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Arithmetic Slices

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ArraysDynamic Programming

An integer array is called arithmetic if it consists of at least three elements and if the difference between any two consecutive elements is the same.

For example, [1,3,5,7,9], [7,7,7,7], and [3,-1,-5,-9] are arithmetic sequences.

Given an integer array nums, return the number of arithmetic subarrays of nums.

A subarray is a contiguous subsequence of the array.

Example 1:

Input: nums = [1,2,3,4]
Output: 3
Explanation: We have 3 arithmetic slices in nums: [1, 2, 3], [2, 3, 4] and [1,2,3,4] itself.

Example 2:

Input: nums = [1]
Output: 0

Constraints:

1 <= nums.length <= 5000
-1000 <= nums[i] <= 1000

Try coding it in LeetCode

Arithmetic Slices

Medium

Asked by:

64 views

Topics:

ArraysDynamic Programming

An integer array is called arithmetic if it consists of at least three elements and if the difference between any two consecutive elements is the same.

For example, [1,3,5,7,9], [7,7,7,7], and [3,-1,-5,-9] are arithmetic sequences.

Given an integer array nums, return the number of arithmetic subarrays of nums.

A subarray is a contiguous subsequence of the array.

Example 1:

Input: nums = [1,2,3,4]
Output: 3
Explanation: We have 3 arithmetic slices in nums: [1, 2, 3], [2, 3, 4] and [1,2,3,4] itself.

Example 2:

Input: nums = [1]
Output: 0

Constraints:

1 <= nums.length <= 5000
-1000 <= nums[i] <= 1000

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the range of values for the numbers within the input array, and can the array contain negative numbers, zero, or floating-point numbers?
What is the minimum size of the array? Is an array of size less than 3 considered a valid input, and if so, what should be returned?
If no arithmetic slices are found in the input array, what should be the returned value (e.g., 0, -1, or null)?
Are the integers in the input array guaranteed to be within the representable range of an integer type to avoid potential overflow issues when calculating differences?
Are there any limitations on the number of arithmetic slices the input array may contain, and if so, is there a particular performance constraint regarding the count of those slices?

Brute Force Solution

Approach

The brute force approach for finding arithmetic slices means we'll examine every possible group of numbers to see if it forms an arithmetic slice. We'll check each group, one by one, to determine if it meets the necessary conditions.

Here's how the algorithm would work step-by-step:

Start by considering all groups of three numbers in the sequence.
For each group, check if the difference between the first two numbers is the same as the difference between the second and third numbers. If they are the same, it's an arithmetic slice.
Next, consider all groups of four numbers. Again, check if the difference between consecutive numbers is constant for the entire group. If so, it's another arithmetic slice.
Continue this process, increasing the group size each time, until you consider all groups up to the size of the entire sequence.
As you go, keep a running count of how many groups you found that are valid arithmetic slices.
The final count represents the total number of arithmetic slices in the sequence.

Code Implementation

def arithmetic_slices_brute_force(number_sequence):
    total_arithmetic_slices = 0

    # Iterate through all possible lengths of subsequences
    for subsequence_length in range(3, len(number_sequence) + 1):

        # Iterate through all possible start indices for subsequences
        for start_index in range(len(number_sequence) - subsequence_length + 1):
            end_index = start_index + subsequence_length
            subsequence = number_sequence[start_index:end_index]

            # Check if the subsequence is an arithmetic slice
            is_arithmetic = True
            difference = subsequence[1] - subsequence[0]

            # Check the difference between each element
            for index in range(2, len(subsequence)):  
                if subsequence[index] - subsequence[index - 1] != difference:
                    is_arithmetic = False
                    break

            # If is_arithmetic is True, increment slices
            if is_arithmetic:
                total_arithmetic_slices += 1

    return total_arithmetic_slices

Big(O) Analysis

Time Complexity

O(n^3) – The algorithm iterates through all possible subarrays of the input array of size n. The outer loop determines the starting index of the subarray, which takes O(n) time. The middle loop determines the length of the subarray, ranging from 3 up to n, also taking O(n) time. The inner loop checks if a given subarray is an arithmetic slice, which requires iterating through the subarray and comparing the differences between consecutive elements, taking O(n) time in the worst case. Therefore, the overall time complexity is O(n * n * n) = O(n^3).

Space Complexity

O(1) – The brute force approach iterates through all possible sub-arrays but does not explicitly create any auxiliary data structures like lists, arrays, or hash maps to store intermediate results. The algorithm primarily utilizes a few integer variables for indexing and comparing differences within the input array. The amount of extra memory required remains constant irrespective of the size N of the input array, therefore the space complexity is O(1).

Optimal Solution

Approach

To find the number of arithmetic slices, the key is to recognize patterns rather than checking every single subsequence. We'll use a dynamic approach, focusing on how many new slices are formed as we move along the sequence, building upon previous results.

Here's how the algorithm would work step-by-step:

Start by looking at the difference between the first two numbers in the sequence.
Check if the difference between the second and third numbers is the same as the first difference. If it is, you've found an arithmetic sequence of length 3.
If the third number continues the sequence, count the number of arithmetic sequences that end at that point. The number of new arithmetic slices you found is the number of arithmetic slices ending at the previous number plus one.
Keep going through the sequence. If a new number continues the arithmetic sequence, update the count of arithmetic slices by adding how many arithmetic slices ended at the previous number, plus one.
If a new number does *not* continue the sequence, reset the current arithmetic sequence counter to zero.
At the end, add up all the counts of arithmetic slices you've accumulated to find the total number of arithmetic slices in the sequence.

Code Implementation

def number_of_arithmetic_slices(sequence):
    sequence_length = len(sequence)
    if sequence_length < 3:
        return 0

    total_arithmetic_slices = 0
    arithmetic_slices_ending_here = 0

    for i in range(2, sequence_length):
        # Check if the current element extends an arithmetic sequence.
        if sequence[i] - sequence[i - 1] == sequence[i - 1] - sequence[i - 2]:
            # If it does, increment the count of slices ending here.
            arithmetic_slices_ending_here += 1
            total_arithmetic_slices += arithmetic_slices_ending_here

        else:
            # If it doesn't, reset the count.
            arithmetic_slices_ending_here = 0

    return total_arithmetic_slices

Big(O) Analysis

Time Complexity

O(n) – The solution iterates through the input array 'nums' of size n once. Within the loop, it performs constant-time operations to check for arithmetic progressions and update the count. There are no nested loops or recursion involved. Therefore, the runtime is directly proportional to the size of the input array, resulting in a time complexity of O(n).

Space Complexity

O(1) – The algorithm uses a single variable to keep track of the number of arithmetic slices ending at the previous index, and another variable to accumulate the total count of arithmetic slices. These variables use constant space regardless of the input array size N. No additional data structures dependent on N are created, so the auxiliary space remains constant. Therefore, the space complexity is O(1).

Edge Cases

Case	How to Handle
Empty or null input array	Return 0 immediately since no arithmetic slices can exist.
Array with length less than 3	Return 0 immediately since an arithmetic slice requires at least 3 elements.
Array with all identical values (e.g., [2, 2, 2, 2])	The solution should correctly count these as arithmetic slices; in this case 3 slices of length 3 or more
Array with a very long arithmetic progression (e.g., [1, 2, 3, ..., 1000])	The solution must handle potentially large counts of slices efficiently, possibly using a dynamic programming approach to avoid redundant calculations.
Integer overflow when calculating differences or summing slice counts	Use appropriate data types (e.g., long) to prevent integer overflows during calculations.
Array with negative numbers and positive numbers that result in a negative difference	The algorithm should correctly handle negative differences, which can occur when the array contains both positive and negative numbers.
Array where the difference between elements is zero	The algorithm should identify these as arithmetic slices as long as there are at least three elements.
Large array size potentially exceeding memory limits	Consider space optimization strategies or algorithm alternatives (if feasible) to handle very large input arrays.

Empty or null input array

How to Handle:

Return 0 immediately since no arithmetic slices can exist.

Array with length less than 3

How to Handle:

Return 0 immediately since an arithmetic slice requires at least 3 elements.

Array with all identical values (e.g., [2, 2, 2, 2])

How to Handle:

The solution should correctly count these as arithmetic slices; in this case 3 slices of length 3 or more

Array with a very long arithmetic progression (e.g., [1, 2, 3, ..., 1000])

How to Handle:

The solution must handle potentially large counts of slices efficiently, possibly using a dynamic programming approach to avoid redundant calculations.

Integer overflow when calculating differences or summing slice counts

How to Handle:

Use appropriate data types (e.g., long) to prevent integer overflows during calculations.

Array with negative numbers and positive numbers that result in a negative difference

How to Handle:

The algorithm should correctly handle negative differences, which can occur when the array contains both positive and negative numbers.

Array where the difference between elements is zero

How to Handle:

The algorithm should identify these as arithmetic slices as long as there are at least three elements.

Large array size potentially exceeding memory limits

How to Handle:

Consider space optimization strategies or algorithm alternatives (if feasible) to handle very large input arrays.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the range of values for the numbers within the input array, and can the array contain negative numbers, zero, or floating-point numbers?
What is the minimum size of the array? Is an array of size less than 3 considered a valid input, and if so, what should be returned?
If no arithmetic slices are found in the input array, what should be the returned value (e.g., 0, -1, or null)?
Are the integers in the input array guaranteed to be within the representable range of an integer type to avoid potential overflow issues when calculating differences?
Are there any limitations on the number of arithmetic slices the input array may contain, and if so, is there a particular performance constraint regarding the count of those slices?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

Start by considering all groups of three numbers in the sequence.
For each group, check if the difference between the first two numbers is the same as the difference between the second and third numbers. If they are the same, it's an arithmetic slice.
Next, consider all groups of four numbers. Again, check if the difference between consecutive numbers is constant for the entire group. If so, it's another arithmetic slice.
Continue this process, increasing the group size each time, until you consider all groups up to the size of the entire sequence.
As you go, keep a running count of how many groups you found that are valid arithmetic slices.
The final count represents the total number of arithmetic slices in the sequence.

Code Implementation

def arithmetic_slices_brute_force(number_sequence):
    total_arithmetic_slices = 0

    # Iterate through all possible lengths of subsequences
    for subsequence_length in range(3, len(number_sequence) + 1):

        # Iterate through all possible start indices for subsequences
        for start_index in range(len(number_sequence) - subsequence_length + 1):
            end_index = start_index + subsequence_length
            subsequence = number_sequence[start_index:end_index]

            # Check if the subsequence is an arithmetic slice
            is_arithmetic = True
            difference = subsequence[1] - subsequence[0]

            # Check the difference between each element
            for index in range(2, len(subsequence)):  
                if subsequence[index] - subsequence[index - 1] != difference:
                    is_arithmetic = False
                    break

            # If is_arithmetic is True, increment slices
            if is_arithmetic:
                total_arithmetic_slices += 1

    return total_arithmetic_slices

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

Start by looking at the difference between the first two numbers in the sequence.
Check if the difference between the second and third numbers is the same as the first difference. If it is, you've found an arithmetic sequence of length 3.
If the third number continues the sequence, count the number of arithmetic sequences that end at that point. The number of new arithmetic slices you found is the number of arithmetic slices ending at the previous number plus one.
Keep going through the sequence. If a new number continues the arithmetic sequence, update the count of arithmetic slices by adding how many arithmetic slices ended at the previous number, plus one.
If a new number does *not* continue the sequence, reset the current arithmetic sequence counter to zero.
At the end, add up all the counts of arithmetic slices you've accumulated to find the total number of arithmetic slices in the sequence.

Code Implementation

def number_of_arithmetic_slices(sequence):
    sequence_length = len(sequence)
    if sequence_length < 3:
        return 0

    total_arithmetic_slices = 0
    arithmetic_slices_ending_here = 0

    for i in range(2, sequence_length):
        # Check if the current element extends an arithmetic sequence.
        if sequence[i] - sequence[i - 1] == sequence[i - 1] - sequence[i - 2]:
            # If it does, increment the count of slices ending here.
            arithmetic_slices_ending_here += 1
            total_arithmetic_slices += arithmetic_slices_ending_here

        else:
            # If it doesn't, reset the count.
            arithmetic_slices_ending_here = 0

    return total_arithmetic_slices

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty or null input array	Return 0 immediately since no arithmetic slices can exist.
Array with length less than 3	Return 0 immediately since an arithmetic slice requires at least 3 elements.
Array with all identical values (e.g., [2, 2, 2, 2])	The solution should correctly count these as arithmetic slices; in this case 3 slices of length 3 or more
Array with a very long arithmetic progression (e.g., [1, 2, 3, ..., 1000])	The solution must handle potentially large counts of slices efficiently, possibly using a dynamic programming approach to avoid redundant calculations.
Integer overflow when calculating differences or summing slice counts	Use appropriate data types (e.g., long) to prevent integer overflows during calculations.
Array with negative numbers and positive numbers that result in a negative difference	The algorithm should correctly handle negative differences, which can occur when the array contains both positive and negative numbers.
Array where the difference between elements is zero	The algorithm should identify these as arithmetic slices as long as there are at least three elements.
Large array size potentially exceeding memory limits	Consider space optimization strategies or algorithm alternatives (if feasible) to handle very large input arrays.

Empty or null input array

How to Handle:

Return 0 immediately since no arithmetic slices can exist.

Array with length less than 3

How to Handle:

Return 0 immediately since an arithmetic slice requires at least 3 elements.

Array with all identical values (e.g., [2, 2, 2, 2])

How to Handle:

The solution should correctly count these as arithmetic slices; in this case 3 slices of length 3 or more

Array with a very long arithmetic progression (e.g., [1, 2, 3, ..., 1000])

How to Handle:

The solution must handle potentially large counts of slices efficiently, possibly using a dynamic programming approach to avoid redundant calculations.

Integer overflow when calculating differences or summing slice counts

How to Handle:

Use appropriate data types (e.g., long) to prevent integer overflows during calculations.

Array with negative numbers and positive numbers that result in a negative difference

How to Handle:

The algorithm should correctly handle negative differences, which can occur when the array contains both positive and negative numbers.

Array where the difference between elements is zero

How to Handle:

The algorithm should identify these as arithmetic slices as long as there are at least three elements.

Large array size potentially exceeding memory limits

How to Handle:

Consider space optimization strategies or algorithm alternatives (if feasible) to handle very large input arrays.