Letter Combinations of a Phone Number

To explain the problem "Letter Combinations of a Phone Number," let's start with the fundamental concept and then explore the algorithm to solve it.

Fundamental Concept

The problem is based on the idea of mapping digits to letters, similar to the layout of a traditional telephone keypad. For example, the digit '2' maps to 'a', 'b', and 'c'. Here's the typical mapping:

  • 2: a, b, c
  • 3: d, e, f
  • 4: g, h, i
  • 5: j, k, l
  • 6: m, n, o
  • 7: p, q, r, s
  • 8: t, u, v
  • 9: w, x, y, z

The challenge is to generate all possible combinations of letters that could represent a given sequence of digits.

Algorithmic Approach: Backtracking

A common approach to solve this problem is to use backtracking. Backtracking is a methodical way of trying out various sequences of decisions until you find one that "works". In the context of this problem, it involves:

  1. Choosing: Start by choosing the first digit and mapping it to all possible letters.
  2. Exploring: For each mapped letter, repeat the process for the next digit.
  3. Unchoosing: Once all digits are processed, or if there's no digit to process, backtrack to explore new possibilities with different choices of letters.

Implementation in Python

Let's implement the solution in Python with detailed comments:

def letterCombinations(digits):
    # Base case: if the input is empty, return an empty list
    if not digits:
        return []

    # Mapping of digits to corresponding letters
    phone_map = {
        "2": "abc",
        "3": "def",
        "4": "ghi",
        "5": "jkl",
        "6": "mno",
        "7": "pqrs",
        "8": "tuv",
        "9": "wxyz"

    def backtrack(index, path):
        # When the path length equals the digits length, we've found a combination
        if len(path) == len(digits):

        # Get the letters that the current digit maps to, and loop through them
        possible_letters = phone_map[digits[index]]
        for letter in possible_letters:
            # Add the letter to our current path
            # Move on to the next digit
            backtrack(index + 1, path)
            # Backtrack by removing the letter before moving onto the next

    # This list will hold the answer
    combinations = []
    # Kick off the backtracking process starting from the first digit
    backtrack(0, [])
    return combinations

# Example usage

Explanation of the Code

  1. Base Case: If digits is empty, return an empty list since there are no combinations to generate.

  2. phone_map: This dictionary maps each digit to its corresponding letters.

  3. backtrack function: This is a recursive function that generates the combinations.

    • If the current path's length is equal to the length of digits, we've found a valid combination.
    • For each digit, iterate over its possible letters, add each letter to the current 'path', and call backtrack for the next digit.
    • After exploring with a letter, remove it (backtrack) to try the next letter.
  4. combinations list: This list stores all valid combinations.

  5. Initial Call to backtrack: The algorithm starts with the first digit (index 0) and an empty path.

This code will generate all the possible letter combinations for a given input of digits, respecting the mapping of a standard phone keypad.