# Rotate Y-Axis of a 2D Vector

So you want to rotate a vector, huh?

The solution I'm going to describe works well with both **two-dimensional** and **multidimensional**
vectors if applied correctly. It is originally based on
John Bentley's Reverse Rotate algorithm but with a
*teeny tiny twist*. So, if you want to understand how this algorithm works, you have to
read that article first.

All I did was invert the indices for `left`

, `up`

and `right`

, `down`

shift operations to
achieve complement sub-vectors $A'$ and $B'$. This makes the algorithm **more
efficient and agile** for dynamic programming.

Note that I have utilized generics in both functions to accept different types.
`[T any]`

allows multiple types as the argument (you may change this to any
concrete type you desire). Additionally, we use an enumeration called
`Direction`

to improve the clarity of the code.

# Shift Up

Suppose you want to shift up all the 2D vector columns by `1`

.

# Shift Down

Suppose you want to shift down all the 2D vector columns by `1`

.

# Y-Axis Column Rotation

If you want to rotate specific columns with **variable** shifts. We need a bit
more than `reverseRotate`

. In such a case, the problem is a bit hard because
when we think of it as columns, each element index is located in a different
array. I devised a 3-step solution, which works for me right now (I want to
keep it super simple).

For example, let's say we have a `3 x 3`

matrix.

Here, the challenge is to shift columns `2`

and `3`

up by 1 and 2 consecutively
instead of rotating the entire facet. Let's take a more complex input
to understand how we can do it.

Suppose you have a `6 x 6`

matrix.

And then, each column of the matrix has to be shifted up a variable amount of `n`

.

We could do this in three easy steps. First, copying the target column to a temporary
vector, then rotating it using `reverseRotate`

, and finally reassigning the values
for the source vector. However, this takes $O(n^2)$ time and uses $O(m)$ space to hold
the temporary vector while iterating each column.

Here's the code for it: -

Also, note that `ReduceShifts`

is a small optimization I've implemented for the algorithm.
For example, if we receive the delta array `[ 4,5,6,6,4,3 ]`

, we can find the minimum
`delta`

and rotate the entire facet $n$ times. Here, the minimum is $3$. So we
rotate the whole facet $3$ times. Once rotated, we can reduce the shifts of each
delta by the minimum rotations we did, and we are left with new changes `[1, 2, 3, 3, 1, 0]`

.
Effectively reducing several iterations of the algorithm.

How to use this function, so? Simple!

# Bonus: Shift Left & Right

If you want to shift in the `x-axis`

, then you have to iterate through each sub-vector
of the `2D`

vector to achieve the outcome. For example, you could do the following: -

Left and right shifts yields a running time of $O(n^2)$ with $O(1)$ space in-place shifting.
But why run sequentially, when you can parallelize?^{*}
**Reverse Rotate** algorithm is an absolute gem.

That's it folks!

# Well, now what?

You can navigate to more writings from here. Connect with me on LinkedIn for a chat.

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