Problem Description

In this problem, you have two integer arrays, nums1 and nums2. The array nums2 is an anagram of nums1, meaning nums2 contains all the elements of nums1, possibly in a different order, and both arrays may have repeated numbers. The goal is to create a mapping from nums1 to nums2 in such a way that for each index i in nums1, mapping[i] gives you the index j in nums2 where the i-th element of nums1 is located. Since nums2 is an anagram of nums1, there is always at least one index j in nums2 for every element in nums1.

Your task is to return any one of these index mapping arrays, as there may be multiple valid mappings due to duplicates.

Intuition

The intuition behind the solution involves two key observations:

1. Since nums1 and nums2 are anagrams with possible duplicates, the elements in nums1 will surely be found in nums2.
2. There can be multiple positions for a single element of nums1 in nums2 due to duplicates.

The approach uses a dictionary data structure to keep track of the indices of elements in nums2. A Python defaultdict is used here with a set as the default data type to handle multiple indices for duplicates in an efficient way. We iterate over nums2 and fill the dictionary such that each unique number in nums2 is a key, and the corresponding value is a set of indices where this number appears in nums2.

After we have this dictionary ready, we iterate over nums1 and for each element, we pop an index from the set in the dictionary. This works well because we can remove any index from the set (since any mapping is valid), and we are guaranteed to have at least one index in the set for each element due to the anagram property. The result is an index mapping list that satisfies the problem requirement.

Solution Approach

Let's dive into the implementation details of the given solution:

1. Data Structures Used:

   • defaultdict from Collections module: A dictionary subclass that calls a factory function to supply missing values. In this case, the factory function is set which holds indices.
   • set: A built-in Python data structure used here to store indices of the elements in nums2. Sets are chosen because they allow efficient addition and removal of elements, and we don't care about the order of indices.

2. Algorithm Steps:

   • Initialize a defaultdict with set as the default factory function: mapper = defaultdict(set).
   • Enumerate over nums2 and fill the mapper: For each num encountered at index i in nums2, add index i to the set corresponding to num in the mapper. This essentially records the indices where each number from nums1 can map to in nums2.
   • Generate the Index Mapping Array: Iterate over nums1, and for each num, pop a value from the set corresponding to num in the mapper. This value is an index in nums2 where num occurs. pop is used because it removes an element from the set, ensuring that an index is not used twice. The index is appended to the resulting list.

3. Complexity Analysis:

   • Time Complexity: O(N) where N is the number of elements in nums1 (or nums2, since they have the same length). It's O(N) because we do a single pass over both nums1 and nums2.
   • Space Complexity: O(N) for the mapper, as it stores indices for elements in nums2. In the worst case, where all elements in nums2 are distinct, the mapper will store one index for each element.

The pattern used in this solution can be identified as Hash Mapping. It utilizes the constant time access feature of the dictionary (hash map) for storing and retrieving element indices.

By following these steps, the function anagramMappings(nums1, nums2) will return the index mapping list that will form a correct anagram mapping from nums1 to nums2.

Example Walkthrough

Consider the following small example to illustrate the solution approach:

Let nums1 be [12, 28, 46, 32, 50] and nums2 be [50, 12, 32, 46, 28].

Firstly, we'll initialize our mapper as a defaultdict of sets. It will look like this after we process nums2:

• 50 appears at index 0 in nums2
• 12 appears at index 1 in nums2
• 32 appears at index 2 in nums2
• 46 appears at index 3 in nums2
• 28 appears at index 4 in nums2

So, our mapper would be filled as follows:

1mapper = {
2    50: {0},
3    12: {1},
4    32: {2},
5    46: {3},
6    28: {4}
7}

Next, we iterate over nums1:

• For 12, we pop an index from mapper[12], which gives us 1. Now mapper[12] is empty.
• For 28, we pop an index from mapper[28], which gives us 4 and mapper[28] becomes empty.
• For 46, we pop an index from mapper[46], which is 3 and mapper[46] is now empty.
• For 32, we pop from mapper[32] giving us 2 and mapper[32] is emptied.
• Finally for 50, pop from mapper[50] gives us 0 and empties mapper[50].

The resulting index mapping array would be [1, 4, 3, 2, 0] which is a valid anagram mapping from nums1 to nums2 since if we position the elements of nums1 using these indices in nums2, we get the same list as nums1.

To summarize, the resulting mapping array is built by popping an index from the set of indices associated with each corresponding element in nums1 from our mapper dictionary. This process ensures that each element from nums1 is uniquely matched to an element in nums2, following the stipulation of the anagram relationship between the two arrays.

Solution Implementation

Time and Space Complexity

The given Python function anagramMappings finds an index mapping from nums1 to nums2, indicating where each number in nums1 appears in nums2. Here is the complexity analysis:

Time Complexity:

The function consists of two main parts: building the mapper dictionary and constructing the result list.

• Constructing mapper: We iterate through nums2 once to build the mapper dictionary. For each element in nums2 of size n, inserting into a set in the dictionary takes amortized O(1) time. Thus, this part has a time complexity of O(n).

• Constructing the result list: For each element in nums1, which also can be size n in the worst case, we pop an element from the corresponding set in mapper. Each pop operation takes O(1) time because it removes an arbitrary element from the set. Thus, this part also has a time complexity of O(n).

The total time complexity of the function is O(n) + O(n) = O(n), where n is the length of nums2.

Space Complexity:

• The space used by mapper: In the worst case, if all elements in nums2 are unique, the dictionary will contain n sets, each with a single index. So, the space complexity for mapper will be O(n).

• The space for the result list: A new list of the same size as nums1 is created for the output. Since nums1 and nums2 can be of size n, this list will also have a space complexity of O(n).

The total space complexity of the function is O(n) [for mapper] + O(n) [for the result list] = O(2n), which simplifies to O(n), where n is the length of nums2.

Note: The space complexity only considers additional space used by the program, not the input and output space. If we consider the inputs and outputs, the space complexity would technically be O(n + n + n), which still simplifies to O(n).

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