Problem Description

The problem presents a scenario where you have two arrays: arr is a single list of distinct integers, and pieces is a list of lists with each sublist also containing distinct integers. The goal is to determine if it is possible to form the arr array by concatenating the subarrays in pieces in any order. However, it is important to note that you cannot reorder the elements within the subarrays in pieces. You must use the subarrays in pieces as they are.

For example, if arr = [1,2,3,4] and pieces = [[2,3], [4], [1]], you could form arr by concatenating [1], [2,3], and [4] in order. However, if pieces were [[2,3], [1,4]], you could not form arr because that would require reordering the integers within a piece, which is not allowed.

The task is to return true if you can form arr by such concatenations or false otherwise.

Intuition

The intuition for solving this problem is to use a hash map to easily find the location of the subarrays in pieces that can potentially be concatenated to match arr. The first element of each subarray in pieces can be used as a unique key in the hash map because we are given that all integers are distinct. This way, you can quickly look up the starting element of each piece to see if you can continue building arr from the current position.

To arrive at the solution, we can follow these steps:

1. Create a hash map (d) where each key is the first integer of a subarray in pieces, and the value is the subarray itself.

2. Iterate over arr, and at each step:

   • Check if the current element exists in our hash map (d). If it doesn’t, it means there's a number in arr that isn't the start of any piece, and thus we cannot form arr fully. Return false.
   • If the present element is in d, fetch the corresponding subarray and check if the subsequent elements in arr match this subarray exactly. If they don’t, return false because the ordering within a piece cannot be altered.
   • If they do match, increment the index (i) by the length of the piece, effectively 'skipping' over these elements in arr.

3. If you reach the end of arr without returning false, it means you've been able to build all of arr using the pieces in the correct order, and hence return true.

The solution demonstrates a greedy approach, building arr incrementally from the start by mapping each element to a potential subarray from pieces, ensuring the order within both arr and pieces remains unchanged.

Solution Approach

The solution approach to this problem uses a hash table (in Python, a dictionary) and array slicing.

1. Creating a hash table: A hash table (d) is created where for every subarray (p) in pieces, there's an entry with the key being the first element of the subarray and the value being the subarray itself. This allows us to quickly look up the piece that could potentially match a segment of arr starting from a given position.

   d = {p[0]: p for p in pieces}

   This is very efficient because finding an item in a hash table has an average-case time complexity of O(1).

2. Iterating over the target array (arr): We then start walking through arr from the first element, intending to find a match in d.

   i, n = 0, len(arr)
   while i < n:

3. Checking existence in the hash table: For each element in arr, we first check whether this element is a key in our hash table d.

   if arr[i] not in d:
       return False

   If it's not present, it means we cannot find a subarray in pieces to continue our concatenation, and the function should return false.

4. Comparing subarrays: If we find the element in d, we fetch the corresponding subarray (p). Then we slice arr starting from the current index i up to the length of the p and compare it with the p.

   p = d[arr[i]]
   if arr[i : i + len(p)] != p:
       return False

   If the sliced portion of arr and the subarray p do not match, it means the current segment cannot be formed with the pieces provided, and we return false.

5. Incrementing the index: If a match is found, we increment our current index (i) by the length of the p to move past the part of arr we just confirmed.

   i += len(p)

6. Returning the result: After the loop, if we haven't returned false, it means arr has been successfully formed by concatenating the subarrays in pieces, we return true.

   return True

This algorithm's overall time complexity is O(n), where n is the number of elements in arr, as we are iterating through each element once and hash map operations are O(1) on average. It's an efficient and effective way to solve the problem provided.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Say we have arr = [5, 6, 7] and pieces = [[7], [5, 6]]. According to the problem, we can only concatenate subarrays from pieces to form arr and cannot change the order within each subarray.

Step 1: Creating a hash table

Firstly, we create a hash table where each key is the first element of the subarray from pieces, and the value is the subarray itself.

d = {7: [7], 5: [5, 6]}

The hash table d is now {5: [5, 6], 7: [7]}.

Step 2: Iterating over the target array (arr)

We aim to match elements in arr to keys in d:

i, n = 0, len(arr)  # i - current index, n - length of arr
while i < n:

Step 3: Checking existence in the hash table

For each element arr[i], check if it is a key in d.

• For i = 0, arr[i] is 5. Since 5 is a key in d, we can continue to the next step.
• If arr[i] were not in d, for example, 8, we would return False.

Step 4: Comparing subarrays

Now, we need to check if the segment of arr starting at index i matches the subarray in d.

• For i = 0, d[arr[i]] is [5, 6]. We compare arr[0:2] ([5, 6]) with [5, 6], and they match.

If they didn't match or if arr had fewer elements than the subarray p, we would return False.

Step 5: Incrementing the index

After finding a match, we skip over the matched part of arr:

• Since the length of [5, 6] is 2, we increment i by 2: i += len([5, 6]).

We then continue to the next iteration of the loop with i = 2.

Step 6: Iterating continues

• Now i = 2 and arr[i] is 7. We look up 7 in the hash table and get [7].
• We compare arr[2:3] ([7]) with [7] and they match.

Step 7: Returning the result

Having reached the end of arr without returning False, we can conclude that arr can be formed by concatenating subarrays in pieces. Therefore, the final result is True.

By following these steps, we demonstrated that arr can be formed from pieces. This approach efficiently utilizes the hash table for quick lookups and array slicing for comparison, resulting in an efficient solution.

Solution Implementation

Time and Space Complexity

The given Python code defines a method canFormArray which determines whether an array can be formed by concatenating the arrays in a given list pieces. Let's analyze the time and space complexity of the code:

Time Complexity

• The construction of the dictionary d has a time complexity of O(m), where m is the total number of elements in pieces, since each of the k sublists in pieces is iterated over once.
• The while loop iterates over each element of arr, which has n elements. Inside the loop, the check if arr[i] is in d is O(1) due to the hash map lookup, and the slice comparison arr[i : i + len(p)] != p is O(l) where l is the length of the current p in pieces.
• In the worst case scenario, all n elements need to be checked, and we might have to compare against each of the k pieces with an average length of l (assuming the pieces are approximately the same size; if they vary significantly, we'd consider an average size for a more accurate analysis).

Putting it all together, the overall time complexity is O(m + n * l), where m is the total number of elements in pieces, n is the number of elements in arr, and l is the average length of the subarrays in pieces.

Space Complexity

• The main extra space usage comes from the dictionary d, which contains at most k entries where k is the number of sublists in pieces. The space complexity for storing d is O(k).
• There is also the space used by variable p which at maximum can hold a list of length l. However, since l is at most the length of arr, this does not exceed O(n) space.
• Therefore, the space complexity of the code is O(k + n).

Considering that the space required for d is dependent on the number k of sublists and each list is stored entirely, whereas the space required for p and other local variables is negligible compared to the space for d, we could simplify the space complexity to O(m) because m includes both the number of sublists and their individual lengths.

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