# 3031. Minimum Time to Revert Word to Initial State II

## Description

You are given a 0-indexed string word and an integer k.

At every second, you must perform the following operations:

• Remove the first k characters of word.
• Add any k characters to the end of word.

Note that you do not necessarily need to add the same characters that you removed. However, you must perform both operations at every second.

Return the minimum time greater than zero required for word to revert to its initial state.

Example 1:

Input: word = "abacaba", k = 3
Output: 2
Explanation: At the 1st second, we remove characters "aba" from the prefix of word, and add characters "bac" to the end of word. Thus, word becomes equal to "cababac".
At the 2nd second, we remove characters "cab" from the prefix of word, and add "aba" to the end of word. Thus, word becomes equal to "abacaba" and reverts to its initial state.
It can be shown that 2 seconds is the minimum time greater than zero required for word to revert to its initial state.


Example 2:

Input: word = "abacaba", k = 4
Output: 1
Explanation: At the 1st second, we remove characters "abac" from the prefix of word, and add characters "caba" to the end of word. Thus, word becomes equal to "abacaba" and reverts to its initial state.
It can be shown that 1 second is the minimum time greater than zero required for word to revert to its initial state.


Example 3:

Input: word = "abcbabcd", k = 2
Output: 4
Explanation: At every second, we will remove the first 2 characters of word, and add the same characters to the end of word.
After 4 seconds, word becomes equal to "abcbabcd" and reverts to its initial state.
It can be shown that 4 seconds is the minimum time greater than zero required for word to revert to its initial state.


Constraints:

• 1 <= word.length <= 105
• 1 <= k <= word.length
• word consists only of lowercase English letters.

## Solutions

### Solution 1

• class Hashing {
private final long[] p;
private final long[] h;
private final long mod;

public Hashing(String word, long base, int mod) {
int n = word.length();
p = new long[n + 1];
h = new long[n + 1];
p[0] = 1;
this.mod = mod;
for (int i = 1; i <= n; i++) {
p[i] = p[i - 1] * base % mod;
h[i] = (h[i - 1] * base + word.charAt(i - 1) - 'a') % mod;
}
}

public long query(int l, int r) {
return (h[r] - h[l - 1] * p[r - l + 1] % mod + mod) % mod;
}
}

class Solution {
public int minimumTimeToInitialState(String word, int k) {
Hashing hashing = new Hashing(word, 13331, 998244353);
int n = word.length();
for (int i = k; i < n; i += k) {
if (hashing.query(1, n - i) == hashing.query(i + 1, n)) {
return i / k;
}
}
return (n + k - 1) / k;
}
}

• class Hashing {
private:
vector<long long> p;
vector<long long> h;
long long mod;

public:
Hashing(string word, long long base, int mod) {
int n = word.size();
p.resize(n + 1);
h.resize(n + 1);
p[0] = 1;
this->mod = mod;
for (int i = 1; i <= n; i++) {
p[i] = (p[i - 1] * base) % mod;
h[i] = (h[i - 1] * base + word[i - 1] - 'a') % mod;
}
}

long long query(int l, int r) {
return (h[r] - h[l - 1] * p[r - l + 1] % mod + mod) % mod;
}
};

class Solution {
public:
int minimumTimeToInitialState(string word, int k) {
Hashing hashing(word, 13331, 998244353);
int n = word.size();
for (int i = k; i < n; i += k) {
if (hashing.query(1, n - i) == hashing.query(i + 1, n)) {
return i / k;
}
}
return (n + k - 1) / k;
}
};

• class Hashing:
__slots__ = ["mod", "h", "p"]

def __init__(self, s: str, base: int, mod: int):
self.mod = mod
self.h = [0] * (len(s) + 1)
self.p = [1] * (len(s) + 1)
for i in range(1, len(s) + 1):
self.h[i] = (self.h[i - 1] * base + ord(s[i - 1])) % mod
self.p[i] = (self.p[i - 1] * base) % mod

def query(self, l: int, r: int) -> int:
return (self.h[r] - self.h[l - 1] * self.p[r - l + 1]) % self.mod

class Solution:
def minimumTimeToInitialState(self, word: str, k: int) -> int:
hashing = Hashing(word, 13331, 998244353)
n = len(word)
for i in range(k, n, k):
if hashing.query(1, n - i) == hashing.query(i + 1, n):
return i // k
return (n + k - 1) // k


• type Hashing struct {
p   []int64
h   []int64
mod int64
}

func NewHashing(word string, base int64, mod int64) *Hashing {
n := len(word)
p := make([]int64, n+1)
h := make([]int64, n+1)
p[0] = 1
for i := 1; i <= n; i++ {
p[i] = (p[i-1] * base) % mod
h[i] = (h[i-1]*base + int64(word[i-1]-'a')) % mod
}
return &Hashing{p, h, mod}
}

func (hashing *Hashing) Query(l, r int) int64 {
return (hashing.h[r] - hashing.h[l-1]*hashing.p[r-l+1]%hashing.mod + hashing.mod) % hashing.mod
}

func minimumTimeToInitialState(word string, k int) int {
hashing := NewHashing(word, 13331, 998244353)
n := len(word)
for i := k; i < n; i += k {
if hashing.Query(1, n-i) == hashing.Query(i+1, n) {
return i / k
}
}
return (n + k - 1) / k
}