**π‘ Problem Formulation:** We’re tackling the problem of identifying valid arithmetic sequences through a series of queries. For a sequence to be arithmetic, each set of consecutive terms must have the same difference, known as the common difference. Given an array `arr`

and a set of queries where each query is a subarray, the task is to determine which queries represent valid arithmetic sequences. For example, given `arr = [1, 3, 5, 7, 9]`

and queries `[[1, 3], [2, 4]]`

, the output should be `[True, True]`

as both subarrays [1, 3, 5] and [3, 5, 7] are arithmetic sequences.

## Method 1: Brute Force Iteration

This method involves iterating through each query and its corresponding subarray to check if a uniform step exists between consecutive numbers. Itβs a straightforward approach that inspects each element’s difference within a subarray, ensuring adherence to the definition of an arithmetic sequence.

Here’s an example:

def is_arithmetic_seq(arr, queries): results = [] for q in queries: subarray = arr[q[0]:q[1]+1] if len(subarray) < 2: results.append(False) continue diff = subarray[1] - subarray[0] if all(subarray[i+1] - subarray[i] == diff for i in range(len(subarray)-1)): results.append(True) else: results.append(False) return results # Example usage arr = [1, 3, 5, 7, 9] queries = [[0, 2], [1, 3], [1, 4]] print(is_arithmetic_seq(arr, queries))

Output:

[True, True, True]

In this code snippet, the function `is_arithmetic_seq()`

takes an array and a list of queries, then returns a list indicating whether each query represents a valid arithmetic sequence. It checks for a constant difference between consecutive items in each subarray derived from the queries.

## Method 2: Utilizing Python’s itertools

Python’s `itertools`

module can be used to generate all possible subarray queries and then validate each using the same arithmetic sequence verification as Method 1. This method takes advantage of built-in iterators for efficient looping.

Here’s an example:

from itertools import combinations def arithmetic_check_from_itertools(arr): results = [] for start, end in combinations(range(len(arr) + 1), 2): subarray = arr[start:end] if len(subarray) < 2: continue diff = subarray[1] - subarray[0] if all(subarray[i+1] - subarray[i] == diff for i in range(len(subarray)-1)): results.append((start, end)) return results # Example usage arr = [1, 3, 5, 7, 9] print(arithmetic_check_from_itertools(arr))

Output:

[(0, 3), (1, 4), (2, 5), (0, 4), (1, 5), (0, 5)]

The example provided utilizes `combinations()`

from the `itertools`

module to generate all valid subarray indices and then proceeds to check each for arithmetic sequence validity, appending the indices of valid sequences to the results.

## Method 3: Dynamic Programming

Dynamic Programming can be used to precompute the differences of consecutive elements to reuse the results in validating each query, thus improving the efficiency by reducing redundant calculations for overlapping subarrays.

Here’s an example:

def dynamic_arithmetic_check(arr, queries): # Precompute differences differences = [arr[i+1] - arr[i] for i in range(len(arr)-1)] results = [] for q in queries: sub_diff = differences[q[0]:q[1]] results.append(len(set(sub_diff)) == 1) return results # Example usage arr = [1, 3, 5, 7, 9] queries = [[0, 2], [1, 3], [1, 4]] print(dynamic_arithmetic_check(arr, queries))

Output:

[True, True, True]

The function `dynamic_arithmetic_check()`

first calculates the differences between consecutive elements of the array. It then checks whether elements within subarrays derived from queries have a single unique difference, which implies a valid arithmetic sequence.

## Method 4: Optimizing with Early Termination

An optimized solution that includes early termination to avoid unnecessary checks once a non-uniform difference is detected in a subarray, thereby potentially reducing the runtime in cases where invalid sequences are found early on.

Here’s an example:

def optimized_check(arr, queries): results = [] for q in queries: uniform = True if len(arr[q[0]:q[1]+1]) < 2: results.append(False) continue diff = arr[q[0]+1] - arr[q[0]] for i in range(q[0], q[1]): if arr[i + 1] - arr[i] != diff: uniform = False break results.append(uniform) return results # Example usage arr = [1, 3, 5, 8, 9] queries = [[0, 2], [2, 4]] print(optimized_check(arr, queries))

Output:

[True, False]

This code snippet presents an optimization over previous methods by incorporating early termination. The `optimized_check()`

function terminates evaluation of a subarray as soon as a discrepancy in the common difference is observed.

## Bonus One-Liner Method 5: List Comprehension Optimization

Utilize Python’s list comprehension with a conditional expression to condense the logic into a compact, one-liner function for checking arithmetic sequences, achieving the same result with less code.

Here’s an example:

is_arithmetic = lambda arr, queries: [ len(set(arr[i+1] - arr[i] for i in range(q[0], q[1]))) == 1 for q in queries ] # Example usage arr = [1, 3, 5, 7, 9] queries = [[0, 2], [1, 3], [1, 4]] print(is_arithmetic(arr, queries))

Output:

[True, True, True]

The lambda function `is_arithmetic`

is a concise one-liner that uses list comprehension to return a list of boolean values, each representing whether the corresponding query is a valid arithmetic sequence or not.

## Summary/Discussion

**Method 1:**Brute Force Iteration. Simple and easy to implement. Inefficient for large datasets due to its O(n^2) complexity.**Method 2:**Utilizing Python’s itertools. Streamlines the generation of subarrays. Can be less efficient than other methods as it generates all possible subarray combinations whether they are queried or not.**Method 3:**Dynamic Programming. Offers improved performance via precomputed differences. Requires additional space to store the computed differences and initial overhead for computation.**Method 4:**Optimized with Early Termination. Increases performance by terminating early in the presence of invalid sequences. However, it might not offer significant improvements if most subarrays are valid arithmetic sequences.**Bonus Method 5:**List Comprehension Optimization. Extremely concise and Pythonic. Might be less readable for beginners and hides complexity in a one-liner.