Let's say I have an array of floating point numbers, in sorted (let's say ascending) order, whose sum is known to be an integer N
. I want to "round" these numbers to integers while leaving their sum unchanged. In other words, I'm looking for an algorithm that converts the array of floating-point numbers (call it fn
) to an array of integers (call it in
) such that:
N
fn[i]
and its corresponding integer in[i]
is less than 1 (or equal to 1 if you really must)fn[i] <= fn[i+1]
), the integers will also be in sorted order (in[i] <= in[i+1]
)Given that those four conditions are satisfied, an algorithm that minimizes the rounding variance (sum((in[i] - fn[i])^2)
) is preferable, but it's not a big deal.
Examples:
[0.02, 0.03, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.11, 0.12, 0.13, 0.14] => [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] [0.1, 0.3, 0.4, 0.4, 0.8] => [0, 0, 0, 1, 1] [0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1] => [0, 0, 0, 0, 0, 0, 0, 0, 0, 1] [0.4, 0.4, 0.4, 0.4, 9.2, 9.2] => [0, 0, 1, 1, 9, 9] is preferable => [0, 0, 0, 0, 10, 10] is acceptable [0.5, 0.5, 11] => [0, 1, 11] is fine => [0, 0, 12] is technically not allowed but I'd take it in a pinch
To answer some excellent questions raised in the comments:
For the curious, here is the test script I used to identify which algorithms worked.
Rounding to the Nearest Integer If the digit in the tenths place is less than 5, then round down, which means the units digit remains the same; if the digit in the tenths place is 5 or greater, then round up, which means you should increase the unit digit by one.
The round() function returns a floating point number that is a rounded version of the specified number, with the specified number of decimals. The default number of decimals is 0, meaning that the function will return the nearest integer.
Another reason to favour integers over floats is performance and efficiency. Integer arithmetic is faster. And for a given range integers consume less memory because integers don't need to represent non-integer values. Another reason is to show intent.
Rounding Numbers: The Common Method The common rounding method says that if the number you are rounding is followed by the number 5 or above (5,6,7,8,or 9) then you should round that number up. For example, you can round the number 27 to 30 because 7 is greater than 5.
One option you could try is "cascade rounding".
For this algorithm you keep track of two running totals: one of floating point numbers so far, and one of the integers so far. To get the next integer you add the next fp number to your running total, round the running total, then subtract the integer running total from the rounded running total:-
number running total integer integer running total 1.3 1.3 1 1 1.7 3.0 2 3 1.9 4.9 2 5 2.2 8.1 3 8 2.8 10.9 3 11 3.1 14.0 3 14
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