Determine if a relation is transitive

Question

Challenge description

Let's start with some definitions:

• a relation is a set of ordered pairs of elements (in this challenge, we'll be using integers)

For instance, [(1, 2), (5, 1), (-9, 12), (0, 0), (3, 2)] is a relation.

• a relation is called transitive if for any two pairs of elements (a, b) and (b, c) in this relation, a pair (a, c) is also present,

• [(1, 2), (2, 4), (6, 5), (1, 4)] is transitive, because it contains (1, 2) and (2, 4), but (1, 4) as well,

• [(7, 8), (9, 10), (15, -5)] is transitive, because there aren't any two pairs (a, b), (c, d) present such that b = c.

• [(5, 9), (9, 54), (0, 0)] is not transitive, because it contains (5, 9) and (9, 54), but not (5, 54)

Given a list of pairs of integers, determine if a relation is transitive or not.

Input / output

You will be given a list of pairs of integers in any reasonable format. Consider a relation

[(1, 6), (9, 1), (6, 5), (0, 0)]

The following formats are equivalent:

[(1, 6), (9, 1), (6, 5), (0, 0)] # list of pairs (2-tuples)
[1, 9, 6, 0], [6, 1, 5, 0] # two lists [x1, x2, ..., xn] [y1, y2, ..., yn]
[[1, 6], [9, 1], [6, 5], [0, 0] # two-dimentional int array
[4, 1, 6, 9, 1, 6, 5, 0, 0] # (n, x1, y1, ..., xn, yn)
[1+6i, 9+i, 6+5i, 0+0i] # list of complex numbers

... many others, whatever best suits golfing purposes

Output: a truthy value for a transitive relation, falsy otherwise. You may assume that the input will consist of at least one pair, and that the pairs are unique.

Answer 1

Haskell, 42 bytes

f x=and[elem(a,d)x|(a,b)<-x,(c,d)<-x,b==c]

Usage example: f [(1,2), (2,4), (6,5), (1,4)]-> True.

(Outer)loop over all pairs (a,b) and (inner)loop over the same pairs, now called (c,d) and every time when b==c check if (a,d)is also an existent pair. Combine the results with logical and.

Answer 2

 Prolog, 66 bytes

t(L):-not((member((A,B),L),member((B,C),L),not(member((A,C),L)))).

The relation is not transitive if we can find (A,B) and (B,C) such that (A,C) doesn't hold.

Answer 3

MATL, 27 25 bytes

7#u2e!l6MX>thl4$XQttY*g<~

Input format is a matrix (using ; as row separator) where each pair of the relation is a column. For example, test cases

[(1, 2), (2, 4), (6, 5), (1, 4)]
[(7, 8), (9, 10), (15, -5)]
[(5, 9), (9, 54), (0, 0)]

are respectively input as

[1 2 6 1; 2 4 5 4]
[7 9 15; 8 10 -5]
[5 9 0; 9 54 0]

Truthy output is a matrix formed by ones. Falsy is a matrix that contains at least one zero.

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Explanation

The code first reduces the input integers to unique, 1-based integer values. From those values it generates the adjacency matrix; matrix-multiplies it by itself; and converts nonzero values in the result matrix to ones. Finally, it checks that no entry in the latter matrix exceeds that in the adjacency matrix.

Answer 4

JavaScript (ES6), 69 67 bytes

a=>(g=f=>a.every(f))(([b,c])=>g(([d,e])=>c-d|!g(([d,c])=>b-d|c-e)))

Saved 2 bytes thanks to an idea by @Cyoce. There were four previous 69-byte formulations:

a=>a.every(([b,c])=>a.every(([d,e])=>c-d|a.some(([d,c])=>b==d&c==e)))
a=>!a.some(([b,c])=>!a.some(([d,e])=>c==d&a.every(([d,c])=>b-d|c-e)))
a=>a.every(([b,c])=>a.every(([d,e])=>c-d|!a.every(([d,c])=>b-d|c-e)))
(a,g=f=>a.every(f))=>g(([b,c])=>g(([d,e])=>c-d|!g(([d,c])=>b-d|c-e)))

Answer 5

Pyth, 14 bytes

!-eMfqFhTCM*_M

Test suite

Input format is expected to be [[0, 0], [0, 1], ... ]

!-eMfqFhTCM*_M
!-eMfqFhTCM*_MQQQ    Variable introduction
            _MQ      Reverse all of the pairs
           *   Q     Cartesian product with all of the pairs
         CM          Transpose. We now have [[A2, B1], [A1, B2]] for each pair
                     [A1, A2], [B1, B2] in the input.
    f                Filter on
       hT            The first element (the middle two values)
     qF              Being equal
  eM                 Take the end of all remaining elements (other two values)
 -              Q    Remove the pairs that are in the input
!                    Negate. True if no transitive pairs were not in the input

Answer 6

Brachylog, 24 bytes

'{psc[A:B:B:C],?'e[A:C]}

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Explanation:

'{psc[A:B:B:C],?'e[A:C]}
'{                     } it is impossible to find
    c                    a flattened
   s                     subset of
  p                      a permutation of the input
     [A:B:B:C]           that has four elements, with the second and third equal
              ,?         and such that the input
                'e       does not contain
                  [A:C]  a list formed of the first and fourth element

In other words, if the input contains pairs [A:B] and [B:C], we can permute the input to put [A:B] and [B:C] at the start, delete all other elements, and produce a list [A:B:B:C]. Then we return truthy from the inner predicate (falsey from the whole program) if [A:C] isn't there.

Answer 7

CJam (22 bytes)

{__Wf%m*{z~~=*}%\-e_!}

Online test suite. This is an anonymous block (function) which takes the elements as a two-level array, but the test suite does string manipulation to put the input into a suitable format first.

Dissection

{         e# Begin a block
  _       e#   Duplicate the argument
  _Wf%    e#   Duplicate again and reverse each pair in this copy
  m*      e#   Cartesian product
  {       e#   Map over arrays of the form [[a b][d c]] where [a b] and [c d]
          e#   are in the relation
    z~~=* e#     b==c ? [a d] : []
  }%
  \-      e#   Remove those transitive pairs which were in the original relation
  e_!     e#   Test that we're only left with empty arrays
}

Answer 8

Mathematica, 49 bytes

#/.{x=___,{a_,b_},x,{b_,c_},x}/;#~FreeQ~{a,c}:>0&

Pure function which takes a list of pairs. If the input list contains {a,b} and {b,c} but not {a,c} for some a, b, c, replaces it with 0. Truthy is the input list, falsy is 0.

Answer 9

C++14, 140 bytes

As unnamed lambda returning via reference parameter. Requires its input to be a container of pair<int,int>. Taking the boring O(n^3) approach.

[](auto m,int&r){r=1;for(auto a:m)for(auto b:m)if (a.second==b.first){int i=0;for(auto c:m)i+=a.first==c.first&&b.second==c.second;r*=i>0;}}

Ungolfed and usage:

#include<vector>
#include<iostream>

auto f=
[](auto m,int&r){
  r=1;                         //set return flag to true
  for(auto a:m)                //for each element
    for(auto b:m)              //check with second element
      if (a.second==b.first){  //do they chain?
        int i=0;               //flag for local transitivity
        for(auto c:m)          //search for a third element
          i+=a.first==c.first&&b.second==c.second;
        r*=i>0;                //multiply with flag>0, resulting in 0 forever if one was not found
      }
}
;

int main(){
 std::vector<std::pair<int,int>> m={
  {1, 2}, {2, 4}, {6, 5}, {1, 4}
 };

 int r;
 f(m,r);
 std::cout << r << std::endl;
 
 m.emplace_back(3,6);
 f(m,r);
 std::cout << r << std::endl;
 
 m.emplace_back(3,5);
 f(m,r);
 std::cout << r << std::endl;

}

Answer 10

Python 2, 91 67 55 bytes

lambda s:all(b-c or(a,d)in s for a,b in s for c,d in s)

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-24 bytes thanks to Leaky Nun
-12 bytes thanks to Bubbler

Answer 11

05AB1E, 14 bytes

íâ€øʒθË}€нIKg_

Port of @isaacg's Pyth answer, so make sure to upvote him/her as well!
Also takes the input as a list of pairs (e.g. [[1,2],[3,4],...]).

Try it online or verify all test cases.

Explanation:

í           # Reverse each pair of the (implicit) input-list
 â          # Get the cartesian product of this list and the (implicit) input-list,
            # creating pairs of pairs
  €         # Map over each pair of pairs:
   ø        #  Zip/transpose; swapping rows/columns
            # (we now have a list of [[x1,y2],[y1,x2]] of each pair in the input-list, 
            # including those where [x1,y1]==[x2,y2])
    ʒ       # Filter this list of 2x2 blocks by:
     θ      #  Get the last row: [y1,x2]
      Ë     #  Check if the values in the pair are the same: y1==x2
    }€      # After the filter: map over each remaining 2x2 block:
      н     #  Leave just the first row: [x1,y2]
       IK   # Remove all input-pairs from this list of remaining pairs
         g  # And then check if the length
          _ # is equal to 0 (1 if the list was empty; 0 if not)
            # (after which the result is output implicitly)

Answer 12

Axiom 103 bytes

c(x)==(for i in x repeat for j in x repeat if i.2=j.1 and ~member?([i.1, j.2],x)then return false;true)

ungolfed:

c(x)==
  for i in x repeat
    for j in x repeat
       if i.2=j.1 and ~member?([i.1, j.2],x) then return false
  true

                                                                   Type: Void

the exercises

(2) -> c([[1,2],[2,4],[6,5],[1,4]])
   Compiling function c with type List List PositiveInteger -> Boolean
   (2)  true
                                                                Type: Boolean
(3) -> c([[7,8],[9,10],[15,-5]])
   Compiling function c with type List List Integer -> Boolean
   (3)  true
                                                            Type: Boolean
(4) -> c([[5,9],[9,54],[0,0]])
   Compiling function c with type List List NonNegativeInteger ->
      Boolean
   (4)  false

Answer 13

Pyth - 22 21 bytes

.Am},hdedQfqFtPTsM^Q2

Test Suite.

Answer 14

Clojure, 56 53 bytes

Update: Instead of using :when I'll just check that for all pairs of [a b] [c d] either b != c or [a d] is found from the input set.

#(every? not(for[[a b]%[c d]%](=[b nil][c(%[a d])])))

Original:

Wow, Clojure for loops are cool :D This checks that the for loop does not generate a falsy value, which occurs if [a d] is not found from the input set.

#(not(some not(for[[a b]%[c d]% :when(= b c)](%[a d]))))

This input has to be a set of two-element vectors:

(f (set [[1, 2], [2, 4], [6, 5], [1, 4]]))
(f (set [[7, 8], [9, 10], [15, -5]]))
(f (set [[5, 9], [9, 54], [0, 0]]))

If input must be list-like then (%[a d]) has to be replaced by ((set %)[a d]) for extra 6 bytes.

Answer 15

Both these solutions are unnamed functions taking a list of ordered pairs as input and returning True or False.

Mathematica, 65 bytes

SubsetQ[#,If[#2==#3,{#,#4},Nothing]&@@@Join@@@#~Permutations~{2}]&

#~Permutations~{2}] creates the list of all ordered pairs of ordered pairs from the input, and Join@@@ converts those to ordered quadruples. Those are then operated upon by the function If[#2==#3,{#,#4},Nothing]&@@@, which has a cool property: if the middle two elements are equal, it returns the ordered pair consisting of the first and last numbers; otherwise it returns Nothing, a special Mathematica token that automatically disappears from lists. So the result is the set of ordered pairs that needs to be in the input for it to be transitive; SubsetQ[#,...] detects that property.

Mathematica, 70 bytes

And@@And@@@Table[Last@i!=#&@@j||#~MemberQ~{#&@@i,Last@j},{i,#},{j,#}]&

Table[...,{i,#},{j,#}] creates a 2D array indexed by i and j, which are taken directly from the input (hence are both ordered pairs). The function of those two indices is Last@i!=#&@@j||#~MemberQ~{#&@@i,Last@j}, which translates to "either the second element of i and the first element of j don't match, or else the input contains the ordered pair consisting of the first element of i and the last element of j". This creates a 2D array of booleans, which And@@And@@@ flattens into a single boolean.

Answer 16

APL(NARS), 39 chars, 78 bytes

{∼∨/{(=/⍵[2 3])∧∼(⊂⍵[1 4])∊w}¨,⍵∘.,w←⍵}

test:

  f←{∼∨/{(=/⍵[2 3])∧∼(⊂⍵[1 4])∊w}¨,⍵∘.,w←⍵}
  f (1 2) (2 4) (6 5) (1 4)
1
  f (7 8) (9 10) (15 ¯5)
1
  f (5 9) (9 54) (0 0)
0

one second 'solution' follow goto ways:

r←q w;i;j;t;v
r←1⋄i←0⋄k←↑⍴w⋄→3
r←0⋄→0
→0×⍳k<i+←1⋄t←i⊃w⋄j←0
→3×⍳k<j+←1⋄v←j⊃w⋄→4×⍳t[2]≠v[1]⋄→2×⍳∼(⊂t[1]v[2])∊w

Answer 17

Common Lisp, 121 bytes

(lambda(x)(not(loop for(a b)in x thereis(loop for(c d)in x do(if(= b c)(return(not(member`(,a ,d) x :test #'equal))))))))

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