13 November 2023 | Challenge 243 |

# Count the Pairs on the Floor

## Task 1: Reverse Pairs

**Submitted by: Mohammad S Anwar**

You are given an array of integers.

Write a script to return the number of reverse pairs in the given array.

A reverse pair is a pair `(i, j)`

where:
a) `0 <= i < j < nums.length`

and
b) `nums[i] > 2 * nums[j]`

.

### Example 1

```
Input: @nums = (1, 3, 2, 3, 1)
Output: 2
(1, 4) => nums[1] = 3, nums[4] = 1, 3 > 2 * 1
(3, 4) => nums[3] = 3, nums[4] = 1, 3 > 2 * 1
```

### Example 2

```
Input: @nums = (2, 4, 3, 5, 1)
Output: 3
(1, 4) => nums[1] = 4, nums[4] = 1, 4 > 2 * 1
(2, 4) => nums[2] = 3, nums[4] = 1, 3 > 2 * 1
(3, 4) => nums[3] = 5, nums[4] = 1, 5 > 2 * 1
```

### Solution

Using the Perl Data Language to solve this task.

First we create a 1-d long ndarray from the given numbers.

```
$nums = long 1, 3, 2, 3, 1;
```

Then we create a sequence in the same shape as `$nums`

, i.e. a 1-d ndarray holding the column indices of `$nums`

and a second sequence as a single column holding the row indices.
When combining these index ndarrays, according to PDL’s broadcasting rules both will be extended by replicating along a dimension to fit each other.
For visualization, these replications may be performed explicitly:

A) Add a dummy dimension 1 to the row and replicate it five times.

```
say sequence(5)->dup(1, 5);
[
[0 1 2 3 4]
[0 1 2 3 4]
[0 1 2 3 4]
[0 1 2 3 4]
[0 1 2 3 4]
]
```

B) Replicate dimension 0 of the column five times.

```
say sequence(1, 5)->dup(0, 5);
[
[0 0 0 0 0]
[1 1 1 1 1]
[2 2 2 2 2]
[3 3 3 3 3]
[4 4 4 4 4]
]
```

Hence we get an upper right triangular matrix of ones when comparing the indices:

```
say sequence($nums) > sequence(1, $nums->dim(0));
[
[0 1 1 1 1]
[0 0 1 1 1]
[0 0 0 1 1]
[0 0 0 0 1]
[0 0 0 0 0]
]
```

In the same manner we can compare `$nums`

as a column with itself as a doubled row:

```
say $nums->dummy(0) > 2 * $nums
[
[0 0 0 0 0]
[1 0 0 0 1]
[0 0 0 0 0]
[1 0 0 0 1]
[0 0 0 0 0]
]
```

The “bit and” of both matrices literally follows the definition of reverse pairs. The sum over the and’ed matrices yields the total number of reverse pairs:

```
((sequence($nums) > sequence(1, $nums->dim(0)))
& ($nums->dummy(0) > 2 * $nums))->sum;
```

## Task 2: Floor Sum

**Submitted by: Mohammad S Anwar**

You are given an array of positive integers (`>=1`

).

Write a script to return the sum of `floor(nums[i] / nums[j])`

where `0 <= i,j < nums.length`

. The `floor()`

function returns the integer part of the division.

### Example 1

```
Input: @nums = (2, 5, 9)
Output: 10
floor(2 / 5) = 0
floor(2 / 9) = 0
floor(5 / 9) = 0
floor(2 / 2) = 1
floor(5 / 5) = 1
floor(9 / 9) = 1
floor(5 / 2) = 2
floor(9 / 2) = 4
floor(9 / 5) = 1
```

### Example 2

```
Input: @nums = (7, 7, 7, 7, 7, 7, 7)
Output: 49
```

### Solution

Again, using PDL.

Creating a 1-d double ndarray from the given numbers:

```
$nums = pdl 2, 5, 9;
```

Divide `$nums`

as row by `$nums`

as column in the same manner as in task 1 and apply `floor()`

:

```
say floor $nums / $nums->dummy(0);
[
[1 2 4]
[0 1 1]
[0 0 1]
]
```

Finally, sum over this matrix:

```
floor($nums / $nums->dummy(0))->sum;
```

This works not only for positive integers but for all non-zero integers.

If you have a question about this post or if you like to comment on it, feel free to open an issue in my github repository.