Construct the Longest New String

Construct the Longest New String - Problem

You are given three integers x, y, and z.

You have x strings equal to "AA", y strings equal to "BB", and z strings equal to "AB". You want to choose some (possibly all or none) of these strings and concatenate them in some order to form a new string. This new string must not contain "AAA" or "BBB" as a substring.

Return the maximum possible length of the new string.

A substring is a contiguous non-empty sequence of characters within a string.

Input & Output

Example 1 — Balanced Case

$ Input: x = 1, y = 1, z = 1

› Output: 6

💡 Note: We have 1 "AA", 1 "BB", 1 "AB". Total A's = 2+1 = 3, Total B's = 2+1 = 3. Since they're equal, we can use all characters: length = 6. One valid arrangement: "AABBAB"

Example 2 — Imbalanced Case

$ Input: x = 2, y = 1, z = 1

› Output: 7

💡 Note: We have 2 "AA", 1 "BB", 1 "AB". Total A's = 4+1 = 5, Total B's = 2+1 = 3. Since |5-3| = 2 > 1, we use formula: 2×min(5,3)+1 = 7. One valid arrangement: "AABABAA"

Example 3 — No AB Strings

$ Input: x = 0, y = 1, z = 0

› Output: 2

💡 Note: We only have 1 "BB" string. Total A's = 0, Total B's = 2. We can use the entire "BB" string since it doesn't create "BBB". Length = 2.

Constraints

1 ≤ x, y, z ≤ 50

Visualization

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Asked in

G Google 15 M Microsoft 12 a Amazon 8

The key insight is counting total A and B characters, then applying a mathematical formula based on character balance. If A's and B's are nearly balanced (difference ≤ 1), use all characters. Otherwise, use formula 2×min(A,B)+1. Best approach is Mathematical Formula with Time: O(1), Space: O(1)

Common Approaches

✓ Brute Force Simulation

⏱️ Time: O(3^(x+y+z)) Space: O(x+y+z)

Generate all possible permutations of using the available AA, BB, and AB strings, then check each arrangement to see if it contains AAA or BBB substrings. Keep track of the maximum valid length found.

Greedy Balancing

⏱️ Time: O(1) Space: O(1)

Calculate the total count of A and B characters from all available strings, then determine the maximum length by balancing them optimally. AB strings help bridge the gap between excess A's and B's.

Mathematical Formula

⏱️ Time: O(1) Space: O(1)

Analyze the problem mathematically by considering the constraints. The key insight is that we need to avoid AAA and BBB, which means we need to alternate or balance A's and B's effectively. This leads to a closed-form solution.

Brute Force Simulation — Algorithm Steps

Generate all possible combinations of using 0 to x AA strings, 0 to y BB strings, 0 to z AB strings
For each combination, try all possible arrangements
Check if the resulting string contains AAA or BBB
Track maximum length of valid strings

Visualization

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Step-by-Step Walkthrough

Generate Combinations

Try different counts of AA, BB, AB

Test Arrangements

Concatenate and check for AAA/BBB

Track Maximum

Keep the longest valid string

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>

int min(int a, int b) {
    return a < b ? a : b;
}

int abs_val(int a) {
    return a < 0 ? -a : a;
}

int solution(int x, int y, int z) {
    // Count total A's and B's
    int totalA = 2 * x + z;
    int totalB = 2 * y + z;
    
    // Special case: if one character type is completely missing
    if (totalA == 0 || totalB == 0) {
        return totalA + totalB;
    }
    
    // Special case: if we have no pure AA or BB blocks, we can be more flexible
    if (x == 0 || y == 0) {
        return totalA + totalB;
    }
    
    // General case: balance the characters
    if (abs_val(totalA - totalB) <= 1) {
        // Nearly balanced - can use all characters
        return totalA + totalB;
    } else {
        // Imbalanced - limited by the minority character
        return 2 * min(totalA, totalB) + 1;
    }
}

int main() {
    int x, y, z;
    scanf("%d", &x);
    scanf("%d", &y);
    scanf("%d", &z);
    printf("%d\n", solution(x, y, z));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(3^(x+y+z))

Exponential combinations of arrangements with validation

✓ Linear Growth

Space Complexity

O(x+y+z)

Space to store current arrangement and validation

✓ Linear Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Brute Force Simulation — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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