Analyzing Non-Decreasing and Non-Increasing Patterns in a List of Numbers

Analyzing Non-Decreasing and Non-Increasing Patterns in a List of Numbers#

In this article, we will explore how to check if a given list of numbers fits a specific pattern: a non-decreasing segment followed by a non-increasing segment. While the terms “non-decreasing” and “non-increasing” may seem unusual, they serve our purpose well, as plateaus (where values remain the same) are acceptable. In other words, we will allow the values to stay constant but not decrease.

Understanding the Concept#

Example: Up and Down Sequence#

Let’s consider a simple example that clearly demonstrates our goal. We will normalize the input list by applying transformations to ensure it adheres to the specified rules. If there are any violations, our modifications will reflect those changes, allowing us to compare the original list with the modified one. If they are identical, the original list follows the rules; if not, it does.

To demonstrate, we will perform a test from the outset, applying functions without naming our arguments. The technique we’ll use involves a “maximum scan.” The maximum function takes two arguments and returns the larger of them, which we will insert between the elements, processing them over prefixes as we go.

The following APL expression performs this left maximum scan:

(⌈\) 1 2 3 4 5 3 1

This results in:

1 2 3 4 5 5 5

Continuing, when we utilize a reverse operation to apply the same logic from the right side, we use the following:

(⌈\∘⌽) 1 2 3 4 5 3 1

which yields:

1 3 5 5 5 5 5

Scanning in the Opposite Direction#

Now, let’s consider what happens if we start our scan from the right. This means after reaching the highest point, we must stay at that maximum value as we move left.

To achieve this, we can preprocess our input by reversing it. For example:

  • Input: 1, 3, 5, 4, 3, 2, 1

  • After reversing: 1, 2, 3, 4, 5

  • Performing the maximum scan now gives us the result:

(⌽⌈\∘⌽) 1 2 3 4 5 3 1

Resulting in:

5 5 5 5 5 3 1

Combining the Two Scans#

To determine whether our original list follows the rules, we need to combine these two maximum scans—one from left to right and one from right to left. We will take the minimum values of the corresponding positions from both scans.

Here’s how it breaks down:

  • Compare the results from both scans:

    • Minimum of 1 (left) and 5 (right) is 1

    • Minimum of 2 (left) and 5 (right) is 2

    • Continue until the scans converge

This method effectively checks both segments of the list for non-decreasing and non-increasing patterns. Using APL, we can illustrate this combination:

(⌈\ ⌊∘⌽ ⌈\∘⌽) 1 2 3 4 5 3 1

This expression combines the results yielding:

1 2 3 4 5 3 1

When introducing a slight variation by adding an extra element:

(⌈\ ⌊∘⌽ ⌈\∘⌽) 1 2 3 4 5 3 1 4

The result becomes:

1 2 3 4 5 4 4 4

Testing for Rule Violations#

We must check if modifications still hold true to the original pattern. If we introduce a change, we will see potential violations. For example, if we analyze the list:

(⊢ ≡ ⌈\ ⌊∘⌽ ⌈\∘⌽) 5 4 3 2 5 7

this results in:

0

indicating a failure in following the pattern. In contrast, for a valid input:

(⊢ ≡ ⌈\ ⌊∘⌽ ⌈\∘⌽) 1 2 3 4 3 1

we receive:

1

showing conformity.

Re-assessing with Variations#

Testing with various configurations (like changing an element in the middle) shows how new values can obscure the original values from both sides.

For instance, if we apply our final check:

F ← ⊢ ≡ ⌈\ ⌊∘⌽ ⌈\∘⌽
F 1 2 3 2 1

this returns:

1

indicating that it respects the pattern.

Final Checks#

Finally, we will compare the collected results with the original input array using an identity function.

This executes checks iteratively:

  • Perform a right scan

  • Perform a left scan

  • Take the minimum of both

  • Compare with the original input list

This systematic approach can confirm whether the original arrangement abides by the non-decreasing/non-increasing rules.

Conclusion#

By applying these scanning techniques and transformations, we can effectively analyze any list of numbers to determine if they follow a specified non-decreasing and non-increasing pattern. Thank you for reading!

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