Problem Description

In this problem, we are given a string word that is comprised of lowercase English letters. The task revolves around the concept that each number key on a telephone keypad can be mapped to a group of distinct lowercase English letters, similar to old feature phones where you would press a number key multiple times to type a letter.

However, the problem allows us to 'remap' the keys numbered 2 through 9 to whatever groups of letters we want. A key point is that any letter can only map to a single key, but a key can have any number of letters assigned to it, and these collections of letters assigned to each key must be distinct from each other.

The ultimate goal is to find the most efficient way to type the given word by remapping these keys in such a way that the total number of key presses is minimized.

Let's consider an example. If the word is "abc", and if we map "a", "b", and "c" all to the number key 2, then we'd press 2 one time to type "a", two times to type "b", and three times to type "c", resulting in a total of 1 + 2 + 3 = 6 presses. The problem asks us to minimize such presses across the whole word by an intelligent mapping of letters to keys.

Intuition

To solve this problem, we follow a greedy algorithm approach combined with sorting. The main intuition behind the solution is that we want to minimize the number of times we press the keys to type the most frequent letters in the word. To achieve this, we analyze the frequency of each letter in the word since the more frequently a letter occurs, the more we can save by minimizing its required key presses.

Firstly, we count the occurrences of each letter using a Counter from the collections module in Python. This gives us the frequency of every distinct letter in the word.

We then sort these frequencies in descending order because we want to assign the highest frequencies to the least number of presses. We map the most frequent letters to single press keys, the next set of most frequent letters to two press keys, and so on. Since there are 8 number keys (2 to 9) available to map, we group every 8 letters together, assigning each group consecutively higher number of key presses.

To calculate the number of presses for the i-th most frequent letter, we divide its rank i (starting from 0) by 8 because there are eight keys that the letters could be assigned to. We then take the integer division result (which represents the group number), add 1 to get the actual number of presses for the group, and then multiply it by the frequency count of the letter. Summing this value across all the letters gives us the total number of key presses required after an optimal remapping of keys.

By following this strategy, we ensure that the most frequent letters in the word are the quickest to type, thus minimizing the total number of presses.

Learn more about Greedy and Sorting patterns.

Solution Approach

The implementation of the provided solution uses a combination of a greedy algorithm, sorting, and a counting mechanism to solve the problem efficiently.

Here's a step-by-step breakdown of the implementation process:

1. Counting the Letter Frequencies:

   • The solution begins by using a Counter from Python's collections module to count the frequency of each letter within the word.
   • The Counter data structure automatically creates a hash table (dictionary-like structure) where the keys are the distinct letters, and the values are the counts of how many times each letter appears in the word.

2. Sorting the Frequencies in Descending Order:

   • The next step is to sort the counted letter frequencies in descending order.
   • This is done because we want to allocate the most frequently occurring letters to the least number of key presses, following the greedy strategy.
   • Sorting is typically achieved through comparison-based algorithms like quicksort or mergesort, which ensure an efficient sorting process—most Python implementations use a variant of Timsort, a hybrid sorting algorithm derived from merge sort and insertion sort.

3. Assigning Letters to Keys:

   • Once sorted, the letters are virtually grouped into sets of 8 because we have 8 keys (2 to 9) that can be remapped to any set of letters.
   • Starting from the most frequent letter (at the beginning of the sorted count values), each letter is assigned a number of key presses that corresponds to the key it would be mapped to.
   • The calculation for the number of presses required for the i-th letter (starting index at 0) is:
     • Divide the letter's rank i by 8 since we have 8 keys, take the integer quotient which represents the group number.
     • Add 1 to translate the group number to the number of presses (since group 0 would be a single press).
     • Multiply this result by the frequency of the letter, to find out the total presses needed for that letter.
   • This is repeated for each letter using a loop.

4. Calculating the Total Minimum Presses:

   • As the last step, each individual letter's presses are added together to yield the total minimum number of key presses required to type the entire word after the optimal remapping.

Using this approach, the Solution class's minimumPushes method ensures we arrive at the minimum number of pushes needed to type the word after remapping the keys. The algorithm is efficient because it leverages frequency counts and sorting to minimize the total number of presses in a very direct way, well-suited to the constraints of the problem.

Example Walkthrough

Let's go through an example to illustrate the solution approach. Suppose our given word is "aaaaaaaaaaabbbccc".

Step 1: Counting the Letter Frequencies: We use the Counter to count the frequency of each letter.

• a: 11 times
• b: 3 times
• c: 3 times

Step 2: Sorting the Frequencies in Descending Order: We sort the letter frequencies:

• a: 11 times
• b: 3 times
• c: 3 times

Since a is the most frequent, followed by b and c (which have equal frequency), we keep this order.

Step 3: Assigning Letters to Keys: We have 8 keys, so let's distribute the letters.

• The letter a gets the first key (Let's assume 2), which needs just 1 press per a. Since a appears 11 times, we have a total of 11 * 1 = 11 presses for a.

• The letters b and c are next in frequency, and they get the second key (Let's assume 3). b requires 2 presses, and we have 3 bs making 3 * 2 = 6 presses for b. The same calculation applies to c, resulting in another 6 presses.

Step 4: Calculating the Total Minimum Presses: We add up the total number of presses:

• a contribution: 11 presses
• b contribution: 6 presses
• c contribution: 6 presses

So the total minimum number of key presses is 11 + 6 + 6 = 23 presses.

Therefore, by remapping the keys as described, it takes a minimum of 23 presses to type "aaaaaaaaaaabbbccc" using a telephone keyboard.

Solution Implementation

Time and Space Complexity

The provided code snippet aims to calculate the minimum number of pushes required for a given word. The time and space complexity analysis is as follows:

Time Complexity

The time complexity of the code consists of two main operations:

1. Counter Construction: Constructing the counter object cnt from the string word. This operation has a time complexity of O(n) where n is the length of the input string word. This is because each character in the string must be visited once to build the frequency count.

2. Sorting and Iteration: The second operation is to sort the values obtained from the counter and then to iteratively calculate the total number of pushes. The sorting operation has a time complexity of O(|\Sigma| * log(|\Sigma|)), where |\Sigma| represents the size of the set of unique characters in the string. The iteration over the sorted counts has a complexity of O(|\Sigma|) since each element is visited once.

Therefore, the combined time complexity is O(n + |\Sigma| * log(|\Sigma|)).

Space Complexity

The space complexity of the code is governed by the additional space used by the data structures within the function:

1. Counter Object: The counter object cnt stores the frequency of each character in the string which requires O(|\Sigma|) space, where |\Sigma| is the number of unique characters in word.

Summing up the space complexities, the total space complexity is O(|\Sigma|).

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