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Topic: Compacting Pairs of Numbers (Read 921 times) |
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Virgilijus
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Compacting Pairs of Numbers
« on: Dec 21st, 2015, 11:37am » |
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After accidentally leaving the door open between the crocodile farm and the nursery, Willy has had to find another job. With his reputation preceding him, he can only find work in the boring and seemingly infinite Number Storage Warehouse.
On his first day, Willy's boss puts him to task; he needs to store every single unique pair of positive integers on the shelf. However, Willy has an idea; if he can take each unique pair and somehow transform them into a unique single number, he'll double the amount of pairs they can store (though his coworker, Mr. Cantor, disagrees) and perhaps he'll get a promotion.
Devise a way to transform a unique pair of positive integers (x,y) into a single, unique positive integer output (z). Note: order does matter to the pair i.e. f(x,y) = z implies f(y,x) =/= z.
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towr
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Re: Compacting Pairs of Numbers
« Reply #1 on: Dec 21st, 2015, 10:34pm » |
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z = frombase10str(tobase9str(x) + '9' + tobase9str(y))
e.g. (123, 345) => 1469556; (987, 654) => 13169806
I bet there's more efficient ways though. I mean, aside from using a large base B and B+1.
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| « Last Edit: Dec 21st, 2015, 10:36pm by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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rmsgrey
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Re: Compacting Pairs of Numbers
« Reply #2 on: Dec 22nd, 2015, 9:11am » |
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A bijective mapping:
1) Convert y in Z to y' in N by taking the absolute value, doubling it, and subtracting 1 if y is negative.
2) Pad the shorter number with leading zeroes so they're the same length, then, starting with the first digit of x, interleave the digits of x and y' to produce a new number, z, with the same sign as x.
To reverse:
2') If z has an odd number of digits, pad it with a leading 0, then take alternate digits and the sign of z as x, and the remaining digits as y'
1') If y' is odd, add 1 and subtract the result from 0. Either way, halve that number to give y.
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markr
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Re: Compacting Pairs of Numbers
« Reply #3 on: Dec 23rd, 2015, 10:45pm » |
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Similar to the diagonal method of mapping fractions to integers:
f(x,y) = (x^2 + y^2 + 2xy - 3x - y + 2) / 2
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