Verify Cyclic Difference Sets

Question

A cyclic difference set is a set of positive integers with a unique property:

1. Let n be the largest integer in the set.
2. Let r be any integer (not necessarily in the set) greater than 0 but less than or equal to n/2.
3. Let k be the number of solutions to (b - a) % n = r where a and b are any members of the set. Each solution is an ordered pair (a,b). (Also note that this version of modulo makes negative numbers positive by adding n to it, unlike the implementations in many languages.)
4. Finally, if and only if this is a cyclic difference set, the value of k does not depend on your choice of r. That is, all values of r give the same number of solutions to the above congruence.

This can be illustrated with the following example:

Cyclic difference set: {4,5,6,8,9,11}
0 < r <= 11/2, so r = 1,2,3,4,5
r=1: (4,5) (5,6) (8,9)
r=2: (4,6) (6,8) (9,11)
r=3: (5,8) (6,9) (8,11)
r=4: (4,8) (5,9) (11,4)  since (4-11)%11=(-7)%11=4
r=5: (4,9) (6,11) (11,5)

Each value of r has the same number of solutions, 3 in this case, so this is a cyclic difference set.

Input

Input will be a list of positive integers. Since this is a set property, assume that input is not sorted. You can assume that n is at least 2, although k may be zero.

Output

Your program/function should output a truthy value if the set is a cyclic difference set, or a falsey value otherwise.

Test Cases

Valid cyclic difference sets:

10,12,17,18,21
7,5,4
57,1,5,7,17,35,38,49
1,24,35,38,40,53,86,108,114,118,135,144,185,210,254,266,273
16,3,19,4,8,10,15,5,6
8,23,11,12,15,2,3,5,7,17,1

(data source, although their convention is different)

Invalid cyclic difference sets:

1,2,3,4,20
57,3,5,7,17,35,38,49
3,4,5,9
14,10,8

Answer 1

Jelly, 14 7 bytes

_þ%ṀṬSE

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How it works

_þ%ṀṬSE  Main link. Argument: A (array of unique elements / ordered set)

_þ       Subtract table; yield a 2D array of all possible differences of two
         (not necessarily distinct) elements of A.
  %Ṁ     Take the differences modulo max(A).
    Ṭ    Untruth; map each array of differences modulo max(A) to a Boolean array
         with 1's at the specified indices. Note that all 0's in the index array
         are ignored, since indexing is 1-based in Jelly.
     S   Take the sum of these arrays, counting occurrences.
      E  Test if all resulting counts are equal.

Answer 2

Husk, 13 bytes

Ë#m%▲¹×-¹¹ḣ½▲

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The three superscript 1s seem wasteful...

Explanation

Ë#m%▲¹×-¹¹ḣ½▲  Input is a list, say x=[7,5,4]
            ▲  Maximum: 7
           ½   Halve: 3.5
          ḣ    Inclusive range from 1: [1,2,3]
Ë              All elements are equal under this function:
                Argument is a number, say n=2.
      ×-¹¹      Differences of all pairs from x: [0,-2,2,-3,0,3,-1,1,0]
  m%▲¹          Map modulo max(x): [0,5,2,4,0,3,6,1,0]
 #              Count occurrences of n: 1

Answer 3

Wolfram Language (Mathematica), 53 52 bytes

SameQ@@Counts@Mod[#-#2&@@@#~Permutations~{2},Max@#]&

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Note, we don't need to divide the max element by two due to symmetry (we may check counts of all modulos 1 to max(input) - 1).

Explanation

#~Permutations~{2}

Take all length-2 permutations of the input.

#-#2&@@@

Find differences of each

Mod[ ... ,Max@#]

Mod the result by the maximal element of the input.

Counts@

Find the frequencies of each element.

SameQ@@

Return whether all of the numbers are the same.

Answer 4

Python 3, 86 84 81 bytes

^{-3 bytes thaks to JungHwan Min}

lambda x:len({*map([(b-a)%max(x)for a in x for b in x].count,range(1,max(x)))})<2

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Answer 5

JavaScript (ES6), 87 bytes

Returns 0 or 1.

a=>a.map(b=>a.map(c=>x[c=(c-b+(n=Math.max(...a)))%n-1]=-~x[c]),x=[])|!x.some(v=>v^x[0])

Test cases

let f =

a=>a.map(b=>a.map(c=>x[c=(c-b+(n=Math.max(...a)))%n-1]=-~x[c]),x=[])|!x.some(v=>v^x[0])

console.log('[Truthy]')
console.log(f([10,12,17,18,21]))
console.log(f([7,5,4]))
console.log(f([57,1,5,7,17,35,38,49]))
console.log(f([1,24,35,38,40,53,86,108,114,118,135,144,185,210,254,266,273]))
console.log(f([16,3,19,4,8,10,15,5,6]))
console.log(f([8,23,11,12,15,2,3,5,7,17,1]))

console.log('[Falsy]')
console.log(f([1,2,3,4,20]))
console.log(f([57,3,5,7,17,35,38,49]))
console.log(f([3,4,5,9]))
console.log(f([14,10,8]))

Answer 6

Perl, 68 67 66 bytes

Includes +2 for ap

perl -apE '\@G[@F];pop@G;s:\d+:$G[$_-$&].=1for@F:eg;$_="@G"=~/^1*( 1*)\1*$/' <<< "4 5 6 8 9 11"

Answer 7

Python 3, 74 bytes

lambda x:len({sum(1+(a+r)%max(x)in x for a in x)for r in range(max(x))})<3

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Answer 8

Ruby, 81 bytes

->s{n=s.max
(1..n/2).map{|r|s.permutation(2).count{|a,b|(b-a)%n==r}}.uniq.size<2}

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Ungolfed:

->s{
  n=s.max
  (1..n/2).map{|r|               # For each choice of r
    s.permutation(2).count{|a,b| # Count the element pairs
      (b-a)%n==r                 #   for which this equality holds
    }
  }.uniq.size<2                  # All counts should be identical.
}

Answer 9

Haskell, 84 bytes

l s=all((g 1==).g)[1..t-1]where t=maximum s;g j=[1|x<-s>>=(`map`s).(-),x==j||x+t==j]

l is the function that does the check. It just computes all differences and counts.