[Medium] 1177. Can Make Palindrome from Substring

Problem Statement

You are given a string s and array queries where queries[i] = [left, right, k]. We may rearrange the substring s[left...right] and then choose up to k of its characters to replace with any lowercase English letter.

If the substring can be made a palindrome after the operations above, the result of the query is true. Otherwise, the result is false.

Return an array answer, where answer[i] is the result of the i-th query queries[i].

Note that: Each letter is counted individually for replacement, so if, for example s[left...right] = "aaa", and k = 2, we can only replace 2 of the letters. Also, note that the initial string s is never modified.

Examples

Example 1:

Input: s = "abcda", queries = [[3,3,0],[1,2,0],[0,3,1],[0,3,2],[0,4,1]]
Output: [true,false,false,true,true]

Explanation:
- queries[0]: substring = "d", could be changed to "d" which is palindrome. true
- queries[1]: substring = "bc", could not get palindrome by rearranging without replacement. false
- queries[2]: substring = "abcd", could not get palindrome by rearranging and replacing 1 character. false
- queries[3]: substring = "abcd", could get palindrome by rearranging to "abba" and replacing 2 characters. true
- queries[4]: substring = "abcda", could get palindrome by rearranging to "aacda" and replacing 1 character. true

Example 2:

Input: s = "lyb", queries = [[0,1,0],[2,2,1]]
Output: [false,true]

Constraints

1 <= s.length, queries.length <= 10^5
0 <= left <= right < s.length
0 <= k <= s.length
s consists of lowercase English letters.

Clarification Questions

Before diving into the solution, here are 5 important clarifications and assumptions to discuss during an interview:

Palindrome construction: What does “can make palindrome” mean? (Assumption: Can rearrange characters and replace at most k characters to form a palindrome)
Substring queries: What are we querying? (Assumption: For each query [left, right, k], check if substring s[left:right+1] can become palindrome with at most k replacements)
Return format: What should we return? (Assumption: Array of booleans - true if substring can become palindrome, false otherwise)
Character replacement: What can we replace characters with? (Assumption: Any lowercase English letter - can change any character)
Rearrangement: Can we rearrange characters? (Assumption: Yes - can rearrange characters freely before replacing)

Interview Deduction Process (20 minutes)

Step 1: Brute-Force Approach (5 minutes)

For each query, extract the substring, count character frequencies, and check if it can form a palindrome. To check palindromicity after rearrangement: count how many characters have odd frequency. We need at most one odd-frequency character for a palindrome. With k replacements, we can fix up to k pairs of odd frequencies. This approach works but requires O(m × q) time where m is substring length and q is number of queries, which can be slow for many queries.

Step 2: Semi-Optimized Approach (7 minutes)

Precompute prefix frequency arrays. For each position, maintain a frequency array counting characters from start to that position. For a query [left, right], calculate frequencies as prefix[right+1] - prefix[left]. Then count odd frequencies and check if odd_count <= 2*k + 1. This reduces substring extraction overhead but still requires O(26 × q) time to process frequencies for each query.

Step 3: Optimized Solution (8 minutes)

Use prefix XOR arrays with bit manipulation. Since we only care about odd/even frequencies (not exact counts), use a bitmask where each bit represents whether a character has odd frequency. Precompute prefix XOR: prefix[i] = XOR of all characters from 0 to i-1. For query [left, right], XOR = prefix[right+1] XOR prefix[left] gives the parity mask. Count set bits (odd-frequency characters) and check if count <= 2*k + 1. This achieves O(n + q) time complexity, optimal for this problem. The key insight is that XOR naturally tracks parity changes, eliminating the need for full frequency counting.

Solution Approach

This problem requires checking if a substring can be rearranged and have at most k characters replaced to form a palindrome. The key insight is that a palindrome can have at most one character with odd frequency (the center character).

Key Insights:

Palindrome Property: A palindrome can have at most 1 character with odd frequency
Bit Manipulation: Use XOR to track character parity (odd/even frequency)
Prefix XOR: Build prefix XOR array to quickly get character frequencies for any range
Count Odd Frequencies: Count bits set in XOR result to find characters with odd frequency
Replacement Formula: Need at most (odd_count - 1) / 2 replacements

Algorithm:

Build Prefix XOR Array: Track character parity using bit manipulation
For Each Query:
- Get XOR of range to find characters with odd frequency
- Count number of odd frequencies
- Check if odd_count <= 2 * k + 1

Solution

Solution: Bit Manipulation with Prefix XOR

class Solution:
    def canMakePaliQueries(self, s, queries):
        n = len(s)

        # prefix bitmask
        count = [0] * (n + 1)

        for i in range(n):
            bit = 1 << (ord(s[i]) - ord('a'))
            count[i + 1] = count[i] ^ bit

        def bit_count(x):
            c = 0
            while x:
                x &= x - 1
                c += 1
            return c

        res = []

        for l, r, k in queries:
            x = count[r + 1] ^ count[l]
            odd = bit_count(x)

            res.append(odd // 2 <= k)

        return res

Algorithm Explanation:

Build Prefix XOR Array (Lines 5-9):
- Initialize: count[0] = 0 (no characters)
- For each character: count[i + 1] = count[i] ^ (1 << (s[i] - 'a'))
- XOR Property: Toggles the bit for character s[i]
- Result: count[i] tracks parity of all characters from index 0 to i-1
  - Bit j is set if character 'a' + j appears odd number of times
Process Each Query (Lines 11-19):
- Get range XOR: x = count[right + 1] ^ count[left]
  - This gives parity of characters in range [left, right]
  - Bit j is set if character 'a' + j appears odd times in range
- Count odd frequencies: Count number of set bits in x using Brian Kernighan’s algorithm
- Check palindrome: bits <= k * 2 + 1
  - A palindrome can have at most 1 odd frequency (center)
  - With k replacements, we can fix 2k odd frequencies (each replacement fixes 2)
  - Plus 1 for the center character if needed

Example Walkthrough:

Initialization: s = "abcda"

s = "a b c d a"
    0 1 2 3 4

Build prefix XOR:
count[0] = 0 (no characters)

i=0: s[0]='a' → count[1] = count[0] ^ (1 << 0) = 0 ^ 1 = 1
     (bit 0 set: 'a' appears odd times)

i=1: s[1]='b' → count[2] = count[1] ^ (1 << 1) = 1 ^ 2 = 3
     (bits 0,1 set: 'a' and 'b' appear odd times)

i=2: s[2]='c' → count[3] = count[2] ^ (1 << 2) = 3 ^ 4 = 7
     (bits 0,1,2 set: 'a', 'b', 'c' appear odd times)

i=3: s[3]='d' → count[4] = count[3] ^ (1 << 3) = 7 ^ 8 = 15
     (bits 0,1,2,3 set: 'a', 'b', 'c', 'd' appear odd times)

i=4: s[4]='a' → count[5] = count[4] ^ (1 << 0) = 15 ^ 1 = 14
     (bits 1,2,3 set: 'b', 'c', 'd' appear odd times, 'a' now even)

count = [0, 1, 3, 7, 15, 14]

Query 1: [3, 3, 0] - substring “d”

x = count[4] ^ count[3] = 15 ^ 7 = 8
Binary: 1000 (bit 3 set: 'd' appears odd times)

Count bits: bits = 1
Check: 1 <= 0 * 2 + 1 = 1 ✓
Result: true (single character is always palindrome)

Query 2: [1, 2, 0] - substring “bc”

x = count[3] ^ count[1] = 7 ^ 1 = 6
Binary: 0110 (bits 1,2 set: 'b' and 'c' appear odd times)

Count bits: bits = 2
Check: 2 <= 0 * 2 + 1 = 1 ✗
Result: false (need at least 1 replacement, but k=0)

Query 3: [0, 3, 1] - substring “abcd”

x = count[4] ^ count[0] = 15 ^ 0 = 15
Binary: 1111 (bits 0,1,2,3 set: all appear odd times)

Count bits: bits = 4
Check: 4 <= 1 * 2 + 1 = 3 ✗
Result: false (need at least 2 replacements, but k=1)

Query 4: [0, 3, 2] - substring “abcd”

x = count[4] ^ count[0] = 15 ^ 0 = 15
Binary: 1111 (bits 0,1,2,3 set: all appear odd times)

Count bits: bits = 4
Check: 4 <= 2 * 2 + 1 = 5 ✓
Result: true (can fix 4 odd frequencies with 2 replacements)

Query 5: [0, 4, 1] - substring “abcda”

x = count[5] ^ count[0] = 14 ^ 0 = 14
Binary: 1110 (bits 1,2,3 set: 'b', 'c', 'd' appear odd times)

Count bits: bits = 3
Check: 3 <= 1 * 2 + 1 = 3 ✓
Result: true (can fix 3 odd frequencies with 1 replacement)

Algorithm Breakdown

Why XOR Works for Parity

XOR has the property that:

x ^ x = 0 (even occurrences cancel out)
x ^ 0 = x (odd occurrence remains)

So when we XOR all characters in a range:

Characters with even frequency → bit becomes 0
Characters with odd frequency → bit becomes 1

Prefix XOR Pattern

Similar to prefix sums, prefix XOR allows O(1) range queries:

count[i] = XOR of characters from 0 to i-1
count[right + 1] ^ count[left] = XOR of characters in range [left, right]

Palindrome Check Formula

For a substring to be a palindrome:

At most 1 character can have odd frequency (the center)
With k replacements, we can fix 2k odd frequencies
Plus 1 for the center: odd_count <= 2k + 1

Why 2k?

Each replacement changes 2 characters’ frequencies
Replacing ‘a’ with ‘b’ decreases ‘a’ frequency by 1 and increases ‘b’ frequency by 1
Net effect: fixes 2 odd frequencies (if both were odd) or creates 2 even frequencies

Brian Kernighan’s Algorithm

Counting set bits efficiently:

bits = 0
while x > 0:
    x &= x - 1   # removes lowest set bit
    bits += 1

Time: O(number of set bits) instead of O(32)

Time & Space Complexity

Time Complexity:
- Initialization: O(n) - build prefix XOR array
- Each Query: O(1) amortized - count bits (at most 26 set bits)
- Total: O(n + q) where q is number of queries
Space Complexity: O(n) - store prefix XOR array

Key Points

Bit Manipulation: Use XOR to track character parity efficiently
Prefix XOR: Similar to prefix sums, enables O(1) range queries
Palindrome Property: At most 1 odd frequency allowed
Replacement Formula: k replacements can fix 2k odd frequencies
Efficient: O(1) per query after O(n) preprocessing

Alternative Approaches

Approach 1: Bit Manipulation (Current Solution)

Time: O(n + q)
Space: O(n)
Best for: Multiple queries, efficient

Approach 2: Frequency Count for Each Query

Time: O(n × q) - count frequencies for each query
Space: O(1)
Use when: Very few queries

Approach 3: 2D Prefix Array

Time: O(n × 26 + q)
Space: O(n × 26)
Overkill: Not needed, bit manipulation is more efficient

Edge Cases

Single character: Always palindrome, bits = 1 <= 2k + 1
All even frequencies: bits = 0 <= 2k + 1 (always true)
All odd frequencies: Need (n - 1) / 2 replacements
k = 0: Only works if substring is already palindrome
Empty substring: Not possible per constraints

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Robina Li

[Medium] 1177. Can Make Palindrome from Substring

[Medium] 1177. Can Make Palindrome from Substring

Problem Statement

Examples

Constraints

Clarification Questions

Interview Deduction Process (20 minutes)

Solution Approach

Key Insights:

Algorithm:

Solution

Solution: Bit Manipulation with Prefix XOR

Algorithm Explanation:

Example Walkthrough:

Algorithm Breakdown

Why XOR Works for Parity

Prefix XOR Pattern

Palindrome Check Formula

Brian Kernighan’s Algorithm

Time & Space Complexity

Key Points

Alternative Approaches

Approach 1: Bit Manipulation (Current Solution)

Approach 2: Frequency Count for Each Query

Approach 3: 2D Prefix Array

Edge Cases

Tags

Related Posts

Recent Posts

[Medium] 1177. Can Make Palindrome from Substring

Problem Statement

Examples

Constraints

Clarification Questions

Interview Deduction Process (20 minutes)

Solution Approach

Key Insights:

Algorithm:

Solution

Solution: Bit Manipulation with Prefix XOR

Algorithm Explanation:

Example Walkthrough:

Algorithm Breakdown

Why XOR Works for Parity

Prefix XOR Pattern

Palindrome Check Formula

Brian Kernighan’s Algorithm

Time & Space Complexity

Key Points

Alternative Approaches

Approach 1: Bit Manipulation (Current Solution)

Approach 2: Frequency Count for Each Query

Approach 3: 2D Prefix Array

Edge Cases

Related Problems

Tags

Related Posts

Recent Posts