Interchanging Adjacent Elements in a Multi-Dimensional Array

Interchanging Adjacent Elements in a Multi-Dimensional Array#

In this article, we will explore how to interchange horizontally adjacent elements within a multi-dimensional array. We will start by using a three-dimensional array as an example, which consists of two layers, each with two rows and five columns. Our objective is to switch adjacent elements within this structure.

Understanding the Problem#

Consider the following example of a three-dimensional array:

  • Array Structure:

    • Layers: 2

    • Rows per Layer: 2

    • Columns per row: 5

The goal is to interchange adjacent elements in such a way that:

  • For example, if the elements are named:

    • Gela is interchanged with Opton

    • Prospero is interchanged with Lysa

    • Toaza remains in place since it has no adjacent element to switch with.

Here’s an example of a multi-dimensional array in APL:

n
⍝ ┌────────┬──────┬────────┬────────┬────────┐
⍝ │Griselda│Upton │Prospero│Delicia │Topaza  │
⍝ ├────────┼──────┼────────┼────────┼────────┤
⍝ │Clyde   │Ellis │Gilles  │Hilary  │Sheridan│
⍝ └────────┴──────┴────────┴────────┴────────┘
⍝ ┌────────┬──────┬────────┬────────┬────────┐
⍝ │Carole  │Clay  │Harley  │Bellance│Nesta   │
⍝ ├────────┼──────┼────────┼────────┼────────┤
⍝ │Dorian  │Sigurd│Clinton │Renee   │Nada    │
⍝ └────────┴──────┴────────┴────────┴────────┘

We can operate on individual rows without worrying about the overall shape of the array for now. Our algorithm will begin by focusing on just the first layer and first row, examining all the columns.

Developing the Algorithm#

Utilizing Partitioning and Function Wrapping#

We will create an anonymous function to address the reordering of adjacent elements. Utilizing the partition function allows us to group these elements for swapping. The function requires a mask to determine where to start a new section (indicating adjacent pairs to flip).

To generate this mask, we will:

  1. Compute the length of the vector using the shape function.

  2. Reshape an array of ones to form our mask.

Once we have our mask, we can partition the array based on these values, finding the adjacent pairs to flip. After reversing the pairs, we then need to merge everything back together. An APL example to flip pairs looks like this:

E←{⊃,/⌽¨⍵⊂⍨1 0⍴⍨⍴⍵}⍤1

This function captures the essence of flipping elements in each row of a multi-dimensional array.

Implementation Example#

Assuming our initial array is defined, we will encapsulate the flipping process within a function:

# Example algorithm to flip adjacent elements
def flip_adjacent(arr):
    # Create a mask and apply partition
    # Flip and merge the results
    pass  # Placeholder for function logic

Applying to Subarrays#

To apply this logic across the entire multi-dimensional array, we define our flipping function to act on rank one arrays. By iterating over subarrays, we can successfully swap adjacent pairs in each layer and row.

An APL illustration for applying the flip function is:

E n
⍝ ┌──────┬────────┬────────┬────────┬────────┐
⍝ │Upton │Griselda│Delicia │Prospero│Topaza  │
⍝ ├──────┼────────┼────────┼────────┼────────┤
⍝ │Ellis │Clyde   │Hilary  │Gilles  │Sheridan│
⍝ └──────┴────────┴────────┴────────┴────────┘

Alternative Approaches#

Using Indexing#

Another method involves creating an indexing approach. Here, we will rebuild the indices of the elements in such a way that they represent the desired order after the required swaps.

  1. Apply the shape to grab dimensions.

  2. Generate shifting indices for the elements, adjusting values to prevent clashes during sorting.

This method can also be encapsulated within a function to process the array:

# Example algorithm using indexing for reordering elements
def index_flip(arr):
    # Build indices and reorder to achieve desired flipping
    pass # Placeholder for function logic

For example, this APL function uses indexing to achieve the necessary order:

I←{⍵[⍋(⍳+⍴∘2 0)⍴⍵]}⍤1

Employing the Stencil Operator#

Lastly, we can consider using the stencil operator, typically associated with cellular automata. This operator processes windows of the array and can be adapted for our need to interchange adjacent pairs.

# Example of flip operation using stencil operator
def stencil_flip(arr):
    # Define stencil window and process
    pass # Placeholder for function logic

In APL, a similar operator can be utilized for adjacent changes:

(⌽⊢⌺(⍪2 2))n[1;1;]

Performance Considerations#

After experimenting with these approaches, we note that using the indexing method generally offers the best performance due to its direct nature of processing the array without unnecessary nesting. We also analyzed performance as we applied these methods to larger arrays, noting significant differences in execution times.

Conclusion#

We have explored multiple methods for interchanging horizontally adjacent elements in a multi-dimensional array, including functional programming techniques, indexing, and stencil operators. Notably, the indexing approach yielded the best performance, showing the power of direct manipulation within array operations.

By treating arrays as a whole rather than iterating over rows individually, we can significantly improve execution speed, a key advantage of array programming languages like APL.

Thank you for reading!

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