Problem Description

The problem requires us to determine whether segments of an array can be rearranged to form an arithmetic sequence. An arithmetic sequence is defined as a sequence where the difference between consecutive elements is constant. This difference is called the common difference and is calculated as s[i+1] - s[i].

Given:

• An array nums with n integers.
• Two arrays l and r each with m integers, representing m range queries. Each query defines a range [l[i], r[i]] on the nums array.

We need to:

• Rearrange each subarray defined by the range queries to see if they can form an arithmetic sequence.
• Return a boolean array, where each element corresponds to a range query, indicating whether the subarray can be rearranged to form an arithmetic progression (true) or not (false).

Intuition

To check if a list of numbers can be rearranged into an arithmetic sequence, a fundamental property can be utilized: an arithmetic sequence must have equal intervals (common differences) between the terms when placed in ascending or descending order.

Here's how we can approach it:

1. For each query, identify the subarray from the original array nums using the given range [l[i], r[i]].
2. Find the minimum (a1) and maximum (an) elements in the subarray.
3. Calculate the common difference d that should exist if the subarray can be formed into an arithmetic sequence. It is found by dividing the range (an - a1) by the number of intervals (n - 1), where n is the length of the subarray.
4. Check two conditions:
   • Whether the computed common difference leaves no remainder, which means it's an exact interval for an arithmetic sequence.
   • Whether all expected terms of the arithmetic sequence exist in the subarray. This is done by checking for the presence of each term a1 + (i - 1) * d for i ranging from 1 to n, within the set of numbers in the subarray.
5. If both conditions are true, the subarray can be rearranged into an arithmetic sequence. Otherwise, it cannot.

The solution code takes these steps and applies them to all queries, returning a list of boolean values as the answer.

Learn more about Sorting patterns.

Solution Approach

The solution implemented in the given Python function uses a helper function named check to assess each query range for the possibility to form an arithmetic sequence. The solution leverages mathematical reasoning to efficiently determine whether a range of numbers in an array could be rearranged into an arithmetic sequence.

Here's a breakdown of the approach:

1. Helper Function: A nested function called check is defined within the checkArithmeticSubarrays method. The check function is responsible for determining whether the subarray can be rearranged into an arithmetic sequence.

2. Subarray Extraction and Set Conversion: For each query, the relevant subarray is carved out using Python's slicing feature (nums[l : l + n]) and immediately converted into a set s.

3. Finding Minimum and Maximum: Python's min and max functions are used to find the smallest and largest elements (a1 and an, respectively) in the subarray. These are critical for calculating the expected arithmetic sequence's common difference.

4. Calculating Common Difference: The common difference d is calculated by dividing the difference between the maximum and minimum elements (an - a1) by one less than the length of the subarray (n - 1). The divmod function is used to simultaneously perform the division and check for a remainder (mod).

5. Arithmetic Sequence Verification: Two checks are performed to verify that an arithmetic sequence can be formed:

   • Zero Remainder Check: The remainder mod from the division must be zero to ensure that the common difference is an integer which is essential for a valid arithmetic sequence.
   • Sequence Formation Check: A comprehension loop runs for each position i from 1 to n, checking whether each term of the theoretical arithmetic series (a1 + (i - 1) * d) exists in the set s.

If both checks pass, the subarray represented by the current query can indeed be rearranged into an arithmetic sequence. This dual check approach makes efficient use of Python's set operations, which offer average constant time complexity for membership testing.

6. Result Compilation: Finally, a list comprehension is utilized to apply the check function to each range [left, right] derived by zipping the lists l and r together. The function returns a list of boolean results that correspond to each range query.

The algorithm has linear complexity with respect to the number of elements in each query [O(n)], but since it needs to be executed for m queries, the overall complexity is O(m * n). The usage of sets and minimal iteration ensures that the solution is optimized within these boundaries.

Here is an illustrative example:

• nums = [4, 6, 5, 9, 3, 7]
• l = [0, 0, 2]
• r = [2, 3, 5]

For the first query range [0, 2], we look at subarray [4, 6, 5]. The minimum is 4, the maximum is 6, and the common difference d should be 1 with no remainder. All numbers between 4 and 6 are present and we can confirm it can form an arithmetic sequence. The check for this and other queries continues similarly using the described approach.

Example Walkthrough

Let's consider small input arrays to illustrate the solution approach described above:

• nums = [3, 1, 4, 1, 5]
• l = [1, 2, 0]
• r = [3, 4, 2]

Following the approach:

1. Query 1: The first range [1, 3] corresponds to the subarray [1, 4, 1]. We sort it to [1, 1, 4] (although for the check we use a set, but to understand let's consider a sorted list). The minimum is 1 and the maximum is 4. The common difference d should be (4 - 1) / (3 - 1) = 3 / 2. Since 3 / 2 is not an integer, we instantly know this subarray cannot form an arithmetic sequence without further checks. The answer is false.

2. Query 2: For the range [2, 4], we extract the subarray [4, 1, 5]. Sorting, we get [1, 4, 5]. The minimum is 1, the maximum is 5, and the common difference d should be (5 - 1) / (3 - 1) = 4 / 2 = 2. We check if each interval is present by adding the common difference to the minimum: 1 + 2 = 3 and 3 + 2 = 5. Since we do not find 3, we can say the original subarray cannot form an arithmetic sequence. The answer is false.

3. Query 3: The range [0, 2] gives us [3, 1, 4]. We sort to get [1, 3, 4]. Here, the minimum is 1, the maximum is 4, and d should be (4 - 1) / (3 - 1) = 3 / 2, which again is not an integer. So, the subarray cannot form an arithmetic sequence. The answer is false.

The result of running our algorithm on these queries would yield [false, false, false] as none of the subarrays can be rearranged into an arithmetic sequence. Note that in an actual implementation, we may use sets directly and bypass the explicit sorting step, but conceptually, it can be instructive to think of the sequences in sorted order.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the check function involves iterating over a subarray of the input nums list and performing operations such as finding the minimum and maximum within this subarray and checking the arithmetic property on each element of the set.

For a single query (on one l to r pair), the steps include:

1. Slicing the subarray, which takes O(k), where k is the length of the subarray (from l to r).
2. Creating a set from the subarray, which takes O(k) in both time for converting a list to a set and potentially less but not better than O(k) for checking if all elements follow the arithmetic sequence.
3. Computing the minimum and maximum values of the subarray, which take O(k) each.
4. The final arithmetic sequence check, which iterates over a range of size k and could take up to O(k) in the worst case.

Since we perform the above steps for each subarray defined by pairs of l and r, if m is the number of queries (the length of lists l and r), the overall time complexity of the checkArithmeticSubarrays method is O(m * k) where each k can vary for different queries but is at most n (the total number of elements in nums). In the worst case where each query spans the whole array, the time complexity becomes O(m * n).

Space Complexity

The space complexity of the provided code includes the space required for the output list and the temporary set used in the check function.

1. The output list that accumulates the boolean results for each query in zip(l, r) has a space complexity of O(m), where m is the number of such queries, corresponding to the number of elements in l and r.

2. The space required for the set s inside the check function which also has a complexity of O(k), where k is the size of the subarray. This set is created for each query and does not grow beyond the size of the largest subarray.

Since sets and the output list do not accumulate across queries but rather are allocated per query (the set is recreated for each query, and the output list is just accumulated once per query), the overall space complexity remains O(n) due to the set potentially growing to store pointers to up to n distinct values in the case of a single subarray spanning the entire array. This assumes that the space taken by the set is the dominant term and that k can reach n for at least one query.

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