API Quest: Generating a Centered Pascal’s Triangle

API Quest: Generating a Centered Pascal’s Triangle#

Introduction#

Welcome to the API Quest! Today, we will delve into a problem from the 2015 round of the APL Problem Solving Competition: generating Pascal’s triangle. While this task is straightforward, our unique challenge is to format the output in a visually appealing way for human readers. Pascal’s triangle, traditionally represented numerically, does not fit neatly into a rectangular array; thus, we will create a character matrix where the lines are centered.

Problem Breakdown#

The task consists of two main steps:

  1. Generating the data.

  2. Centering the lines.

Let’s explore these components in detail.

Generating Pascal’s Triangle#

Pascal’s triangle is elegantly defined by the binomial coefficients. It can be generated using an outer product on all numbers from 0 to n. However, a typical representation may show the triangle sideways, padded with zeros.

To improve this, we can employ two main methods: the flat approach and the nested approach.

Flat Approach#

The flat approach constructs the triangle data as follows:

⍝ GENERATION
F←{⍕⍤~∘0⍤1⍉∘.!⍨0,⍳⍵}

This function generates a flat representation of Pascal’s triangle.

Nested Approach#

Alternatively, we can adopt a nested approach. This involves:

  1. Generating all indices up to n.

  2. Performing a half outer product to build the triangle, which results in progressively longer rows.

The nested method can be represented as:

⍝ Nested:
i←0,⍳
N←(⍕i!⊢)¨i

This method constructs a nested format with included ranges for our indices.

Centering the Rows#

With our data generated in either the flat or nested format, we proceed to center the rows.

Strategy for Centering#

One method involves calculating the number of leading spaces in each line, which informs how we will manipulate the row layout:

  1. Count the leading spaces for each row.

  2. Take half of those spaces to determine how many will be moved to the front.

  3. Implement the rotation of the entire row accordingly.

We can rotate the matrix using the following function:

⍝ Rotate matrix:
RF←{⍵⌽⍨-⌊2÷⍨+/∧' '=⌽⍵} F  ⍝ flat
RN←(↑⌽⍨∘(⌈2÷⍨⊢-⊢/)≢¨) N    ⍝ nested

By rotating the rows appropriately, we produce a visually centered character matrix.

Further Techniques#

In addition to rotation, we can explore using the take function, which pads leading spaces while maintaining alignment. The full implementation for this step can be found in:

⍝ (Over-)Take nested:
TN←{↑⍵↑¨⍨(⌈2÷⍨--⊢/)≢¨⍵} N

Simplification and Additional Approaches#

We can simplify our process through algebraic manipulation, realizing that certain operations can be combined for efficiency.

Performance Comparison#

To understand how our different approaches compare, we can run efficiency tests. By increasing the print precision, we can evaluate triangle generations up to order 50. Here’s a brief overview of the findings, corresponding APL code for comparison:

⎕PP←17
'cmpx'⎕CY'dfns'
n←⍳50
cmpx'RF¨n' 'RN¨n' 'TN¨n'

This will give outputs, such as:

⍝ Results:
⍝ RF¨n → 8.8E¯3 |   0% ⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕
⍝ RN¨n → 4.1E¯3 | -54% ⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕
⍝ TN¨n → 4.3E¯3 | -51% ⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕⎕

  • The nested approaches generally perform approximately half as fast as the flat approach.

  • This aligns with our expectations, as the nested method computes less of the triangle while still achieving the same result.

Conclusion#

In conclusion, while traditional wisdom suggests that staying flat is often best in APL, we must recognize when flat processing does not deliver efficiency, particularly when computing each row separately.

Thank you for watching our exploration into generating a centered Pascal’s triangle!

Contents