Letter Tile Possibilities

Start by counting each letter's frequency to handle duplicates, then perform a backtracking DFS to explore every possible sequence. Use pruning to skip invalid paths and avoid redundant sequences. This approach guarantees an accurate count while maintaining performance within the problem's constraints.

Problem Statement

You are given a string tiles representing lettered tiles you can use. Each tile has exactly one uppercase letter, and you can use any tile only once per sequence. Determine how many unique non-empty sequences can be formed by arranging these tiles in different orders.

For example, given tiles = "AAB", the possible sequences include "A", "B", "AA", "AB", "BA", "AAB", "ABA", and "BAA", resulting in 8 sequences. Your solution must efficiently handle up to 7 tiles and account for duplicate letters to avoid counting identical sequences multiple times.

Examples

Example 1

Input: tiles = "AAB"

Output: 8

The possible sequences are "A", "B", "AA", "AB", "BA", "AAB", "ABA", "BAA".

Example 2

Input: tiles = "AAABBC"

Output: 188

Example details omitted.

Example 3

Input: tiles = "V"

Output: 1

Example details omitted.

Constraints

• 1 <= tiles.length <= 7
• tiles consists of uppercase English letters.

Solution Approach

Count Letter Frequencies

First, build a hash map to count occurrences of each letter. This helps manage duplicates and allows the backtracking function to know which letters are available at each step.

Backtracking DFS with Pruning

Use a recursive DFS to build sequences by choosing each available letter and decreasing its count. Skip letters with zero count to prune invalid paths and prevent generating duplicate sequences.

Aggregate Sequence Counts

At each step of DFS, increment the total count for the current sequence. Continue recursively until all sequences are explored. Finally, return the total count representing all unique non-empty sequences.

Complexity Analysis

Metric	Value
Time	O(2^n \cdot n)
Space	O(2^n \cdot n)

Time complexity is O(2^n * n) because each subset of tiles may generate sequences up to length n, and space complexity is O(2^n * n) due to storing counts during recursion.

What Interviewers Usually Probe

• Pay attention to duplicate letters and avoid counting identical sequences multiple times.
• Consider using a hash map for tracking available letters instead of generating permutations blindly.
• Focus on pruning paths early to keep recursion efficient for the maximum of 7 tiles.

Common Pitfalls or Variants

Common pitfalls

• Failing to account for duplicate letters, leading to overcounting sequences.
• Not decrementing letter counts correctly in recursion, causing incorrect sequence totals.
• Trying to generate all permutations explicitly instead of pruning with backtracking.

Follow-up variants

• Count sequences allowing repeated usage of the same tile multiple times.
• Find the lexicographically smallest sequence of a given length using the tiles.
• Return all unique sequences instead of just counting them.

FAQ

What is the main approach to solve Letter Tile Possibilities?

Use backtracking DFS combined with a hash map to track letter counts and prune invalid sequences efficiently.

How do I handle duplicate letters in tiles?

Count the frequency of each letter and decrement counts during recursion to avoid generating identical sequences multiple times.

What is the time complexity for this problem?

The time complexity is O(2^n * n), accounting for all subsets and permutations of tile sequences.

Can I generate all sequences instead of counting?

Yes, modify the DFS to append each sequence to a result list instead of just incrementing a counter.

Why is pruning important in backtracking for this problem?

Pruning prevents exploring invalid paths and avoids redundant computations, especially when tiles contain duplicates.