Problem Description

The problem provides an array called nums, which is a zero-based permutation. This means that nums is an array where each number is unique and ranges from 0 to nums.length - 1, with no duplicates and no numbers out of this range. The task is to create a new array ans with the same length as nums. The value at each index i in the new array, ans[i], should be determined by finding the value at index nums[i] within nums. More clearly, for every index i from 0 to nums.length - 1, ans[i] is equal to nums[nums[i]]. Overall, the objective is to re-map the array based on the current values pointing to other indices within the same array.

Intuition

To arrive at the solution, we do not need to modify the original array or use additional data structures. The problem can be solved with a straightfoward approach:

• Iterate through each element of the array nums.
• For each index i, find the element at the index specified by the value of nums[i], which is nums[nums[i]].
• Append this element to our answer array ans.

In the language of the given solution in Python, this translates into a simple list comprehension, which is a concise and efficient way of creating a list in Python from another iterable while applying some condition or operation to each element.

The line return [nums[num] for num in nums] is Python's way of saying, "for each element num in the list nums, take nums[num], and put it in a new list," which is exactly the new array ans we want to return.

So the intuition behind the solution is simply following the instructions given by the problem in a direct manner, utilizing the language's features to write less code that does more.

Solution Approach

Since the Reference Solution Approach section provided is empty, I'll elaborate based on the solution code given. The implementation of the problem's solution is straightforward:

• Algorithm: The algorithm is a simple iteration over the original array. The permutation property of the array assures us that we will not have out-of-bounds errors or duplicates.

• Data Structures: The only data structure used is the list itself in Python (List[int]). There is no need for additional data structures like stacks, queues, or hash maps because we're not dealing with operations that require such complexity.

• Pattern Used: The pattern used here is direct addressing where indices of the array directly correspond to the values because of the permutation's nature.

Here's a walk-through of the implementation in the given solution:

• We start by using the list comprehension feature in Python, which allows us to construct a new list in one line of code.

• The expression within the list comprehension [nums[num] for num in nums] is the core of our solution. for num in nums is a loop that goes through each number (as index) in the nums array.

• For each iteration, the value nums[num] is calculated. Due to the nature of the permutation array (every value in the array is a valid index into the array), num is always a valid index of nums, and thus nums[num] is always a valid operation.

• Each result of nums[num] is collected into a new list. This constructed list is immediately returned as the final answer without the necessity of a temporary holder.

The absence of nested loops or complicated branching indicates that this operation is linear in time complexity, essentially O(n), where n is the number of elements in the nums array. Space complexity is also O(n) due to the creation of the ans array which holds the same number of elements as nums.

Nothing beyond the list itself and basic iteration is used to arrive at the solution, making this a cleanly implemented answer for the given problem.

Example Walkthrough

Let's walkthrough the solution approach with a given example. Suppose we have an input array nums defined as follows:

nums = [2, 0, 1]

Following the problem description, the result array ans will be constructed where each of its elements ans[i] must be equal to nums[nums[i]]. Here's the step-by-step logic to get ans:

1. Start with an empty array ans.

2. For the first element nums[0] = 2, find nums[nums[0]], which is nums[2]. Since nums[2] = 1, we add 1 to the ans array, making it now ans = [1].

3. Move to the second element nums[1] = 0, find nums[nums[1]], which is nums[0]. Since nums[0] = 2, we add 2 to the ans array, now ans = [1, 2].

4. Lastly, for the third element nums[2] = 1, find nums[nums[2]], which is nums[1]. Since nums[1] = 0, we add 0 to the ans array, resulting in ans = [1, 2, 0].

The final ans array is [1, 2, 0], which directly corresponds to the permutation of indices defined by the original nums array. Thus, by iterating over the indices of nums and accessing the values in the order given by the elements of nums, we applied the permutation to create a new array that matches the problem's requirements.

We can convert this step-by-step process into a one-liner in Python using list comprehension:

ans = [nums[num] for num in nums]

This single line of code loops through nums, takes each element as an index (num), and fetches the value of nums at that index, appending it to the new list. It carries out the steps we manually processed earlier but does so in a concise and efficient way, which is why it is the optimal solution for this problem.

Solution Implementation

Time and Space Complexity

The given Python code takes an input list nums and returns a new list based on the values from nums where the i-th element in the new list is nums[nums[i]].

Time Complexity:

The time complexity of the code is determined by the single loop which iterates through the list nums. Since we are going through each element exactly once to construct the new list, the time complexity is O(n), where n is the number of elements in nums.

Space Complexity:

The space complexity includes the space needed for the input and the additional space required by the program. Since the input nums is given, we typically don't consider this in the calculation.

The code creates a new list as a result. No additional data structures were created that would depend on the size of the input. Therefore, the space complexity is O(n), where n is the length of the resultant list, which is the same as the input nums.

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