbab | aba".
[Naive Approach] Using Recursion - O(n*2^n) Time and O(n) Space
In this approach, we will try to apply all possible partitions and at the end return the correct combination of partitions.
This approach is similar to that of Matrix Chain Multiplication problem.
In this approach, we recursively evaluate the following conditions:
- Base Case: If the current string is a palindrome, then we simply return 0, no Partitioning is required.
- Else, like the Matrix Chain Multiplication problem,
- we try making cuts at all possible places,
- recursively calculate the cost for each cut
- return the minimum value.
C++
// C++ Program for Palindrome Partitioning Problem
// using Recursion
#include <iostream>
#include <vector>
#include <limits.h>
using namespace std;
// Function to Check if a substring is a palindrome
bool isPalindrome(string& s, int i, int j) {
while (i < j) {
if (s[i] != s[j])
return false;
i++;
j--;
}
return true;
}
// Recursive Function to find the minimum number of
// cuts needed for palindrome partitioning
int palPartitionRec(string& s, int i, int j) {
// Base case: If the substring is empty or
// a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return 0;
int res = INT_MAX, cuts;
// Iterate through all possible partitions and
// find the minimum cuts needed
for (int k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k)
+ palPartitionRec(s, k + 1, j);
res = min(res, cuts);
}
return res;
}
int palPartition(string &s) {
return palPartitionRec(s, 0, s.size()-1);
}
int main() {
string s = "geek";
cout << palPartition(s) << endl;
return 0;
}
// C Program for Palindrome Partitioning Problem
// using Recursion
#include <stdio.h>
#include <string.h>
#include <limits.h>
// Function to check if a substring is a palindrome
int isPalindrome(char* s, int i, int j) {
while (i < j) {
if (s[i] != s[j])
return 0;
i++;
j--;
}
return 1;
}
// Recursive function to find the minimum number
// of cuts needed for palindrome partitioning
int palPartitionRec(char* s, int i, int j) {
// Base case: If the substring is empty
// or a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return 0;
int res = INT_MAX, cuts;
// Iterate through all possible partitions
// and find the minimum cuts needed
for (int k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k) + palPartitionRec(s, k + 1, j);
if (cuts < res) res = cuts;
}
return res;
}
int palPartition(char* s) {
return palPartitionRec(s, 0, strlen(s) - 1);
}
int main() {
char s[] = "geek";
printf("%d\n", palPartition(s));
return 0;
}
// Java Program for Palindrome Partitioning Problem
// using Recursion
import java.util.*;
class GfG {
// Function to check if a substring is a palindrome
static boolean isPalindrome(String s, int i, int j) {
while (i < j) {
if (s.charAt(i) != s.charAt(j))
return false;
i++;
j--;
}
return true;
}
// Recursive function to find the minimum number of
// cuts needed for palindrome partitioning
static int palPartitionRec(String s, int i, int j) {
// Base case: If the substring is empty
// or a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return 0;
int res = Integer.MAX_VALUE, cuts;
// Iterate through all possible partitions
// and find the minimum cuts needed
for (int k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k) +
palPartitionRec(s, k + 1, j);
res = Math.min(res, cuts);
}
return res;
}
static int palPartition(String s) {
return palPartitionRec(s, 0, s.length() - 1);
}
public static void main(String[] args) {
String s = "geek";
System.out.println(palPartition(s));
}
}
# Python Program for Palindrome Partitioning Problem
# using Recursion
import sys
# Function to check if a substring is a palindrome
def isPalindrome(s, i, j):
while i < j:
if s[i] != s[j]:
return False
i += 1
j -= 1
return True
# Recursive function to find the minimum number
# of cuts needed for palindrome partitioning
def palPartitionRec(s, i, j):
# Base case: If the substring is empty
# or a palindrome, no cuts needed
if i >= j or isPalindrome(s, i, j):
return 0
res = sys.maxsize
# Iterate through all possible partitions
# and find the minimum cuts needed
for k in range(i, j):
cuts = 1 + palPartitionRec(s, i, k) \
+ palPartitionRec(s, k + 1, j)
res = min(res, cuts)
return res
def palPartition(s):
return palPartitionRec(s, 0, len(s) - 1)
if __name__ == "__main__":
s = "geek"
print(palPartition(s))
// C# Program for Palindrome Partitioning Problem
// using Recursion
using System;
class GfG {
// Function to check if a substring is a palindrome
static bool isPalindrome(string s, int i, int j) {
while (i < j) {
if (s[i] != s[j])
return false;
i++;
j--;
}
return true;
}
// Recursive function to find the minimum number of
// cuts needed for palindrome partitioning
static int palPartitionRec(string s, int i, int j) {
// Base case: If the substring is empty
// or a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return 0;
int res = int.MaxValue, cuts;
// Iterate through all possible partitions
// and find the minimum cuts needed
for (int k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k)
+ palPartitionRec(s, k + 1, j);
res = Math.Min(res, cuts);
}
return res;
}
static int palPartition(string s) {
return palPartitionRec(s, 0, s.Length - 1);
}
static void Main() {
string s = "geek";
Console.WriteLine(palPartition(s));
}
}
// JavaScript Program for Palindrome Partitioning Problem
// using Recursion
// Function to check if a substring is a palindrome
function isPalindrome(s, i, j) {
while (i < j) {
if (s[i] !== s[j])
return false;
i++;
j--;
}
return true;
}
// Recursive function to find the minimum number
// of cuts needed for palindrome partitioning
function palPartitionRec(s, i, j) {
// Base case: If the substring is empty
// or a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return 0;
let res = Number.MAX_SAFE_INTEGER, cuts;
// Iterate through all possible partitions
// and find the minimum cuts needed
for (let k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k)
+ palPartitionRec(s, k + 1, j);
res = Math.min(res, cuts);
}
return res;
}
function palPartition(s) {
return palPartitionRec(s, 0, s.length - 1);
}
// Driver Code
const s = "geek";
console.log(palPartition(s));
[Better Approach 1] Using Top-Down DP (Memoization) - O(n^3) Time and O(n^2) Space
The above recursive approach has overlapping subproblems, leading to redundant computations thereby resulting in exponential time complexity. This redundant computation can be solved by using Memoization.
To avoid redundant computations, we can memoize results of each subproblem and reuse them as needed. A 2D array can serve as a memoization table to store solutions for overlapping subproblems. The size of this memo table will be n*n, as there are n possible starting indices and n possible ending indices for any subarray.
C++
// C++ Program for Palindrome Partitioning Problem
// Using Top-Down DP
#include <iostream>
#include <vector>
#include <limits.h>
using namespace std;
// Function to Check if a substring is a palindrome
bool isPalindrome(string& s, int i, int j) {
while (i < j) {
if (s[i] != s[j])
return false;
i++;
j--;
}
return true;
}
// Recursive Function to find the minimum number of
// cuts needed for palindrome partitioning
int palPartitionRec(string& s, int i, int j,
vector<vector<int>>& memo) {
// check in memo for
if (memo[i][j] != -1)
return memo[i][j];
// Base case: If the substring is empty or
// a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return memo[i][j] = 0;
int res = INT_MAX, cuts;
// Iterate through all possible partitions and
// find the minimum cuts needed
for (int k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k, memo)
+ palPartitionRec(s, k + 1, j, memo);
res = min(res, cuts);
}
return memo[i][j] = res;
}
int palPartition(string &s) {
int n = s.size();
// declare a memo array to store the result
// and initialise it with -1
vector<vector<int>> memo(n, vector<int> (n, -1));
return palPartitionRec(s, 0, n - 1, memo);
}
int main() {
string s = "geek";
cout << palPartition(s) << endl;
return 0;
}
// Java Program for Palindrome Partitioning Problem
// Using Top-Down DP
import java.util.Arrays;
class GfG {
// Function to check if a substring is a palindrome
static boolean isPalindrome(String s, int i, int j) {
while (i < j) {
if (s.charAt(i) != s.charAt(j))
return false;
i++;
j--;
}
return true;
}
// Recursive function to find the minimum number of
// cuts needed for palindrome partitioning
static int palPartitionRec(String s, int i, int j, int[][] memo) {
// check in memo for previously computed results
if (memo[i][j] != -1)
return memo[i][j];
// Base case: If the substring is empty or
// a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return memo[i][j] = 0;
int res = Integer.MAX_VALUE, cuts;
// Iterate through all possible partitions and
// find the minimum cuts needed
for (int k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k, memo)
+ palPartitionRec(s, k + 1, j, memo);
res = Math.min(res, cuts);
}
return memo[i][j] = res;
}
static int palPartition(String s) {
int n = s.length();
int[][] memo = new int[n][n];
// Initialize memo array with -1
for (int[] row : memo)
Arrays.fill(row, -1);
return palPartitionRec(s, 0, n - 1, memo);
}
public static void main(String[] args) {
String s = "geek";
System.out.println(palPartition(s));
}
}
# Python Program for Palindrome Partitioning Problem
# Using Top-Down DP
import sys
# Function to check if a substring is a palindrome
def isPalindrome(s, i, j):
while i < j:
if s[i] != s[j]:
return False
i += 1
j -= 1
return True
# Recursive function to find the minimum number of
# cuts needed for palindrome partitioning
def palPartitionRec(s, i, j, memo):
# Check memo for previously computed results
if memo[i][j] != -1:
return memo[i][j]
# Base case: If the substring is empty or
# a palindrome, no cuts needed
if i >= j or isPalindrome(s, i, j):
memo[i][j] = 0
return 0
res = sys.maxsize
# Iterate through all possible partitions and
# find the minimum cuts needed
for k in range(i, j):
cuts = 1 + palPartitionRec(s, i, k, memo) \
+ palPartitionRec(s, k + 1, j, memo)
res = min(res, cuts)
memo[i][j] = res
return res
def palPartition(s):
n = len(s)
memo = [[-1 for _ in range(n)] for _ in range(n)]
return palPartitionRec(s, 0, n - 1, memo)
if __name__ == "__main__":
s = "geek"
print(palPartition(s))
// C# Program for Palindrome Partitioning Problem
// Usin Top-Down DP
using System;
class GfG {
// Function to check if a substring is a palindrome
static bool isPalindrome(string s, int i, int j) {
while (i < j) {
if (s[i] != s[j])
return false;
i++;
j--;
}
return true;
}
// Recursive function to find the minimum number of
// cuts needed for palindrome partitioning
static int palPartitionRec(string s, int i,
int j, int[,] memo) {
// Check memo for previously computed results
if (memo[i, j] != -1)
return memo[i, j];
// Base case: If the substring is empty
// or a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return memo[i, j] = 0;
int res = int.MaxValue, cuts;
// Iterate through all possible partitions
// and find the minimum cuts needed
for (int k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k, memo)
+ palPartitionRec(s, k + 1, j, memo);
res = Math.Min(res, cuts);
}
return memo[i, j] = res;
}
static int palPartition(string s) {
int n = s.Length;
int[,] memo = new int[n, n];
// Initialize memo array with -1
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
memo[i, j] = -1;
return palPartitionRec(s, 0, n - 1, memo);
}
static void Main() {
string s = "geek";
Console.WriteLine(palPartition(s));
}
}
// JavaScript Program for Palindrome Partitioning Problem
// Using Top-Down DP
// Function to check if a substring is a palindrome
function isPalindrome(s, i, j) {
while (i < j) {
if (s[i] !== s[j])
return false;
i++;
j--;
}
return true;
}
// Recursive function to find the minimum number of
// cuts needed for palindrome partitioning
function palPartitionRec(s, i, j, memo) {
// Check memo for previously computed results
if (memo[i][j] !== -1)
return memo[i][j];
// Base case: If the substring is empty or
// a palindrome, no cuts needed
if (i >= j || isPalindrome(s, i, j))
return memo[i][j] = 0;
let res = Infinity, cuts;
// Iterate through all possible partitions
// and find the minimum cuts needed
for (let k = i; k < j; k++) {
cuts = 1 + palPartitionRec(s, i, k, memo)
+ palPartitionRec(s, k + 1, j, memo);
res = Math.min(res, cuts);
}
return memo[i][j] = res;
}
function palPartition(s) {
const n = s.length;
const memo = Array.from({ length: n },
() => Array(n).fill(-1));
return palPartitionRec(s, 0, n - 1, memo);
}
// Driver Code
const s = "geek";
console.log(palPartition(s));
[Better Approach 2] Using Bottom-Up DP (Tabulation) - O(n^3) Time and O(n^2) Space
The approach is similar to the previous one; just instead of breaking down the problem recursively, we iteratively build up the solution by calculating in bottom-up manner.
Here, we can use two 2D array dp[][] and isPalin[][], for storing the computed result.
- dp[i][j] stores the minimum cuts for palindrome partitioning of the substring s[i ... j]
- isPalin[i][j] tells us whether substring s[i ... j] is a palindromic string or not.
We starts with smaller substrings and gradually builds up to the result for entire string.
C++
// C++ Solution for Palindrome Partitioning Problem
// using Bottom Up Dynamic Programming
#include <iostream>
#include <vector>
#include <climits>
using namespace std;
// Function to find the minimum number of cuts
// needed to partition a string such that
// every part is a palindrome
int palPartition(string& s) {
int n = s.length();
// dp[i][j] = Minimum number of cuts needed for
// palindrome partitioning of substring s[i..j]
int dp[n][n];
// isPalin[i][j] = true if substring s[i..j]
// is palindrome,else false
bool isPalin[n][n];
// Every substring of length 1 is a palindrome
for (int i = 0; i < n; i++) {
isPalin[i][i] = true;
dp[i][i] = 0;
}
for (int len = 2; len <= n; len++) {
// Build solution for all substrings s[i ... j]
// of length len
for (int i = 0, j = i + len - 1; j < n ; i++, j++) {
// If len is 2, then we just need to
// compare two characters.
if (len == 2)
isPalin[i][j] = (s[i] == s[j]);
// Else need to check two corner characters
// and value of isPalin[i+1][j-1]
else
isPalin[i][j] = (s[i] == s[j]) &&
isPalin[i + 1][j - 1];
// IF s[i..j] is palindrome, then dp[i][j] is 0
if (isPalin[i][j] == true)
dp[i][j] = 0;
else {
// Make a cut at every possible location starting
// from i to j, and get the minimum cost cut.
dp[i][j] = INT_MAX;
for (int k = i; k <= j - 1; k++)
dp[i][j] = min(dp[i][j], 1 +
dp[i][k] + dp[k + 1][j]);
}
}
}
// Return the min cut value for
// complete string. i.e., s[0..n-1]
return dp[0][n - 1];
}
int main() {
string s = "geek";
cout << palPartition(s) << endl;
return 0;
}
// Java Solution for Palindrome Partitioning Problem
// using Bottom Up Dynamic Programming
import java.util.Arrays;
class GfG {
// Function to find the minimum number of cuts
// needed to partition a string such that
// every part is a palindrome
static int palPartition(String s) {
int n = s.length();
// dp[i][j] = Minimum number of cuts needed for
// palindrome partitioning of substring s[i..j]
int[][] dp = new int[n][n];
// isPalin[i][j] = true if substring s[i..j]
// is palindrome, else false
boolean[][] isPalin = new boolean[n][n];
// Every substring of length 1 is a palindrome
for (int i = 0; i < n; i++) {
isPalin[i][i] = true;
dp[i][i] = 0;
}
for (int len = 2; len <= n; len++) {
// Build solution for all substrings s[i ... j]
// of length len
for (int i = 0, j = i + len - 1; j < n; i++, j++) {
// If len is 2, then we just need to
// compare two characters.
if (len == 2)
isPalin[i][j] = (s.charAt(i) == s.charAt(j));
// Else need to check two corner characters
// and value of isPalin[i+1][j-1]
else
isPalin[i][j] = (s.charAt(i) == s.charAt(j))
&& isPalin[i + 1][j - 1];
// IF s[i..j] is palindrome, then dp[i][j] is 0
if (isPalin[i][j])
dp[i][j] = 0;
else {
// Make a cut at every possible location starting
// from i to j, and get the minimum cost cut.
dp[i][j] = Integer.MAX_VALUE;
for (int k = i; k <= j - 1; k++)
dp[i][j] = Math.min(dp[i][j], 1 +
dp[i][k] + dp[k + 1][j]);
}
}
}
// Return the min cut value for
// complete string. i.e., s[0..n-1]
return dp[0][n - 1];
}
public static void main(String[] args) {
String s = "geek";
System.out.println(palPartition(s));
}
}
# Python Solution for Palindrome Partitioning Problem
# using Bottom Up Dynamic Programming
# Function to find the minimum number of cuts
# needed to partition a string such that
# every part is a palindrome
def palPartition(s):
n = len(s)
# dp[i][j] = Minimum number of cuts needed for
# palindrome partitioning of substring s[i..j]
dp = [[0] * n for _ in range(n)]
# isPalin[i][j] = true if substring s[i..j]
# is palindrome, else false
isPalin = [[False] * n for _ in range(n)]
# Every substring of length 1 is a palindrome
for i in range(n):
isPalin[i][i] = True
dp[i][i] = 0
for length in range(2, n + 1):
# Build solution for all substrings s[i ... j]
# of length len
for i in range(n - length + 1):
j = i + length - 1
# If len is 2, then we just need to
# compare two characters.
if length == 2:
isPalin[i][j] = (s[i] == s[j])
# Else need to check two corner characters
# and value of isPalin[i+1][j-1]
else:
isPalin[i][j] = (s[i] == s[j]) and isPalin[i + 1][j - 1]
# IF s[i..j] is palindrome, then dp[i][j] is 0
if isPalin[i][j]:
dp[i][j] = 0
else:
# Make a cut at every possible location starting
# from i to j, and get the minimum cost cut.
dp[i][j] = min(1 + dp[i][k] + dp[k + 1][j] for k in range(i, j))
# Return the min cut value for
# complete string. i.e., s[0..n-1]
return dp[0][n - 1]
if __name__ == "__main__":
s = "geek"
print(palPartition(s))
// C# Solution for Palindrome Partitioning Problem
// using Bottom Up Dynamic Programming
using System;
class GfG {
// Function to find the minimum number of cuts
// needed to partition a string such that
// every part is a palindrome
static int palPartition(string s) {
int n = s.Length;
// dp[i][j] = Minimum number of cuts needed for
// palindrome partitioning of substring s[i..j]
int[,] dp = new int[n, n];
// isPalin[i][j] = true if substring s[i..j]
// is palindrome, else false
bool[,] isPalin = new bool[n, n];
// Every substring of length 1 is a palindrome
for (int i = 0; i < n; i++) {
isPalin[i, i] = true;
dp[i, i] = 0;
}
for (int len = 2; len <= n; len++) {
// Build solution for all substrings s[i ... j]
// of length len
for (int i = 0, j = i + len - 1; j < n; i++, j++) {
// If len is 2, then we just need to
// compare two characters.
if (len == 2)
isPalin[i, j] = (s[i] == s[j]);
// Else need to check two corner characters
// and value of isPalin[i+1][j-1]
else
isPalin[i, j] = (s[i] == s[j]) &&
isPalin[i + 1, j - 1];
// IF s[i..j] is palindrome, then dp[i][j] is 0
if (isPalin[i, j])
dp[i, j] = 0;
else {
// Make a cut at every possible location starting
// from i to j, and get the minimum cost cut.
dp[i, j] = int.MaxValue;
for (int k = i; k <= j - 1; k++)
dp[i, j] = Math.Min(dp[i, j], 1 +
dp[i, k] + dp[k + 1, j]);
}
}
}
// Return the min cut value for
// complete string. i.e., s[0..n-1]
return dp[0, n - 1];
}
static void Main() {
string s = "geek";
Console.WriteLine(palPartition(s));
}
}
// JavaScript Solution for Palindrome Partitioning Problem
// using Bottom Up Dynamic Programming
// Function to find the minimum number of cuts
// needed to partition a string such that
// every part is a palindrome
function palPartition(s) {
let n = s.length;
// dp[i][j] = Minimum number of cuts needed for
// palindrome partitioning of substring s[i..j]
let dp = Array.from({ length: n },
() => Array(n).fill(0));
// isPalin[i][j] = true if substring s[i..j]
// is palindrome, else false
let isPalin = Array.from({ length: n },
() => Array(n).fill(false));
// Every substring of length 1 is a palindrome
for (let i = 0; i < n; i++) {
isPalin[i][i] = true;
dp[i][i] = 0;
}
for (let len = 2; len <= n; len++) {
// Build solution for all substrings s[i ... j]
// of length len
for (let i = 0; i <= n - len; i++) {
let j = i + len - 1;
// If len is 2, then we just need to
// compare two characters.
if (len === 2)
isPalin[i][j] = (s[i] === s[j]);
// Else need to check two corner characters
// and value of isPalin[i+1][j-1]
else
isPalin[i][j] = (s[i] === s[j]) &&
isPalin[i + 1][j - 1];
// IF s[i..j] is palindrome, then dp[i][j] is 0
if (isPalin[i][j])
dp[i][j] = 0;
else {
// Make a cut at every possible location starting
// from i to j, and get the minimum cost cut.
dp[i][j] = Infinity;
for (let k = i; k <= j - 1; k++)
dp[i][j] = Math.min(dp[i][j], 1 +
dp[i][k] + dp[k + 1][j]);
}
}
}
// Return the min cut value for
// complete string. i.e., s[0..n-1]
return dp[0][n - 1];
}
// Driver Code
let s = "geek";
console.log(palPartition(s));
[Expected Approach] Using Optimized Bottom-Up DP (Tabulation) - O(n^2) Time and O(n^2) Space
Here we precompute a palindrome table isPalin[][], where isPalin[i][j] = true if s[i..j] is a palindrome. This helps efficiently check palindrome partitions. Then we fill dp[], where dp[i] stores the minimum cuts needed to partition s[0..i] into palindromic substrings. If s[0..i] is a palindrome, no cut is needed (dp[i] = 0). Otherwise, we check all valid partitions and update dp[i] accordingly.
To fill dp[i], we find the suffix starting from j and ending at index i, (1 <= j <= i <= n - 1), which are palindromes. Hence, we can make a cut here that requires 1 + min cut from rest substring [0, j – 1]. For all such palindromic suffixes starting at j and ending at i, keep minimizing in dp[i]. Similarly, we need to compute results for all such i. (1 <= i <= n – 1) and finally, dp[n - 1] will be the minimum number of cuts needed for palindrome partitioning of the given string.
C++
// C++ Solution for Palindrome Partitioning Problem
// using Optimised Bottom Up Dynamic Programming
#include <iostream>
#include <vector>
#include <climits>
using namespace std;
// Function to fill isPalin array such that isPalin[i][j]
// stores whether substring s[i, j] is a palindrome or not
void generatePal(string& s, vector<vector<bool>>& isPalin) {
int n = s.size();
// Substring s[i .. i] of len 1
// is always palindromic
for (int i = 0; i < n; i++) {
isPalin[i][i] = true;
}
// Iterate over different lengths of substrings
for (int len = 2; len <= n; len++) {
for (int i = 0, j = i + len - 1; j < n ; i++, j++) {
// Check whether s[i] == s[j] and the
// substring between them is a palindrome
if (s[i] == s[j] && (len == 2
|| isPalin[i + 1][j - 1])) {
// Mark the substring from i to j as a
// palindrome
isPalin[i][j] = true;
}
}
}
}
// Function to calculate the minimum number of cuts required
// to make all substrings of 's' palindromic
int palPartition(string& s) {
int n = s.size();
// 2D array to store whether substring
// s[i, j] is a palindrome or not
vector<vector<bool>> isPalin(n, vector<bool>(n, false));
generatePal(s, isPalin);
// dp[i] stores minimum cuts for Palindrome
// Partitioning of substring s[0...i]
vector<int> dp(n, n);
// There is no cut required for single character
// as it is always palindrome
dp[0] = 0;
// Iterate over the given string
for (int i = 1; i < n; i++) {
// Check if string 0 to i is palindrome.
if (isPalin[0][i]) {
// if palindrome then cuts required is 0
dp[i] = 0;
}
else {
for (int j = i; j >= 1; j--) {
// if substring s[j...i] is palindromic
// then we can make a cut over here
if (isPalin[j][i]) {
// update dp[i] with minimum cuts
dp[i] = min(dp[i], 1 + dp[j-1]);
}
}
}
}
// Return the minimum cuts required
// for the entire string 's'
return dp[n - 1];
}
int main() {
string s = "geek";
cout << palPartition(s) << endl;
return 0;
}
// Java Solution for Palindrome Partitioning Problem
// using Optimised Bottom Up Dynamic Programming
import java.util.Arrays;
class GfG {
// Function to fill isPalin array such that isPalin[i][j]
// stores whether substring s[i, j] is a palindrome or not
static void generatePal(String s, boolean[][] isPalin) {
int n = s.length();
// Substring s[i .. i] of len 1
// is always palindromic
for (int i = 0; i < n; i++) {
isPalin[i][i] = true;
}
// Iterate over different lengths of substrings
for (int len = 2; len <= n; len++) {
for (int i = 0, j = i + len - 1; j < n; i++, j++) {
// Check whether s[i] == s[j] and the
// substring between them is a palindrome
if (s.charAt(i) == s.charAt(j) &&
(len == 2 || isPalin[i + 1][j - 1])) {
// Mark the substring from i to j as a
// palindrome
isPalin[i][j] = true;
}
}
}
}
// Function to calculate the minimum number of cuts required
// to make all substrings of 's' palindromic
static int palPartition(String s) {
int n = s.length();
// 2D array to store whether substring
// s[i, j] is a palindrome or not
boolean[][] isPalin = new boolean[n][n];
generatePal(s, isPalin);
// dp[i] stores minimum cuts for Palindrome
// Partitioning of substring s[0...i]
int[] dp = new int[n];
Arrays.fill(dp, n);
// There is no cut required for single character
// as it is always palindrome
dp[0] = 0;
// Iterate over the given string
for (int i = 1; i < n; i++) {
// Check if string 0 to i is palindrome.
if (isPalin[0][i]) {
// if palindrome then cuts required is 0
dp[i] = 0;
} else {
for (int j = i; j >= 1; j--) {
// if substring s[j...i] is palindromic
// then we can make a cut over here
if (isPalin[j][i]) {
// update dp[i] with minimum cuts
dp[i] = Math.min(dp[i], 1 + dp[j - 1]);
}
}
}
}
// Return the minimum cuts required
// for the entire string 's'
return dp[n - 1];
}
public static void main(String[] args) {
String s = "geek";
System.out.println(palPartition(s));
}
}
# Python Solution for Palindrome Partitioning Problem
# using Optimised Bottom Up Dynamic Programming
# Function to fill isPalin array such that isPalin[i][j]
# stores whether substring s[i, j] is a palindrome or not
def generatePal(s, isPalin):
n = len(s)
# Substring s[i .. i] of len 1
# is always palindromic
for i in range(n):
isPalin[i][i] = True
# Iterate over different lengths of substrings
for length in range(2, n + 1):
for i in range(n - length + 1):
j = i + length - 1
# Check whether s[i] == s[j] and the
# substring between them is a palindrome
if s[i] == s[j] and (length == 2 or isPalin[i + 1][j - 1]):
# Mark the substring from i to j as a
# palindrome
isPalin[i][j] = True
# Function to calculate the minimum number of cuts required
# to make all substrings of 's' palindromic
def palPartition(s):
n = len(s)
# 2D array to store whether substring
# s[i, j] is a palindrome or not
isPalin = [[False] * n for _ in range(n)]
generatePal(s, isPalin)
# dp[i] stores minimum cuts for Palindrome
# Partitioning of substring s[0...i]
dp = [n] * n
# There is no cut required for single character
# as it is always palindrome
dp[0] = 0
# Iterate over the given string
for i in range(1, n):
# Check if string 0 to i is palindrome.
if isPalin[0][i]:
# if palindrome then cuts required is 0
dp[i] = 0
else:
for j in range(i, 0, -1):
# if substring s[j...i] is palindromic
# then we can make a cut over here
if isPalin[j][i]:
# update dp[i] with minimum cuts
dp[i] = min(dp[i], 1 + dp[j - 1])
# Return the minimum cuts required
# for the entire string 's'
return dp[n - 1]
if __name__ == "__main__":
s = "geek"
print(palPartition(s))
// C# Solution for Palindrome Partitioning Problem
// using Optimised Bottom Up Dynamic Programming
using System;
class GfG {
// Function to fill isPalin array such that isPalin[i][j]
// stores whether substring s[i, j] is a palindrome or not
static void GeneratePal(string s, bool[,] isPalin) {
int n = s.Length;
// Substring s[i .. i] of len 1
// is always palindromic
for (int i = 0; i < n; i++) {
isPalin[i, i] = true;
}
// Iterate over different lengths of substrings
for (int len = 2; len <= n; len++) {
for (int i = 0, j = i + len - 1; j < n; i++, j++) {
// Check whether s[i] == s[j] and the
// substring between them is a palindrome
if (s[i] == s[j] && (len == 2 || isPalin[i + 1, j - 1])) {
// Mark the substring from i to j as a
// palindrome
isPalin[i, j] = true;
}
}
}
}
// Function to calculate the minimum number of cuts required
// to make all substrings of 's' palindromic
static int palPartition(string s) {
int n = s.Length;
// 2D array to store whether substring
// s[i, j] is a palindrome or not
bool[,] isPalin = new bool[n, n];
GeneratePal(s, isPalin);
// dp[i] stores minimum cuts for Palindrome
// Partitioning of substring s[0...i]
int[] dp = new int[n];
Array.Fill(dp, n);
// There is no cut required for single character
// as it is always palindrome
dp[0] = 0;
// Iterate over the given string
for (int i = 1; i < n; i++) {
// Check if string 0 to i is palindrome.
if (isPalin[0, i]) {
// if palindrome then cuts required is 0
dp[i] = 0;
} else {
for (int j = i; j >= 1; j--) {
// if substring s[j...i] is palindromic
// then we can make a cut over here
if (isPalin[j, i]) {
// update dp[i] with minimum cuts
dp[i] = Math.Min(dp[i], 1 + dp[j - 1]);
}
}
}
}
// Return the minimum cuts required
// for the entire string 's'
return dp[n - 1];
}
static void Main() {
string s = "geek";
Console.WriteLine(palPartition(s));
}
}
// JavaScript Solution for Palindrome Partitioning Problem
// using Optimised Bottom Up Dynamic Programming
// Function to fill isPalin array such that isPalin[i][j]
// stores whether substring s[i, j] is a palindrome or not
function generatePal(s, isPalin) {
let n = s.length;
// Substring s[i .. i] of len 1
// is always palindromic
for (let i = 0; i < n; i++) {
isPalin[i][i] = true;
}
// Iterate over different lengths of substrings
for (let len = 2; len <= n; len++) {
for (let i = 0, j = i + len - 1; j < n; i++, j++) {
// Check whether s[i] == s[j] and the
// substring between them is a palindrome
if (s[i] === s[j] && (len === 2 || isPalin[i + 1][j - 1])) {
// Mark the substring from i to j as a
// palindrome
isPalin[i][j] = true;
}
}
}
}
// Function to calculate the minimum number of cuts required
// to make all substrings of 's' palindromic
function palPartition(s) {
let n = s.length;
// 2D array to store whether substring
// s[i, j] is a palindrome or not
let isPalin = Array.from({ length: n },
() => Array(n).fill(false));
generatePal(s, isPalin);
// dp[i] stores minimum cuts for Palindrome
// Partitioning of substring s[0...i]
let dp = new Array(n).fill(n);
// There is no cut required for single character
// as it is always palindrome
dp[0] = 0;
// Iterate over the given string
for (let i = 1; i < n; i++) {
// Check if string 0 to i is palindrome.
if (isPalin[0][i]) {
// if palindrome then cuts required is 0
dp[i] = 0;
} else {
for (let j = i; j >= 1; j--) {
// if substring s[j...i] is palindromic
// then we can make a cut over here
if (isPalin[j][i]) {
// update dp[i] with minimum cuts
dp[i] = Math.min(dp[i], 1 + dp[j - 1]);
}
}
}
}
// Return the minimum cuts required
// for the entire string 's'
return dp[n - 1];
}
// Driver Code
let s = "geek";
console.log(palPartition(s));
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