# 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):
combinations.append("".join(path))
return

# 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
path.append(letter)
# Move on to the next digit
backtrack(index + 1, path)
# Backtrack by removing the letter before moving onto the next
path.pop()

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

# Example usage
print(letterCombinations("23"))


### 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.