Problem Description

In this problem, we are given two arrays: names and heights. Both arrays have the same length, n, and for any given index i, names[i] represents the name of a person, and heights[i] represents their respective height. The heights array is made up of distinct positive integers, which means no two people can have the same height.

Our task is to sort the people by their heights in descending order, i.e., from the tallest to the shortest person. The output should be a reordered names array that reflects this sorting by height.

Understanding the problem involves recognizing that we need to maintain the association between each person's name and their height while sorting, which is commonly known as a "sort by keys" problem where the heights serve as keys, and the names are the values associated with those keys.

Intuition

The solution to this problem requires a sorting strategy. Since each person's height is unique (there are no duplicates), we can sort the people based on their heights in descending order. However, rather than sorting heights directly, we need to sort indices that can then be used to reorder the names.

Here's the intuitive breakdown of the solution:

1. Generate a list of indices idx from 0 to n-1 where n is the length of the heights array.
2. Sort the idx list, but instead of sorting it in the usual ascending order, we use a custom sorting key. The key is a function that takes an index i and returns the negative of heights[i]. By using the negative value, we effectively sort the indices in order of descending heights since the usual sorting is in ascending order.
3. After sorting the indices in this way, we can create a new list with names in the proper order. We do this by iterating through the sorted idx list and fetching the corresponding name from the names list for each sorted index.

This approach allows us to sort the names list according to the descending order of the heights without losing the association between names and their corresponding heights.

Learn more about Sorting patterns.

Solution Approach

The implementation of the solution involves a few key steps, and it employs basic list manipulation and sorting techniques in Python. Here's a step-by-step walkthrough of the algorithm and patterns used:

1. Initialization of index list: We create a list called idx which contains indices from 0 to n-1, where n is the length of the heights array. This is accomplished with the range function in Python, which returns a sequence of numbers. We convert it to a list using the built-in list function.

   idx = list(range(len(heights)))

2. Custom Sorting: We want to sort the indices based on the values in heights array but in descending order. To do this, we use the sort method of the list. The sort method is modified by a custom key function, which is passed as a lambda function. The lambda function takes each index i and returns -heights[i], which ensures that we are sorting in descending order based on height values.

   idx.sort(key=lambda i: -heights[i])

3. Reordering names by sorted indices: With the indices sorted in the correct order, the final step is to reorder the names according to these indices. We accomplish this by iterating over the idx and using a list comprehension to create a new list. For each i in idx, we fetch names[i] and add it to the new list.

   return [names[i] for i in idx]

This approach leverages a key aspect of Python's sorting capabilities, where sort is stable and can be customized easily with a key function. By inverting the heights in the key function (using negative heights), we avoid the need to write a custom comparison function; the sort method automatically handles it for us.

This solution is efficient because it does not involve any complicated data structures or algorithms beyond basic list operations and sorting, and it maintains a time complexity of O(n log n), where n is the number of people in the list. The custom key function does not significantly affect this complexity, as it simply changes the values used for comparison during the sort process.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have the following lists of names and heights:

names = ['Alice', 'Bob', 'Charlie', 'Dana']
heights = [155, 165, 160, 170]

We can see that Dana is the tallest, followed by Bob, then Charlie, and finally Alice. We want to sort the names array so that it matches the descending order of heights. Following the steps in the solution approach:

1. Initialization of index list: We create a list of indices which will correspond to the indices of the heights list.

   idx = list(range(len(heights)))  # idx will be [0, 1, 2, 3]

2. Custom Sorting: Now we sort idx by the heights in descending order by using the negative heights as the key for sorting.

   idx.sort(key=lambda i: -heights[i])  # After sorting, idx will be [3, 1, 2, 0]

   The index 3 comes first as heights[3] (which is 170) is the greatest, and so on.

3. Reordering names by sorted indices: Using our sorted idx, we reorder names by mapping each index to the corresponding name.

   sorted_names = [names[i] for i in idx]  # This gives us ['Dana', 'Bob', 'Charlie', 'Alice']

Now, when we look at sorted_names, we have the names ordered by descending height:

print(sorted_names)  # Output: ['Dana', 'Bob', 'Charlie', 'Alice']

This matches our expectations based on the heights. The sorted_names list is the resultant list where the corresponding names are ordered from the tallest to the shortest person. This example clearly demonstrates the efficacy of the solution approach for the problem described.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is primarily determined by the sorting operation. The sort method on the idx list is using a custom key, which is based on the values of the heights list. Sorting in Python is implemented with the Timsort algorithm, which has a time complexity of O(n log n) in the average and worst case, where n is the number of elements to be sorted.

In the given code, the list idx has a length equal to that of names and heights, so there will be n items to sort, where n represents the length of either names or heights.

Hence, the overall time complexity is O(n log n).

Space Complexity

The space complexity of this function can be analyzed by looking at the additional memory used by the program.

1. The list idx is created which will hold n integers. This contributes O(n) space.
2. The sorting operation may temporarily require additional space to hold elements when sorting, contributing an additional O(log n) space due to the recursion stack of Timsort.
3. After sorting, a new list comprehension is used to generate the result list based on the sorted indices. This list will also be of length n, which means an additional O(n) space.

As the dominant term in space complexity is O(n) (the space used for storing indices and the final output list), we can conclude the overall space complexity of the given code is O(n).

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