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Pythagorean.cpp
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Pythagorean.cpp
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// Oliver Kullmann, 12.6.2016 (Swansea)
/* Copyright 2016, 2017, 2019 Oliver Kullmann
This file is part of the OKlibrary. OKlibrary is free software; you can redistribute
it and/or modify it under the terms of the GNU General Public License as published by
the Free Software Foundation and included in this library; either version 3 of the
License, or any later version. */
/*!
\file Satisfiability/Transformers/Generators/Pythagorean.cpp
\brief Generator for Pythagorean triples and tuples, and the related
SAT problems (as CNFs in DIMACS format). The SAT-problems formulate the
m-colouring of the hypergraph of tuples.
See https://en.wikipedia.org/wiki/Pythagorean_triple for explanations on
triples/tuples, and see
https://en.wikipedia.org/wiki/Boolean_satisfiability_problem#SAT_problem_format
for the DIMACS format. The 2-colouring (boolean) problem is discussed at
https://en.wikipedia.org/wiki/Boolean_Pythagorean_triples_problem
This program is available at
https://github.com/OKullmann/oklibrary/blob/master/Satisfiability/Transformers/Generators/Pythagorean.cpp
USAGE:
In the following, tuples are considered with components from 1 to n.
For the boolean problems for triples, use
> ./Pythagorean n 3 0 2
or
> ./Pythagorean n 3 1 2
(doesn't matter here), which creates files "Pyth_n-3-0-2-SB.cnf" resp.
"Pyth_n-3-1-2-SB.cnf". This is with symmetry-breaking, but if it shall be
excluded, use
> ./Pythagorean n 3 0 2 sb=off
or
> ./Pythagorean n 3 1 2 sb=off
, creating files "Pyth_n-3-0-2.cnf" resp. "Pyth_n-3-1-2.cnf"
(Note that ">" denotes the command-line prompt, and that the command is
issued in the same directory where the executable "Pythagorean" has been
compiled.)
For the boolean problem for quadruples, use
> ./Pythagorean n 4 0 2
while the injective form (all components different) is obtained by
> ./Pythagorean n 4 1 2
creating files Pyth_n-4-0-2-SB.cnf resp. Pyth_n-4-1-2-SB.cnf.
To deactivate symmetry-breaking, use e.g.
> ./Pythagorean n 4 0 2 sb=off
creating file Pyth_n-4-0-2.cnf
The second parameter is K >= 3, the length of the Pythagorean tuple.
For the third parameter d >= 0, also larger values can be used, which enforce
a respective minimum distance between the (sorted) components of the tuples
x_1^2 + ... + x_{K-1}^2 = x_K^2,
that is, x_i + d <= x_{i+1} for 1 <= i <= K-1.
An optional fourth parameter is "format=SD" (the default) resp.
"format=D" (for "strict DIMACS" versus "DIMACS"): in the former case,
variables are renamed, so that they start with 1, without gaps.
Furthermore, "format=R" is possible, which replaces the hypergraph
with a random hypergraph having the same hyperedge-sizes and vertex-degrees
(after subsumptiion-elimination, and for m >= 2 after core-reduction).
The fourth/fifth parameter m >= 0 is the number of colours, with
- m = 0: only output the max-occurring vertex, the number hn of hyperedges,
and hn / (n^(K-2) * log(n)) (the factor in the estimation).
- m = 1: output the hypergraph
- m = 2: output the boolean problem
- m >= 3: currently strong or weak direct/nested translation available.
In case of m=0, K=3, dist=0, the computation of the count uses a
factorisation table for the natural numbers until n: This is much faster,
but uses more memory (4 bytes * n).
The 4-bytes-size comes from the 32-bit type Factorisation::base_t, which
restricts the maximal n here to roughly 2*10^9. When using a 64-bit type
instead, then this restrictions goes away, but then 8 bytes are used
"per vertex".
In case of m >= 3, the fifth/sixth parameter specifies the translation to a
boolean clause-set; currently we have the following possibilities:
- "S" the strong direct translation (with ALOAMO-clauses)
- "W" the weak direct translation (only with ALO-clauses)
- "N" the (weak) nested translation (no AMO-clauses)
- "NS" the strong nested translation (with AMO-clauses)
An optional fifth/sixth/seventh parameter can be "sb=off", which deactivates
symmetry-breaking, while "sb=on" is the default.
A further optional parameter can be "-", in which case output is put to
standard output, or "filename", in which case a file is created.
Default output is to file "Pyth_n-K-d-m.cnf" for 1 <= m <= 2 resp.
"Pyth_n-K-d-m-T.cnf" for m >= 3, where T is the acronym for the
translation-type, in both cases without symmetry-breaking, while with
we get "Pyth_n-K-d-m-SB.cnf" resp. "Pyth_n-K-d-m-T-SB.cnf".
So the complete usage-description is
Pythagorean n K d [format=SD|D] m [S|W|N|NS] [sb=on|off] [-|filename]
where the parameter after m is needed if m >= 3, while for m <= 2 this
parameter can not be used. As usual, square-brackets mean optional
arguments, while "|" means an alternative.
Subsumption-elimination is first applied, and then for m >= 2 elimination
of hyperedges containing a vertex occurring at most m-1 times (repeatedly),
that is, the "m-core" of the hypergraph is computed.
The output contains additional information for m >= 1 (as comments):
- Library and version information.
- Information on parameters.
- Information on number of hyperedges.
- Information on vertex-degrees (m >= 1) and variable-degrees (m >= 2).
The variable-degrees ignore additional clauses by symmetry-breaking or
the non-boolean translation.
After the comments (started with "c ") then comes the parameter line:
- for SAT problems "p cnf max c", with "max" the maximal occurring
variable-index, and with "c" the number of clauses;
- for the hypergraph (m=1) "p hyp max h", with "max" the maximal occurring
vertex (the largest hypotenuse), and with "h" the number of hyperedges.
The CNF-output puts related clauses on the same line (note that in DIMACS,
a clause is completed by "0", and thus you can have as many clauses on a
line as you like).
The order of the clauses is anti-lexicographically.
COMPILATION:
Requires C++11. Compile with
> g++ -Wall --std=c++17 -Ofast -o Pythagorean Pythagorean.cpp
resp. (now asserts disabled)
> g++ -Wall --std=c++17 -Ofast -DNDEBUG -o Pythagorean Pythagorean.cpp
resp. (fastest, and without superfluous warnings)
> g++ -Wall -Wno-dangling-else -Wno-catch-value --std=c++17 -Ofast -funroll-loops -DNDEBUG -o Pythagorean Pythagorean.cpp
If on the other hand a debugging version is needed, use:
> g++ -Wall --std=c++17 -g -o Pythagorean Pythagorean.cpp
FURTHER WORK:
TODO: Enable to consider large n (64 bits), at least for counting.
On a 64-bit machine, just using std::uint_least64_t for base_t
does the job. But this needs to be made consistent.
TODO: Implement output-"format" "R", which replaces the hypergraph with
a random hypergraph with the given hypergraph-edge-sizes and
-vertex-degrees. The seed should also be an input; perhaps here
"R0", "R177" etc.
TODO: implement "super-strict" DIMACS output, with all clauses on their own
line.
TODO: prove that subsumption does not happen for K=4.
TODO: implement arbitrary K.
TODO: implement multi-threaded computation.
Easy with std::async, just dividing up the outer loops for computing
the tuples (enumeration or counting). Since smaller numbers are easier,
a bit of thought on an equal splitting is needed.
The solution for splitting is, not to split the load into contiguous
blocks, but to use "slicing": for the number nt >= 1 of threads,
just use the congruence classes {1,...,n} mod i, i = 0,...,nt-1
for the threads. It seems the computation of the factorisation-table
can not be parallelised.
nt=0 could be used for outputting just the header? Or just the
estimation?!
It seems then that "nt=.." should become another optional parameter
(default-value 1).
nt should not become part of the default filename.
TODO: implement mixed k_i.
Likely best here to use for example "[3,4]", that is, enclosing the
list into square brackets on the parameter line; and demanding, that
this is one parameter, so that when spaces are used, then the whole
must be quoted.
TODO: Check that the translations are in line with the OKlibrary at Maxima
level.
TODO: Estimate number N of tuples for arbitrary K; the conjecture is
N ~ factor_K * n^(k-2) * log(n).
factor3 ~ 0.1494 (n <= 2^32-2); The Sierpinski formula yields for
factor3 = 1/2pi = 0.15915494309189533.., with two additional terms
"+ B n" with B as below, and "+ O(n^(1/2))" (using newer estimates).
However this is quite a bit bigger than the above factor -- our n's
are "too small".
The original Sierpinski-formula needs to be divided by 8=2^3, since
it considers a^2+b^2=c^2 with a,b arbitrary integers, and thus we
have 2^2 sign-possibilities, and the swap a <-> b.
According to "Mathematical Constants By Steven R. Finch", in Maxima-code the
computation of B:
G : 1/2/%pi*beta(1/4,1/2); // https://en.wikipedia.org/wiki/Gauss%27s_constant
S : log(exp(2*%gamma)/2/G^2); // K / pi for https://en.wikipedia.org/wiki/Sierpi%C5%84ski%27s_constant
A : 1.2824271291006226368753425688; // http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html http://oeis.org/A074962
zp2 : 1/6*%pi^2*(%gamma + log(2*%pi) - 12*log(A)); // zeta'(2) http://mathworld.wolfram.com/RiemannZetaFunction.html
Sh : %gamma + S + 12 / %pi^2 * zp2 + log(2)/3 - 1/%pi;
B : Sh / 2 / %pi;
float(B);
.027511221451371352
factor4 ~ 0.005668 (n <= 2*10^4)
factor5 ~ 0.000788 (n <= 3200)
factor6 ~ 0.00010193 (n <= 800)
factor7 ~ 1.0329e-05 (n <= 400)
While for K=3 the factor seems to be increasing, for K >= 4 it seems
decreasing, and they seem to converge. Would be good to have a more
precise approximation (perhaps having another term "+n^(k-2)" ?).
And would be good to have for K>3 faster computation.
TODO: implement intelligent methods for K>3. Start with the generalised
Dickson-method for K=4.
TODO: Implement reductions triggered by the symmetry-breaking clauses.
We have unit-clause propagation and subsumption. Possibly this enables
further m-core-reductions (if correct?)?
FURTHER DISCUSSIONS:
Hyperedge-counting links:
- https://oeis.org/A224921 for the number of Pythagorean triples (K=3)
up to n-1.
This sequence is obtained for, say, the first 73 elements, on the
command-line via (using Bash)
> for ((n=0; n<=72; ++n)); do ./Pythagorean $n 3 0 0 - | cut -f2 -d" " | tr "\n" ","; done; echo
(This computation is rather wasteful, since instead of running just once
through n=1,...,72, computing the triples with hypotenuse =n, and adding
tehm up, it uses a quadratic effort, by starting again and again at the
beginning. One could easily add a special output mode, which avoids this,
if needed. But for small numbers it is very fast anyway. For example
> for ((n=1000000; n<=1000010; ++n)); do ./Pythagorean $n 3 0 0 - | cut -f2 -d" " | tr "\n" ","; done; echo
which produces
23471475,23471475,23471476,23471477,23471478,23471479,23471479,23471480,
23471481,23471481,23471494,
takes 16s on an older 2.4 GHz machine.)
Bigger examples:
- The number of triples up to 2*10^9 is 6,380,787,008,
obtained by "./Pythagorean 2000000000 3 0 0" in 335 sec, using 7.5 GB
(on a standard 64-bit machine with 32 GB RAM). This yields factor3 =
0.14897 (see above).
- While the number of triples for n=4,294,967,294=2^32-2 is
14,225,080,520, obtained by "./Pythagorean 4294967294 3 0 0" in
960 sec (same machine), using 16 GB. factor3 = 0.14932.
- Using std::uint_least64_t for base_t, n=10*10^9 yields
34,465,432,859, using 74.5 GB and 22m13s. factor3 = 0.149681.
- And n=20*10^9 yields 71,137,221,952, using 150 GB and 44m47s.
factor3 = 0.149958.
- n=30*10^9 yields 108,641,785,354, using 220 GB and 68m6s.
factor3 = 0.150113.
- Number of Pythagorean quadruples (K=4) or quintuples (K=5): not yet
in OEIS.
The number of quadruples up to 20000 is 22,452,483, obtained by
"./Pythagorean 20000 4 0 0" in 134m10s (2.4 GHz) (using very
little memory; faster methods are needed here).
The number of quintuples up to 3200 is 208,319,099, obtained by
"./Pythagorean 3200 5 0 0" in 258m42s (as above).
The number of sixtuples up to 800 is 279,072,340, obtained by
"./Pythagorean 800 6 0 0" in 87m52s (as above).
The number of seventuples up to 400 is 633,708,884, obtained by
"./Pythagorean 400 7 0 0" in 96m38s (as above).
Pythagorean numbers Ptn(...) established, generalising
http://link.springer.com/chapter/10.1007%2F978-3-319-40970-2_15
https://arxiv.org/abs/1605.00723
Solving and Verifying the boolean Pythagorean Triples problem
via Cube-and-Conquer
Ptn(k_1, ..., k_m) is the smallest n >= 1 (if it exists, otherwise infinity),
such that for every partition of {1,...,n} into m parts A_1, ..., A_m, there
is 1 <= i <= m such that A_i contains a Pythagorean k_i-tuple.
Experimental results are as follows.
In square brackets [h;h';c], number of hyperedges before/after reduction and
number of clauses; if "=" is used, then the reductions don't do anything
here; algorithms "vw1" and "g2wsat" are from the UBCSAT suite of local-search
algorithms, while "SplittingViaOKsolver" is the basic C&C-implementation in
the OKlibrary; except where stated "W", the strong direct translation is
used for >= 3 colours, and symmetry-breaking is used iff stated ("SB"):
- Ptn(3,3) = 7825 SB [9,472; 7,336; 14,673]
http://cs.swan.ac.uk/~csoliver/papers.html#PYTHAGOREAN2016C
Finding a solution for 7824 SB [9,465; 7,326; 14,653]: strongest seems
ddfw from ubcsat, with the following success-rate for the cutoff-values:
10^6 -> 2%, 5*10^6 -> 12%, 10^7 -> 21%, 10^8 -> 49%.
Without SB:
10^6 -> 0%, 5*10^6 -> 10%, 10^7 -> 15%, 10^8 -> 60%.
For 7825, maxsat is all-except-of-one-clause (with and without SB).
- Ptn(3,3,3) > 1.4 * 10^7 N [33,609,851; 26,652,251; 79,956,753], with
12,121,598 occurring variables: g2wsat, runs with cutoff=6*10^9
success at run 2 (seed 2950083789, steps 3917132950).
It seems this lower bound is still far away from the truth. So this
problem is excessively hard.
- Ptn(4,4) = 105 SB [639; 638; 1277] (known)
- Ptn_i(4,4) = 163 SB [545; 544; 1089]
- Ptn(4,4,4) > 1724 SB [167,077; =; 508130]; n = 3 * 1724 = 5172.
Best seems saps with S-SB.
1680: S-SB [158,627; =; 482,604], cutoff=10^7 has success rate 17%, and
5*10^7 has 28%.
1687: S-SB 10^7 -> 10%, 1710: S-SB 10^7 -> 1%.
1718: S-SB 2*10^7 -> 1%.
1719: S-SB 2*10^7 -> 1%, 200 runs.
1724: S-SB 2*10^7 -> 0.2%, 500 runs; 5*10^7 -> 0.3%, 1000 runs;
10^8 -> 0.6%, 500 runs.
1725: S-SB 2*10^7 -> 0%, 500 runs; 5*10^7 -> 0%, 1000 runs,
10^8 -> 0%, 1000 runs.
W-SB: 5*10^7 -> 0%, walksat-tabu, 20000 runs (min = 6).
Conjecture: Ptn(4,4,4) = 1725.
Seems to be a very hard problem, too hard for current methods.
- Ptn(5,5) = 37 SB [404; 254; 509] (known)
- Ptn_i(5,5) = 75 SB [2,276; =; 4,553]
- Ptn(5,5,5) = 191 S-SB [46,633; 41,963; 126,656]
Satisfying for 190, cutoff=10^5, success-rate:
vw1 for S: 15%, S-SB: 5%;
vw1 for W: 14%, for W-SB: 8%;
walksat-tabu N: 8%, N-SB: 1%;
walksat-tabu NS: 47%, NS-SB: 36%.
C&C via SplittingViaOKsolver, S-SB, with D=20 and minisat-2.2.0 for 191:
total run-time around 15 min (while for D=30 around 54 min);
for N-SB, D=20, around 26 min, and for NS-SB around 36 min).
- Ptn_i(5,5,5) > 468
saps with S-SB seems best:
410 NS with g2wsat, cutoff=10^6: 18% success.
410 W with vw1, cutoff=10^6: 0% success.
410 S-SB with saps, cutoff=10^6: 31% success.
467: cutoff=10^6: 7% success.
468: cutoff=2*10^6: 500 runs, 8.4% success.
469: cutoff=10^6: 300 runs, 0% success. 2*10^6, 700 runs: 0%.
Conjecture: Ptn_i(5,5,5) = 469.
- Ptn(6,6) = 23 SB [311; 267; 535] (known)
- Ptn_i(6,6) = 61 SB [6,770; =; 13,541]
- Ptn(6,6,6) = 121;
120 S-SB [154,860; 151,105; 453,798] found satisfiable with saps:
cutoff=10^5 has success-rate 2%.
121 [159,697; 155,857; 468,058] found
unsatisfiable with C&C and D=20 and solver="lingelingala-b02aa1a-121013",
370 min.
- Ptn(7,7) = 18 SB [306; 159; 319] (known)
- Ptn_i(7,7) = 65 SB [44,589; =; 89,179]
- Ptn(7,7,7) = 102 S-SB [789,310; 694,898; 2,085,104]; for 101 C&C with D=20
and glucose-2.2, splitting into ~ 1500 instances, instance 151
satisfiable; also D=20 for 102, same C&C, 4,390 min.
The sequence Ptn(k,k) for k=2,..., (which is 1, 7825, 105, 37, 23, 18, ...)
is https://oeis.org/A250026 .
The sequence Ptn_i(k,k) for k=3,..., is also of interest (7825,163,75,61,
65). This can be further extended.
The sequence Ptn(k,k,k) for k=3,... (?,?,191,121,102) is of interest, since
these are the only 3-colour-problems solved until now. Can likely be further
extended (with good effort).
*/
#include <iostream>
#include <string>
#include <limits>
#include <cmath>
#include <vector>
#include <algorithm>
#include <cassert>
#include <fstream>
#include <forward_list>
#include <chrono>
#include <map>
#include <cstdint>
#include <set>
#include <utility>
#include <stdexcept>
#include <iomanip>
#include <random>
namespace {
template <class C> using val_t = typename C::value_type;
template <class C> using siz_t = typename C::size_type;
template <class C> using it_t = typename C::iterator;
}
namespace Factorisation {
typedef std::uint_least32_t base_t;
typedef std::uint_least8_t exponent_t;
// Compute table T, such that for 1 <= i <= n, T[i] is the
// prime-factorisation of i, in map-form (basis -> exponent):
template <typename E = exponent_t>
std::vector<std::map<base_t,E>> table_factorisations(const base_t n) {
std::vector<std::map<base_t,E>> T(n+1);
std::vector<base_t> rem; rem.reserve(n+1);
for (base_t i = 2; i <= n; ++i) rem[i] = i;
for (base_t i = 2; i <= n; ++i)
if (rem[i] != 1) {
const base_t b = (T[i].empty()) ? i : T[i].begin()->first;
for (base_t j = i; j <= n; j+=i) { ++T[j][b]; rem[j] /= b; }
}
return T;
}
// This function is not used here, since the following implicit
// representation is faster and uses less memory.
// Now the table T only contains one prime factor (the largest):
template <typename B = base_t>
std::vector<B> table_factor(const base_t n) {
typedef siz_t<std::vector<B>> size_t;
assert(n <= std::numeric_limits<size_t>::max() / 2);
std::vector<B> T(size_t(n)+1);
for (B i = 2; i <= n; ++i)
if (T[i] == 0) for (size_t j = i; j <= n; j+=i) T[j] = i;
return T;
}
// Various functions for working with tables produced by table_factor:
// Extracting the exponents of prime factors congruent 1 mod 4:
template <class V>
inline std::vector<val_t<V>> extract_exponents_1m4(const V& T, val_t<V> n) {
assert(n >= 2);
assert(n < T.size());
typedef val_t<V> B;
typedef std::vector<B> R;
R res;
siz_t<R> next = 0;
B old_f = 0;
do {
const B f = T[n];
if (f%4==1)
if (f==old_f) ++res[next-1];
else {res.push_back(1); ++next;}
n /= f; old_f = f;
} while (n != 1);
return res;
}
// Extracting the prime factorisation:
template <class V, typename B, typename E = exponent_t>
inline std::map<B,E> extract_factorisation(const V& T, B n) {
assert(n >= 2);
assert(n < T.size());
std::map<B,E> res;
do {
const B f = T[n];
++res[f];
n /= f;
} while (n != 1);
return res;
}
// The list of factors <= bound, given factorisation F:
template <class V, typename B>
inline std::vector<B> bounded_factors(const V& F, const B bound) {
typedef std::vector<B> R;
R res;
typedef siz_t<R> size_t;
size_t num_factors = 1;
for (const auto& p : F) num_factors *= p.second+1;
res.reserve(num_factors);
const auto begin = res.cbegin();
res.push_back(1);
for (const auto p : F) {
const auto end = res.cend();
const B b = p.first;
const auto max_e = p.second;
for (auto i = begin; i != end; ++i) {
const auto f = *i;
B power = 1;
for (decltype(p.second) e = 0; e < max_e; ++e) {
power *= b;
const auto new_f = f*power;
if (new_f > bound) break;
else res.push_back(new_f);
}
}
}
return res;
}
}
namespace Pythagorean {
constexpr double factor3 = 0.1494; // yields an upper bound for n <= 2^32-2.
constexpr double pi = 2 * std::asin(1);
constexpr double factor3_theory = 1.0 / 2 / pi;
template <typename C>
constexpr double estimating_triples(const C n) noexcept {
return factor3 * n * std::log(n);
}
template <typename C1, typename C2>
constexpr long double estimating_tuples_factor(const C1 n, const C1 K, const C2 N) noexcept {
return N / (std::pow((long double)n,K-2) * std::log((long double)n));
}
// Counting triples:
template <typename C1, typename C2>
void triples_c(const C1 n, C1& max, C2& hn) {
using namespace Factorisation;
const auto T = table_factor(n);
assert(T.size() == n+1);
for (C1 i = 5; i <= n; ++i) {
C2 prod = 1;
for (const auto e : extract_exponents_1m4(T,i)) prod *= 2*e + 1;
if (prod != 1) {max = std::max(max, i); hn += (prod-1)/2;}
}
}
// Counting triples with minimum distance between (sorted) components
// (much slower than above, but minimal space usage):
template <typename C1, typename C2>
void triples_c(const C1 n, const C1 dist, C1& max, C2& hn) noexcept {
assert(dist >= 1);
for (C1 r = 2; r <= C1(n/(1+std::sqrt(2))); r+=2) {
const C1 rs = r*(r/2);
for (C1 s = dist; s <= C1(std::sqrt(rs)); ++s)
if (rs % s == 0) {
const C1 t = rs / s;
if (t >= n or t < s+dist) continue;
const C1 c = r+s+t;
if (c <= n) {max = std::max(max,c); ++hn;}
}
}
}
// Generating triples:
template <class V, typename C1>
V triples_e(const C1 n, const C1 dist, C1& max) {
V res; res.reserve(Pythagorean::estimating_triples(n));
const C1 max_r = n/(1+std::sqrt(2));
const auto T = Factorisation::table_factor(max_r);
assert(T.size() == max_r+1);
for (C1 r = 2; r <= max_r; r+=2) {
auto F = Factorisation::extract_factorisation(T, r);
for (auto& p : F) p.second *= 2;
--F[2];
const C1 rs = r*(r/2);
const C1 bound = std::sqrt(rs);
for (const C1 s : Factorisation::bounded_factors(F,bound)) {
if (s < dist) continue;
const C1 t = rs / s;
if (t >= n or t < s+dist) continue;
const C1 c = r+s+t;
if (c <= n and s+dist <= t) {
res.push_back({{r+s,r+t,c}});
max = std::max(max,c);
}
}
}
return res;
}
// Counting quadruples :
template <typename C1, typename C2>
void quadruples_c(const C1 n, const C1 dist, C1& max, C2& hn) noexcept {
const C1 n2 = n*n;
for (C1 a = 1; a < n; ++a) {
const C1 a2 = a*a;
const C1 bbound = std::sqrt(n2-a2);
for (C1 b = a+dist; b <= bbound; ++b) {
const C1 b2 = a2+b*b;
const C1 cbound = std::sqrt(n2-b2);
for (C1 c = b+dist; c <= cbound; ++c) {
const C1 d2 = b2 + c*c;
const C1 d = std::sqrt(d2);
if (d*d == d2 and d >= c+dist) {max = std::max(max,d); ++hn;}
}
}
}
}
// Generating quadruples:
template <class V, typename C1>
V quadruples_e(const C1 n, const C1 dist, C1& max) {
V res;
const C1 n2 = n*n;
for (C1 a = 1; a < n; ++a) {
const C1 a2 = a*a;
const C1 bbound = std::sqrt(n2-a2);
for (C1 b = a+dist; b <= bbound; ++b) {
const C1 b2 = a2+b*b;
const C1 cbound = std::sqrt(n2-b2);
for (C1 c = b+dist; c <= cbound; ++c) {
const C1 d2 = b2 + c*c;
const C1 d = std::sqrt(d2);
if (d*d == d2 and d >= c+dist) {
res.push_back({{a,b,c,d}});
max = std::max(max,d);
}
}
}
}
return res;
}
}
namespace Container {
template <class C>
void remove_empty_elements(C& v) noexcept {
v.erase(std::remove_if(v.begin(), v.end(),
[](const val_t<C>& x){return x.empty();}), v.end());
}
}
namespace Subsumption {
// Selecting some element from a non-empty sequence:
template <class V>
inline val_t<V> select(const V& t) noexcept {
assert(not t.empty());
return t[0];
}
// In general a random choice is better for algorithm min_elements, and best
// should be a choice of minimum degree, however here it is not worth the effort.
// Remove subsumed elements from vector v, where the elements of v are
// ordered sequences of unsigned integral type C1, with maximal value max:
template <class V, typename C1>
void min_elements(V& v, const C1 max) {
if (v.empty()) return;
const auto begin = v.begin(), end = v.end();
typedef val_t<V> tuple_t;
std::sort(begin, end,
[](const tuple_t& x, const tuple_t& y) {return x.size() < y.size();});
typedef it_t<V> it_t;
std::vector<std::forward_list<it_t>> occ(max+1);
it_t i = begin;
for (const auto size = begin->size(); i != end and i->size()==size; ++i)
occ[select(*i)].push_front(i);
if (i == end) return;
while (true) {
const auto size = i->size();
const it_t old_i = i;
for (; i < end and i->size()==size; ++i) {
for (const auto x : *i)
for (const auto j : occ[x])
if (std::includes(i->begin(),i->end(), j->begin(),j->end())) {
i->clear();
goto Outer;
}
Outer :;
}
if (i == end) break;
for (auto j = old_i; j != i; ++j)
if (not j->empty()) occ[select(*j)].push_front(j);
}
Container::remove_empty_elements(v);
}
}
namespace Reduction {
// Iteratively removing all hyperedges containing some vertex occurring
// at most m-1 time, computing the (final) vertex-degrees in deg:
template <class V, class SV, typename C1>
void core_red(V& hyp, SV& deg, const C1 m) noexcept {
for (const auto& h : hyp) for (const auto v : h) ++deg[v];
if (m <= 1) return;
typedef val_t<V> tuple_t;
typedef val_t<tuple_t> vert_t;
bool changed;
do {
changed = false;
for (auto& h : hyp)
if (std::find_if(h.begin(), h.end(),
[&](const vert_t v){return deg[v] < m;}) != h.end()) {
changed = true;
for (const auto v : h) --deg[v];
h.clear();
}
} while (changed);
Container::remove_empty_elements(hyp);
}
}
namespace Random {
// Choose randomly k elements, strictly ascending, from the range
// begin to end:
template <typename UInt, class RGen>
std::vector<UInt> random_choice(const UInt begin, UInt end, const UInt k, RGen& rgen) {
assert(k <= end-begin);
std::vector<UInt> res;
res.reserve(k);
const auto vbegin = res.begin();
for (UInt i = 0; i < k; ++i) {
const auto vend = res.end();
auto r = std::uniform_int_distribution<typename RGen::result_type>(begin, end-1)(rgen);
auto prev = vbegin;
{auto it = std::upper_bound(prev, vend, r);
while (it != prev) {
r += it - prev;
prev = it;
it = std::upper_bound(prev, vend, r);
}}
res.insert(prev,r);
--end;
}
return res;
}
template <class Hyp, class LDist, typename vertex_t, class Gen>
void randomise(Hyp& hyp, const LDist& hc, const vertex_t max, Gen& rgen) {
assert(not hyp.empty());
assert(not hc.empty());
assert(max >= 1);
typedef val_t<LDist> count_t;
std::vector<std::map<vertex_t, count_t>> enh_deg(max+1);
for (const auto& h : hyp)
for (const auto v : h) ++enh_deg[v][h.size()];
const auto K = hc.size()-1;
typedef std::pair<count_t,count_t> range_t;
std::vector<range_t> length_ranges(K+1);
Begin :
{count_t next = 0;
for (vertex_t k = 0; k <= K; ++k)
if (hc[k] != 0) {
const auto new_next = next + hc[k];
length_ranges[k] = {next, new_next};
next = new_next;
}
assert(next == hyp.size());
}
for (auto& h : hyp) h.clear();
for (vertex_t v = 1; v <= max; ++v) {
const auto& deg_m = enh_deg[v];
for (const auto& deg_p : deg_m) {
const auto k = deg_p.first;
const auto count = deg_p.second;
auto& range = length_ranges[k];
const auto begin = range.first;
auto& end = range.second;
if (count > end-begin) goto Begin;
const auto rvec = random_choice(begin, end, count, rgen);
for (const auto r : rvec) hyp[r].push_back(v);
for (auto it = rvec.crbegin(); it != rvec.crend(); ++it) {
const auto r = *it;
if (hyp[r].size() == k) std::swap(hyp[r], hyp[--end]);
}
}
}
}
}
namespace Translation {
enum class Type {
direct_strong,
direct_weak,
nested,
nested_strong,
none,
failure
};
// explicit output:
std::ostream& operator <<(std::ostream& out, const Type t) {
switch (t) {
case Type::direct_strong : out << "direct-strong"; break;
case Type::direct_weak : out << "direct-weak"; break;
case Type::nested : out << "nested"; break;
case Type::nested_strong : out << "nested-strong"; break;
case Type::none : out << "no-translation"; break;
case Type::failure : out << "FAILURE"; break;
}
return out;
}
// from and to type-abbreviations:
Type get_type(const std::string& arg) noexcept {
if (arg == "S") return Type::direct_strong;
else if (arg == "W") return Type::direct_weak;
else if (arg == "N") return Type::nested;
else if (arg == "NS") return Type::nested_strong;
else return Type::failure;
}
std::string type_abbr(const Type t) noexcept {
switch (t) {
case Type::direct_strong : return "S";
case Type::direct_weak : return "W";
case Type::nested : return "N";
case Type::nested_strong : return "NS";
default : return "";
}
}
std::string list_abbr() noexcept { return "\"S\", \"W\", \"N\" or \"NS\""; }
constexpr bool is_direct(const Type t) noexcept {
return t == Type::direct_weak or t == Type::direct_strong;
}
constexpr bool is_nested(const Type t) noexcept {
return t == Type::nested or t == Type::nested_strong;
}
template <typename C1, typename C2>
void pline_output(std::ostream* const out, const C2 n, const C2 c,
const C1 m) {
assert(*out);
assert(m >= 1);
switch (m) {
case 1 :
*out << "p hyp " << n << " " << c << "\n"; break;
default :
*out << "p cnf " << n << " " << c << "\n";
}
}
template <typename C2, typename C1>
inline C2 num_var(const C1 max, const C1 m, const Type t) noexcept {
assert(m >= 2);
assert(t != Type::failure);
switch (t) {
case Type::direct_strong : return C2(m) * max;
case Type::direct_weak : return C2(m) * max;
case Type::nested : return C2(m-1) * max;
case Type::nested_strong : return C2(m-1) * max;
default : return max;
}
}
template <typename C1, typename C2>
inline C2 num_cl(const C1 occ_n, const C1 m, const C2 hn, const Type t, const bool sb) noexcept {
assert(m >= 2);
assert(t != Type::failure);
const C2 m2 = m;
const C2 sbc = (not sb) ? 0 :
((is_nested(t)) ? m2-1 : (m2*(m2-1))/2);
switch (t) {
case Type::direct_weak : return m2 * hn + occ_n + sbc;
case Type::direct_strong :
return m2 * hn + occ_n + occ_n*(m2*(m2 - 1)) / 2 + sbc;
case Type::nested_strong :
return m2 * hn + occ_n*((m2-1)*(m2 - 2)) / 2 + sbc;
default : return m2 * hn + sbc;
}
}
// For the direct translation (strong or weak), translate vertex v and
// colour col into non-boolean variable var(v,m,col),
// expressing that v does not get colour col:
template <typename C2, typename C1>
inline C2 var_d(const C1 v, const C1 m, const C1 col) noexcept {
assert(v >= 1);
assert(m >= 3);
assert(col < m);
return C2(v-1) * m + col + 1;
}
// The variables related to vertex v for the nested translation:
template <typename C2, typename C1>
inline C2 var_n(const C1 v, const C1 m, const C1 i) noexcept {
assert(v >= 1);
assert(m >= 3);
assert(i < m-1);
return C2(v-1) * (m-1) + i + 1;
}
// Output the literals for the nested translation; the UHIT(1) clause-set is
// {-v_1}, {v_1,-v_2}, {v_1,v_2,-v_3}, ..., {v_1,...,v_{m-2},-v_{m-1}},
// {v_1,...,v_{m-1}} (all literals flipped, compared to literature, due to
// minimising "-"-symbols in output).
template <typename C2, typename C1>
inline void lits_n(const C1 v, const C1 m, const C1 col, std::ostream* const out) {
assert(v >= 1);
assert(col < m);
if (col == 0) *out << "-" << var_n<C2,C1>(v,m,0);
else {
*out << var_n<C2,C1>(v,m,0);
for (C1 col2 = 1; col2 < col; ++col2) *out << " " << var_n<C2>(v,m,col2);
if (col < m-1) *out << " -" << var_n<C2>(v,m,col);
}
}
template <class Hyp, typename C1, typename C2, class Deg, class T>
void output_colouring_problem(std::ostream* const out, const Hyp& G, const C1 m,
const C2 max, const C2 c, const Deg deg, const C1 md_v,
const Type t, const bool sb, const T& re) {
assert(m >= 1);
bool r = not re.empty();
pline_output(out, max, c, m);
if (m == 1) {
assert(not sb);
for (const auto& H : G) {
for (const auto v : H) *out << ((r)?re[v]:v) << " ";
*out << "0\n";
}
}
else if (m == 2) {
if (sb) *out << ((r)?re[md_v]:md_v) << " 0\n";
for (const auto& H : G) {
for (const auto v : H) *out << ((r)?re[v]:v) << " ";
*out << "0 ";
for (const auto v : H) *out << "-" << ((r)?re[v]:v) << " ";
*out << "0\n";
}
}
else {
if (sb) {
// First handling of the max-vertex:
if (is_direct(t))
for (C1 col = 1; col < m; ++col)
*out << var_d<C2>((r)?re[md_v]:md_v,m,col) << ((col==m-1) ? std::string(" 0") : std::string(" 0 "));
else *out << var_n<C2,C1>((r)?re[md_v]:md_v,m,0) << " 0";
// Find the other m-2 vertices of highest degree and put into "store":
std::vector<C1> store; store.reserve(m-2);
{std::set<C1> avoid; avoid.insert(md_v);
const auto size = deg.size();
for (C1 i = 0; i < m-2; ++i) {
C2 max_d = 0; C1 max_v = 0;
const auto end = avoid.end();
for (C1 v = 1; v < size; ++v) {
const auto d = deg[v];
if (d > max_d and avoid.find(v) == end) {
max_d = d; max_v = v;
}
}
avoid.insert(max_v);
store.push_back((r)?re[max_v]:max_v);
}}
// Finally handling of the other vertices:
if (is_direct(t)) {
C1 exclude_col = 2;
for (const auto v : store) {
for (C1 col = exclude_col; col < m; ++col)
*out << " " << var_d<C2>(v,m,col) << " 0";
++exclude_col;
}
*out << "\n";
} else {
C1 include_col = 1;
for (const auto v : store) {
for (C1 col = 0; col <= include_col; ++col)
*out << " " << var_n<C2>(v,m,col);
*out << " 0";
++include_col;
}
*out << "\n";
}
}
// Translating the clauses:
if (is_direct(t))
for (const auto& H : G) {
for (C1 col = 0; col < m; ++col) {
for (const auto v : H) *out << var_d<C2>((r)?re[v]:v,m,col) << " ";
*out << "0"; if (col != m-1) *out << " ";
}
*out << "\n";
}
else {
for (const auto& H : G) {
for (C1 col = 0; col < m; ++col) {
for (const auto v : H) {lits_n<C2>((r)?re[v]:v,m,col,out); *out << " ";}
*out << "0"; if (col != m-1) *out << " ";
}
*out << "\n";
}
}
// Adding the support-clauses:
if (is_direct(t)) {
for (C1 i = 1; i < deg.size(); ++i)
if (deg[i] != 0) {
const auto ir = (r)?re[i]:i;
// ALO:
for (C1 col = 0; col < m; ++col)
*out << "-" << var_d<C2>(ir,m,col) << " ";
*out << "0";
// AMO:
if (t == Type::direct_strong)
for (C1 col1 = 0; col1 < m; ++col1)
for (C1 col2 = col1+1; col2 < m; ++col2)
*out << " " << var_d<C2>(ir,m,col1) << " " <<
var_d<C2>(ir,m,col2) << " 0";
*out << "\n";
}
}
else if (t == Type::nested_strong) {
// AMO:
for (C1 i = 1; i < deg.size(); ++i)