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What is the time complexity of the optimal backtracking solution for letter case permutation on a string of length n with k letters (non-digit characters)?

medium🪤 Complexity Trap Q13 of Q15

Subsets & Combinations - Letter Case Permutation

What is the time complexity of the optimal backtracking solution for letter case permutation on a string of length n with k letters (non-digit characters)?

AO(2^k * n) because only letters toggle case and each permutation requires joining n characters

BO(2^k * k) because only letters are toggled and digits are skipped

CO(n * 2^n) because each character doubles the permutations

DO(n^2) because of nested recursion and string concatenation

Step-by-Step Solution

Solution:

Step 1: Identify branching factor
Only letters (k of them) cause branching, each with 2 choices, so total permutations are 2^k.
Step 2: Calculate work per permutation
Each permutation requires joining n characters to form a string, so O(n) per permutation.
Final Answer:
Option A -> Option A
Quick Check:
Time is exponential in letters, linear in string length [OK]

Quick Trick: Complexity depends on letters, not total length [OK]

Common Mistakes:

MISTAKES

Assuming all characters double permutations
Ignoring join cost per permutation

Trap Explanation:

PITFALL

Confusing n and k leads to overestimating complexity as O(n*2^n).

Interviewer Note:

CONTEXT

Tests understanding of branching factor and cost per recursive call.

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