* of any interest to the user of the library. */
/*
- * GiNaC Copyright (C) 1999-2016 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
}
}
+/** Base class for generating all bounded combinatorial partitions of an integer
+ * n with exactly m parts in non-decreasing order.
+ */
+class basic_partition_generator {
+protected:
+ // Partitions n into m parts, not including zero parts.
+ // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
+ // FXT library)
+ struct mpartition2
+ {
+ // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
+ std::vector<unsigned> x;
+ unsigned n; // n>0
+ unsigned m; // 0<m<=n
+ mpartition2(unsigned n_, unsigned m_)
+ : x(m_+1), n(n_), m(m_)
+ {
+ for (unsigned k=1; k<m; ++k)
+ x[k] = 1;
+ x[m] = n - m + 1;
+ }
+ bool next_partition()
+ {
+ unsigned u = x[m]; // last element
+ unsigned k = m;
+ unsigned s = u;
+ while (--k) {
+ s += x[k];
+ if (x[k] + 2 <= u)
+ break;
+ }
+ if (k==0)
+ return false; // current is last
+ unsigned f = x[k] + 1;
+ while (k < m) {
+ x[k] = f;
+ s -= f;
+ ++k;
+ }
+ x[m] = s;
+ return true;
+ }
+ };
+ mpartition2 mpgen;
+ basic_partition_generator(unsigned n_, unsigned m_)
+ : mpgen(n_, m_)
+ { }
+};
+
+/** Generate all bounded combinatorial partitions of an integer n with exactly
+ * m parts (including zero parts) in non-decreasing order.
+ */
+class partition_with_zero_parts_generator : public basic_partition_generator {
+private:
+ unsigned m; // number of parts 0<m
+ mutable std::vector<unsigned> partition; // current partition
+ mutable bool current_updated; // whether partition vector has been updated
+public:
+ partition_with_zero_parts_generator(unsigned n_, unsigned m_)
+ : basic_partition_generator(n_, 1), m(m_), partition(m_), current_updated(false)
+ { }
+ // returns current partition in non-decreasing order, padded with zeros
+ const std::vector<unsigned>& get() const
+ {
+ if (!current_updated) {
+ for (unsigned i = 0; i < m - mpgen.m; ++i)
+ partition[i] = 0; // pad with zeros
+
+ for (unsigned i = m - mpgen.m; i < m; ++i)
+ partition[i] = mpgen.x[i - m + mpgen.m + 1];
+
+ current_updated = true;
+ }
+ return partition;
+ }
+ bool next()
+ {
+ current_updated = false;
+ if (!mpgen.next_partition()) {
+ if (mpgen.m == m || mpgen.m == mpgen.n)
+ return false; // current is last
+ // increment number of parts
+ mpgen = mpartition2(mpgen.n, mpgen.m + 1);
+ }
+ return true;
+ }
+};
+
+/** Generate all bounded combinatorial partitions of an integer n with exactly
+ * m parts (not including zero parts) in non-decreasing order.
+ */
+class partition_generator : public basic_partition_generator {
+private:
+ mutable std::vector<unsigned> partition; // current partition
+ mutable bool current_updated; // whether partition vector has been updated
+public:
+ partition_generator(unsigned n_, unsigned m_)
+ : basic_partition_generator(n_, m_), partition(m_), current_updated(false)
+ { }
+ // returns current partition in non-decreasing order, padded with zeros
+ const std::vector<unsigned>& get() const
+ {
+ if (!current_updated) {
+ for (unsigned i = 0; i < mpgen.m; ++i)
+ partition[i] = mpgen.x[i + 1];
+
+ current_updated = true;
+ }
+ return partition;
+ }
+ bool next()
+ {
+ current_updated = false;
+ return mpgen.next_partition();
+ }
+};
+
+/** Generate all compositions of a partition of an integer n, starting with the
+ * compositions which has non-decreasing order.
+ */
+class composition_generator {
+private:
+ // Generates all distinct permutations of a multiset.
+ // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
+ // Multiset Permutations using a Constant Number of Variables by Prefix
+ // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
+ struct coolmulti {
+ // element of singly linked list
+ struct element {
+ unsigned value;
+ element* next;
+ element(unsigned val, element* n)
+ : value(val), next(n) {}
+ ~element()
+ { // recurses down to the end of the singly linked list
+ delete next;
+ }
+ };
+ element *head, *i, *after_i;
+ // NB: Partition must be sorted in non-decreasing order.
+ explicit coolmulti(const std::vector<unsigned>& partition)
+ : head(nullptr), i(nullptr), after_i(nullptr)
+ {
+ for (unsigned n = 0; n < partition.size(); ++n) {
+ head = new element(partition[n], head);
+ if (n <= 1)
+ i = head;
+ }
+ after_i = i->next;
+ }
+ ~coolmulti()
+ { // deletes singly linked list
+ delete head;
+ }
+ void next_permutation()
+ {
+ element *before_k;
+ if (after_i->next != nullptr && i->value >= after_i->next->value)
+ before_k = after_i;
+ else
+ before_k = i;
+ element *k = before_k->next;
+ before_k->next = k->next;
+ k->next = head;
+ if (k->value < head->value)
+ i = k;
+ after_i = i->next;
+ head = k;
+ }
+ bool finished() const
+ {
+ return after_i->next == nullptr && after_i->value >= head->value;
+ }
+ } cmgen;
+ bool atend; // needed for simplifying iteration over permutations
+ bool trivial; // likewise, true if all elements are equal
+ mutable std::vector<unsigned> composition; // current compositions
+ mutable bool current_updated; // whether composition vector has been updated
+public:
+ explicit composition_generator(const std::vector<unsigned>& partition)
+ : cmgen(partition), atend(false), trivial(true), composition(partition.size()), current_updated(false)
+ {
+ for (unsigned i=1; i<partition.size(); ++i)
+ trivial = trivial && (partition[0] == partition[i]);
+ }
+ const std::vector<unsigned>& get() const
+ {
+ if (!current_updated) {
+ coolmulti::element* it = cmgen.head;
+ size_t i = 0;
+ while (it != nullptr) {
+ composition[i] = it->value;
+ it = it->next;
+ ++i;
+ }
+ current_updated = true;
+ }
+ return composition;
+ }
+ bool next()
+ {
+ // This ugly contortion is needed because the original coolmulti
+ // algorithm requires code duplication of the payload procedure,
+ // one before the loop and one inside it.
+ if (trivial || atend)
+ return false;
+ cmgen.next_permutation();
+ current_updated = false;
+ atend = cmgen.finished();
+ return true;
+ }
+};
+
+/** Compute the multinomial coefficient n!/(p1!*p2!*...*pk!) where
+ * n = p1+p2+...+pk, i.e. p is a partition of n.
+ */
+const numeric
+multinomial_coefficient(const std::vector<unsigned> & p);
+
// Collection of `construct on first use' wrappers for safely avoiding
// internal object replication without running into the `static
git://www.ginac.de / ginac.git / blobdiff