# Numerical Join of two sets of vectors ignoring repeats

I have two sets of numerical vectors which can be either Real or Complex and I want to combine them together, ignoring repeats. Up until now I have been joining the two sets together and using Oleksander R.'s compiledUnion from How to remove duplicates from set of machine precision 2D points?.

Example code for what I'm doing at the moment is something like:

a = RandomReal[1, {100, 2}]
a = RandomChoice[a, 100]
b = RandomReal[1, {100, 2}]
b = RandomChoice[b, 100]
compiledUnion[Join[a,b]]


I think I can actually get out a little bit more efficiency from this however. I know that my first set of vectors already has repeats removed, and is generally larger than the second set. So I was thinking that I could somehow use this as a reference and then I would only have to compare the elements from the second set against the first set rather than comparing all elements together. I guess it would make sense to also sort the shorter second set.

So setting up roughly the variables I'm working with would be something like this:

a = RandomReal[1, {1000, 2}];
a = RandomChoice[a, 1000];
a = compiledUnion[a];
b = RandomReal[1, {10, 2}];
b = RandomChoice[b, 10];


Then what I would guess is that I would also run compiledUnion on b, and then I would want something like a compiledUnionJoin[a,b] which reduces the total number of operations compared to just compiledUnion[Join[a,b]]

I think that actually implementing this is a bit beyond my current programming skills however. Does anybody know of a good way to do it? Or perhaps my solution to this problem is not optimal and somebody else can think of a faster way to do it?

You could use slightly adapted version of merge step of merge sort. For lists with lengths $n_1$ and $n_2$, where first is already sorted, sorting second list and merging it with first should run in $O(n_1 + n_2\log n_2)$ time.

Since we'll be compiling two versions of our function for Real and for Complex arguments, and each requires slightly different comparisons let's use simple quoting to simplify code building.

ClearAll[quote, unquote, eval]
SetAttributes[{quote, unquote}, HoldAllComplete]
quote@expr : Except@_Symbol :=
Unevaluated@expr /. {x : _unquote | _quote :> x, s_Symbol -> quote@s}
unquote@args___ := args
eval = # /. HoldPattern@quote@s_ :> s &;


Some type dependent code generating functions:

inline // ClearAll
inline // Attributes = HoldAllComplete;

inline[a_[[i_]] > b_[[j_]], Real] :=
quote[CompileGetElement[a, i, 1] > CompileGetElement[b, j, 1]]
inline[a_[[i_]] > b_[[j_]], Complex] := quote@With[
{aEl = CompileGetElement[a, i, 1], bEl = CompileGetElement[b, j, 1]},
Re@aEl > Re@bEl || Re@aEl == Re@bEl && Abs@Im@aEl > Abs@Im@bEl
]

inline[a_ != b_, eps_, Real] :=
quote[Abs@Subtract[a, b] > eps Max[Abs@a, Abs@b]]
inline[a_ != b_, eps_, Complex] := quote[
Abs@Subtract[Re@a, Re@b] > eps Max[Abs@Re@a, Abs@Re@b] ||
Abs@Subtract[Im@a, Im@b] > eps Max[Abs@Im@a, Abs@Im@b]
]

inline[a_[[i_]] != b_[[j_]], eps_, m_, type_] :=
quote@Module[{bool = False, aEl, bEl, l},
Do[
aEl = CompileGetElement[a, i, l];
bEl = CompileGetElement[b, j, l];
If[unquote@inline[aEl != bEl, eps, type],
bool = True;
Break[]
],
{l, 1, m}
];
bool
]

inline[a_[[i_]] = b_[[j_]], m_] := quote@Module[{l},
Do[a[[i, l]] = CompileGetElement[b, j, l], {l, 1, m}]
]


In above function "comparison with tolerance" was implemented "manually" as shown by Carl Woll.

Library function performing merging for given types:

compiledUnionJoin // ClearAll
compiledUnionJoin[type : Real | Complex] := compiledUnionJoin@type =
eval@quote@Last@Compile[{{set, _type, 2}, {list, _type, 2}, {eps, _Real}},
Module[{i, j, k, n, nSet, nSorted, m, lastFromSet, sorted, res},
nSorted = Length@list;
sorted = Sort@list;
nSet = Length@set;
n = nSet + nSorted;
lastFromSet = False;
i = j = k = 1;
If[nSet > 0,
m = Length@CompileGetElement[set, i];
If[nSorted > 0,
m = Min[m, Length@CompileGetElement[sorted, j]]
]
(* else *),
If[nSorted > 0,
m = Length@CompileGetElement[sorted, j];
(* else *),
m = 0;
]
];
res = Table[unquote@Replace[type, {Real -> 0., Complex -> I 0.}], {n}, {m}];
If[nSet > 0,
If[nSorted > 0 && unquote@inline[set[[i]] > sorted[[j]], type],
unquote@inline[res[[k]] = sorted[[j]], m];
++j
(* else *),
unquote@inline[res[[k]] = set[[i]], m];
lastFromSet = True;
++i
]
(* else *),
If[nSorted > 0,
unquote@inline[res[[k]] = sorted[[j]], m];
++j
(* else *),
k = 0
]
];
While[i <= nSet && j <= nSorted,
If[unquote@inline[set[[i]] > sorted[[j]], type],
If[unquote@inline[res[[k]] != sorted[[j]], eps, m, type],
++k;
unquote@inline[res[[k]] = sorted[[j]], m];
lastFromSet = False
];
++j
(* else *),
If[lastFromSet || unquote@inline[res[[k]] != set[[i]], eps, m, type],
++k;
unquote@inline[res[[k]] = set[[i]], m];
lastFromSet = True
];
++i
]
];
If[i <= nSet,
If[lastFromSet || unquote@inline[res[[k]] != set[[i]], eps, m, type],
++k;
unquote@inline[res[[k]] = set[[i]], m]
]
];
Do[
++k;
unquote@inline[res[[k]] = set[[l]], m]
,
{l, i + 1, nSet}
];
Do[
If[unquote@inline[res[[k]] != sorted[[l]], eps, m, type],
++k;
unquote@inline[res[[k]] = sorted[[l]], m]
],
{l, j, nSorted}
];
Take[res, k]
],
RuntimeOptions -> "Speed", CompilationTarget -> "C"
]


For comparison let's add compiledUnion from answer by Oleksandr R. adapted to both types of arguments:

compiledUnion // ClearAll
compiledUnion[type : Real | Complex, tol_] := compiledUnion[type, tol] =
Block[{Internal$EqualTolerance = tol}, Compile[{{r, _type, 2}}, Block[{sorted = Sort@r, output, seen, current}, output = InternalBag[seen = First@sorted, 1]; Do[ If[i != seen, InternalStuffBag[output, seen = i, 1]], {i, sorted} ]; Partition[InternalBagPart[output, All], Length@seen] ], RuntimeOptions -> {"Speed", "CompareWithTolerance" -> True}, CompilationTarget -> "C" ] ]  Let's pre-compile both functions for both argument types. compiledUnion requires comparison tolerance at compile time, we'll use tol = 15: compiledUnionJoin@Real; compiledUnionJoin@Complex; tol = 15; eps = 10^(tol -$MachinePrecision);

compiledUnion[Real, tol];
compiledUnion[Complex, tol];


Our test data, both real and complex:

SeedRandom@0
n = 2 10^6;
dataR = RandomChoice[RandomReal[1, {n, 2}], n];
dataC = RandomChoice[RandomComplex[1, {n, 2}], n];


Calling compiledUnionJoin with empty list as first argument is equivalent to compiledUnion:

res1 = compiledUnion[Real, tol]@dataR; // AbsoluteTiming // First
res2 = compiledUnionJoin[Real][{}, dataR, eps]; // AbsoluteTiming // First
res1 === res2
res2 // Length
(* 0.780 *)
(* 0.671 *)
(* True *)
(* 1123205 *)

res1 = compiledUnion[Complex, tol]@dataC; // AbsoluteTiming // First
res2 = compiledUnionJoin[Complex][{}, dataC, eps]; // AbsoluteTiming // First
res1 === res2
res2 // Length
(* 0.9354 *)
(* 0.8026 *)
(* True *)
(* 1124283 *)


compiledUnionJoin is slightly faster than compiledUnion when normalizing single list.

Now task from OP, joining two lists from which first is already sorted and free of duplicates.

a = RandomReal[1, {1000, 2}];
a = RandomChoice[a, 1000];
a = compiledUnion[Real, tol][a];
b = RandomReal[1, {10, 2}];
b = RandomChoice[b, 10];

res1 = compiledUnion[Real, tol]@Join[a, b]; // AbsoluteTiming // First
res2 = compiledUnionJoin[Real][{}, Join[a, b], eps]; // AbsoluteTiming // First
res3 = compiledUnionJoin[Real][a, b, eps]; // AbsoluteTiming // First
res1 === res2 === res3
res3 // Length
(* 0.000079 *)
(* 0.0000385 *)
(* 0.0000189 *)
(* True *)
(* 565 *)


For our bigger test data:

a = Take[dataR, Round[.99 n]];
b = Drop[dataR, Round[.99 n]];
a = compiledUnionJoin[Real][{}, a, eps];
Length@a
Length@b
(* 1116705 *)
(*   20000 *)

res1 = compiledUnion[Real, tol]@Join[a, b]; // AbsoluteTiming // First
res2 = compiledUnionJoin[Real][{}, Join[a, b], eps]; // AbsoluteTiming // First
res3 = compiledUnionJoin[Real][a, b, eps]; // AbsoluteTiming // First
res1 === res2 === res3
res3 // Length
(* 0.166 *)
(* 0.0897 *)
(* 0.0425 *)
(* True *)
(* 1122434 *)

a = Take[dataC, Round[.99 n]];
b = Drop[dataC, Round[.99 n]];
a = compiledUnionJoin[Complex][{}, a, eps];
Length@a
Length@b
(* 1117685 *)
(*   20000 *)

res1 = compiledUnion[Complex, tol]@Join[a, b]; // AbsoluteTiming // First
res2 = compiledUnionJoin[Complex][{}, Join[a, b], eps]; // AbsoluteTiming // First
res3 = compiledUnionJoin[Complex][a, b, eps]; // AbsoluteTiming // First
res1 === res2 === res3
res3 // Length
(* 0.230 *)
(* 0.123 *)
(* 0.060 *)
(* True *)
(* 1123507 *)


If non-normalized list is about two orders of magnitude smaller than normalized one, as in OP, merging them using compiledUnionJoin is about four times faster than using compiledUnion on Join[a, b].

• Thankyou jkuczm for an excellent solution to my problem. I have one further query; I didn't specify this but I need to be able to work with either Real or Complex numbers. I tried wrapping in compiledUnionJoin[domain_] := compiledUnionJoin[domain] = (*your code with _Real -> _domain, and a choice of Complex/Real modification for the "comparison with tolerance" part*), but this runs nearly twice as slowly. I don't understand why that is, shouldn't this just define two separate compiled functions and choose one as necessary? Do you have a solution for this that doesn't slow the code down?
– Jojo
Dec 19, 2017 at 17:55
• @Joe I've added version for complex arguments. Maybe you could edit your question to incorporate you argument types requirements. In my tests version for complex arguments is about 40% slower than version for real arguments when joining a and b sets, since comparison of complex numbers is more complicated this slowdown is expected. Dec 20, 2017 at 23:56
• Thanks again for your help. I edited the question. I wasn't very specific in my comment, sorry; I know that the complex code should be a bit slower, but I was finding even the real code to be slower when wrapped up as I wrote above. As far as I can see this shouldn't change anything but maybe I've misunderstood f[x_] := f[x] = (*f here*) When I run your code I get a bunch of compile errors, but perhaps this is because I'm on Mathematica 8? I will try on a newer version also when I can. compiledUnionJoin[Real][{{0.1}}, {{0.101}}, 0.1] for me evaluates to Sequence[{{0.1}}, {{0.101}}, 0.1]
– Jojo
Dec 22, 2017 at 15:15
• @Joe I've adapted code to version 8. Just two small tweaks were needed in compiled code. SameQ is compilable from version 9, so it needed to be changed to Equal. Table with number as iterator is allowed since version 10.2, in previous versions iterator always needed to be a list. Dec 22, 2017 at 20:26

Try this:

DeleteDuplicates[Union[a,b]]