Problem Description

The given problem presents a creative challenge of naming a company by generating names from a list of string ideas. The procedure to create a company name is quite unique. We must take two distinct names from the ideas list, which we will refer to as ideaA and ideaB. Then we swap the first letters of each idea. After swapping, we need to check if the new names, obtained after swapping the first characters, do not already exist in the original ideas list. If they do not exist, the concatenation of ideaA and ideaB separated by a space creates a valid company name. The main task is to count how many distinct valid company names can be produced following these rules.

Intuition

The primary insight in solving this problem lies in the realization that direct computation for each pair would be inefficient due to the sheer potential number of pairs; for n ideas, there are O(n^2) pairs to consider.

Hence, the solution employs a clever technique to reduce the complexity of the problem. It uses a two-dimensional list f, where f[i][j] keeps a count of how many strings we can form that have their first letter as the jth letter of the alphabet and do not exist in the original list ideas, given that we swap the first letter with the ith letter of the alphabet.

Approach to the solution:

1. Create a set s from the list ideas for constant-time lookups to see if a string is part of the original list.
2. Initialize a 2D list f with dimensions 26x26 to zero, which represents all possible swaps between letters.
3. For each idea, compute the index of its first letter i (0 for 'a', 1 for 'b', etc.). Then for each possible first letter j (also 0-25), check if swapping the first letter of the idea with letter j results in a string not in the set s. If it isn't, increment f[i][j] by one. This tells us how many valid swaps we can make when the first letter of the original word is letter i and we swap it with letter j.
4. After filling the 2D list f, we iterate again through each idea. For each idea, we look at the potential first letter after swapping i with every other letter j. We use our pre-computed f[j][i] values to add to the total count of valid company names. This is possible because if swapping the first letter of one idea does not result in a collision with the original set s, then all the other names that can be formed by replacing their first letter with the current idea's first letter (and also not resulting in a collision) can form a valid company name pair with the current idea.
5. Return the final count of valid company names.

The solution is efficient because it transforms the problem into a counting problem that avoids considering every pair individually by precomputing possible valid swap counts.

Solution Approach

The implementation of the solution approach is as follows:

1. Creating a Set for Fast Lookups: A set s is created from the input list ideas. Sets in Python provide O(1) average time complexity for lookups, which is crucial for checking if the swapped names already exist.

s = set(ideas)

2. Initializing a 2D List for Swap Counts: A 2D list f with dimensions 26x26 is initialized to zero. This list will store counts of possible swaps for every pair of letters in the alphabet.

f = [[0] * 26 for _ in range(26)]

3. Populating the 2D List: For each idea, we determine the index of its first letter in the alphabet (i). Then, for each alphabet letter (j), we construct a new string by swapping the first letter and check if it's not in s. If it's not, we increment f[i][j] by one.

for v in ideas:
    i = ord(v[0]) - ord('a')
    t = list(v)
    for j in range(26):
        t[0] = chr(ord('a') + j)
        if ''.join(t) not in s:
            f[i][j] += 1

This step uses two key Python functions: ord() to obtain the ASCII value of a character, and chr() to convert an ASCII value back to a character.

4. Counting Valid Names: We iterate through each idea again. This time, for every possible swap that results in a name not in s, we add the pre-computed value of f[j][i] to a counter ans. This represents all the valid unique company names that can be formed by swapping the first character of idea with the jth character of the alphabet.

ans = 0
for v in ideas:
    i = ord(v[0]) - ord('a')
    t = list(v)
    for j in range(26):
        t[0] = chr(ord('a') + j)
        if ''.join(t) not in s:
            ans += f[j][i]

This works under the premise that if swapping ideaA's first letter with ideaB's first letter results in a string that does not collide with the original list, then ideaB can form a valid pair with all ideaA-like strings (strings that would be formed by making the same letter-swap on ideaA's first letter).

5. Return the Result: Lastly, we return ans, which by now holds the count of all distinct valid company names.

return ans

This approach encapsulates an efficient way to compute the number of distinct valid names by using a combination of (a) a hash set for fast string existence checks, (b) a 2D count array to avoid redundant checks, and (c) optimized iteration over the ideas list and the English alphabet to leverage those structures.

Example Walkthrough

Let's walk through a small example to illustrate the approach.

Assume we have the following list of string ideas for our company naming task:

ideas = ["alpha", "bravo", "charlie", "delta"]

1. Creating a Set for Fast Lookups: First, we create a set s from ideas:

   s = set(["alpha", "bravo", "charlie", "delta"])

2. Initializing a 2D List for Swap Counts: We initialize a 2D list f with dimensions 26x26 to zero. This will store counts of possible name swaps for every pair of letters.

   f = [[0] * 26 for _ in range(26)]

3. Populating the 2D List: Populating f involves iterating through ideas. Take "alpha" as an example: Its first letter index i is 0 ('a' - 'a'). For every other alphabet letter index j from 0 to 25, we try to swap 'a' with the jth letter: When j is 1 (which corresponds to 'b'), the swapped idea is "blpha" which isn't in set s. So, f[0][1] is incremented. This process is repeated for all ideas and for all characters in the alphabet.

4. Counting Valid Names: With f populated, we now count valid company names. Go through each idea again; for "alpha": Index i is 0 ('a' - 'a'). For every possible other first letter j (i.e., 'b' to 'z'), we perform swaps and if the new name is not in s, we add the pre-computed value f[j][i] to our answer ans.

   For "alpha", swapping 'a' with 'b' would look at f[1][0] to see how many strings not in s can become "alpha" by swapping 'b' to 'a'. Since "blpha" is not in s, we have at least one valid pair: "blpha alpha".

5. Return the Result: After iterating through all ideas and all possible first letters, ans will be the total count of valid company name pairs. For this example, let's say ans equals 6 (which is for illustration; the actual count will depend on the content of f after computing).

The takeaway from the illustration is we avoid checking all possible pairs directly by instead preparing a lookup that tells us potential valid pairs and using that to count efficiently.

Solution Implementation

Time and Space Complexity

The given Python script is designed to find the number of distinct pairs of names that can be formed by switching the first letters of two different strings from a provided list, provided that the new strings do not appear in the original list.

Time Complexity

The time complexity of the above code is O(n * k + 26 * 26) where n is the number of ideas (strings) in the input ideas list, and k is the average length of the ideas.

• The first for loop runs through all the given ideas which is O(n). Inside this loop, it runs another loop 26 times (for each letter in the alphabet), which brings this part to O(n * 26).
• In the nested loop, it constructs a new string which is O(k) operation as it depends on the length of the word.
• The construction of f using two lists (each of size 26) for counting is O(26 * 26) which is a constant time operation, hence considered O(1) in the overall complexity.

Therefore, the first portion of code dealing with populating f is O(n * k).

• The second for-loop is similar to the first part, running for each idea n and internally iterating 26 times to try out each alphabet, also creating a new string every time. So it performs O(n * k) operations as well.

• The increment operation inside the second loop is O(1). After the nested loops, it sums up the computed values, which is O(26 * 26) since we are iterating over all possible pairs of letters.

This results in the second for-loop also having a complexity of O(n * k).

Since O(n * k) dominates over O(26 * 26), the total time complexity of the script is O(n * k).

Space Complexity

The space complexity of the code is O(n + 26 * 26).

• We have a set s that is storing all the ideas, so in the worst case, it's O(n) for space.
• The list f is a 2D list with 26 * 26 integers since we have 26 possible starting letters and we are counting possibilities for each pair.

As O(n) dominates over O(26 * 26), the overall space complexity is O(n) which is based on the number of unique ideas in the input list.

In summary, the code provided is quite efficient, with linear time complexity depending on the number of input strings and their average length, and also linear space complexity depending on just the number of input strings.

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