17. Letter Combinations of a Phone Number

Problem Description

The problem presents a scenario where we're given a string that contains digits from 2-9. The task is to generate all possible letter combinations that the number could represent, similar to the way multi-press text entry worked on old phone keypads. Each number corresponds to a set of letters. For instance, '2' maps to "abc", '3' maps to "def", and so on, just as they do on telephone buttons. We should note that the digit '1' does not map to any letters. The goal here is to return all the possible letter combinations in any order.

Intuition

The intuition behind the solution is that we can solve the problem using a form of backtracking—generating all the possible combinations by traversing over the input digits and mapping them to the corresponding letters. We start with an initial list containing just an empty string, which represents the starting point of our combination. For each digit in the input string, we look up the corresponding string of characters it can represent and then generate new combinations by appending each of these letters to each of the combinations we have so far.

The solution uses a bread-first search-like approach (without using a queue). Initially, since we only have one combination (an empty string), for the first digit, we will generate as many new combinations as there are letters corresponding to that digit. For each subsequent digit, we take all the combinations generated so far and expand each by the number of letters corresponding to the current digit. This process is repeated until we have processed all digits in the input. The beauty of this approach is that it incrementally builds up the combination list with valid letter combinations corresponding to the digits processed so far, and it ensures that all combinations of letters for the digits are covered by the end.

Learn more about Backtracking patterns.

Solution Approach

The implemented solution makes use of a list comprenhension, which is a compact way to generate lists in Python. Here's a step-by-step walk-through of the approach:

1. Initialize a list d with strings representing the characters for each digit from '2' to '9'. For example, d[0] is 'abc' for the digit '2', d[1] is 'def' for the digit '3', and so on.

2. Start with a list ans containing an empty string. The empty string serves as a starting point for building the combinations.

3. Loop through each digit in the given digits string.

   • Convert the digit to an integer and subtract 2 to find the corresponding index in the list d (since the list d starts with the letters for '2').
   • Retrieve the string s representing the possible characters for that digit.
   • Generate new combinations by taking every element a in ans and concatenating it with each letter b in the string s. This is done using a list comprehension: [a + b for a in ans for b in s].
   • Replace ans with the newly generated list of combinations.

4. After the loop finishes, ans contains all of the letter combinations that the input number can represent, and the function returns ans.

It's interesting to note that this solution has a linear time complexity with respect to the number of digits in the input, but the actual complexity depends on the number of letters that each digit can map to, since the size of ans can grow exponentially with the number of digits.

Example Walkthrough

Let's take "23" as an example to illustrate the solution approach.

1. We initialize a list d representing the characters for each digit from '2' to '9'. Thus, d[0] = 'abc' for '2' and d[1] = 'def' for '3'.

2. We start with a list ans containing one element, an empty string: ans = [""].

3. Loop through each digit in "23":

   • For the first digit '2':

     • Convert '2' to an integer and subtract 2 to find the corresponding index in list d which is 0.
     • Retrieve the string s which is 'abc'.
     • Generate new combinations by appending each letter in 'abc' to the empty string in ans. So ans becomes ["a", "b", "c"].

   • For the second digit '3':

     • Convert '3' to an integer and subtract 2 to find the corresponding index in list d which is 1.
     • Retrieve the string s which is 'def'.
     • Generate new combinations by concatenating each element in ans ("a", "b", "c") with each letter in 'def'. Using list comprehension, the new ans becomes:

       1[
       2  "a" + "d", "a" + "e", "a" + "f",
       3  "b" + "d", "b" + "e", "b" + "f",
       4  "c" + "d", "c" + "e", "c" + "f"
       5]

     • So now ans is ["ad", "ae", "af", "bd", "be", "bf", "cd", "ce", "cf"].

4. After processing both digits, ans contains all possible letter combinations for "23". Hence, the final output is ["ad", "ae", "af", "bd", "be", "bf", "cd", "ce", "cf"].

This example demonstrates the backtracking nature of the solution where each step builds upon the previous steps' results, expanding them with all possible letter combinations for the next digit.

Solution Implementation

Time and Space Complexity

The given Python code generates all possible letter combinations that each number on a phone map to, based on a string of digits. Here is an analysis of its time and space complexity:

• Time Complexity: Let n be the length of the input string digits. The algorithm iterates through each digit, and for each digit, iterates through all the accumulated combinations so far to add the new letters.

  For a digit i, we can have at most 4 letters (e.g., for digits mapping to 'pqrs', 'wxyz') which increases the number of combinations by a factor of up to 4 each time. Thus, in the worst case scenario, the time complexity can be expressed as O(4^n), as each step could potentially multiply the number of combinations by 4.

• Space Complexity: The space complexity is also O(4^n) because we store all the possible combinations of letters. Although the list ans is being used to keep track of intermediate results, its size grows exponentially with the number of digits in the input. At each step, the new list of combinations is up to 4 times larger than the previous one until all digits are processed.

Learn more about how to find time and space complexity quickly using problem constraints.