1154 lines
42 KiB
Python
1154 lines
42 KiB
Python
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"""
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Algorithms and classes to support enumerative combinatorics.
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Currently just multiset partitions, but more could be added.
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Terminology (following Knuth, algorithm 7.1.2.5M TAOCP)
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*multiset* aaabbcccc has a *partition* aaabc | bccc
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The submultisets, aaabc and bccc of the partition are called
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*parts*, or sometimes *vectors*. (Knuth notes that multiset
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partitions can be thought of as partitions of vectors of integers,
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where the ith element of the vector gives the multiplicity of
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element i.)
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The values a, b and c are *components* of the multiset. These
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correspond to elements of a set, but in a multiset can be present
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with a multiplicity greater than 1.
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The algorithm deserves some explanation.
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Think of the part aaabc from the multiset above. If we impose an
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ordering on the components of the multiset, we can represent a part
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with a vector, in which the value of the first element of the vector
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corresponds to the multiplicity of the first component in that
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part. Thus, aaabc can be represented by the vector [3, 1, 1]. We
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can also define an ordering on parts, based on the lexicographic
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ordering of the vector (leftmost vector element, i.e., the element
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with the smallest component number, is the most significant), so
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that [3, 1, 1] > [3, 1, 0] and [3, 1, 1] > [2, 1, 4]. The ordering
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on parts can be extended to an ordering on partitions: First, sort
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the parts in each partition, left-to-right in decreasing order. Then
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partition A is greater than partition B if A's leftmost/greatest
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part is greater than B's leftmost part. If the leftmost parts are
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equal, compare the second parts, and so on.
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In this ordering, the greatest partition of a given multiset has only
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one part. The least partition is the one in which the components
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are spread out, one per part.
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The enumeration algorithms in this file yield the partitions of the
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argument multiset in decreasing order. The main data structure is a
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stack of parts, corresponding to the current partition. An
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important invariant is that the parts on the stack are themselves in
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decreasing order. This data structure is decremented to find the
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next smaller partition. Most often, decrementing the partition will
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only involve adjustments to the smallest parts at the top of the
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stack, much as adjacent integers *usually* differ only in their last
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few digits.
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Knuth's algorithm uses two main operations on parts:
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Decrement - change the part so that it is smaller in the
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(vector) lexicographic order, but reduced by the smallest amount possible.
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For example, if the multiset has vector [5,
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3, 1], and the bottom/greatest part is [4, 2, 1], this part would
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decrement to [4, 2, 0], while [4, 0, 0] would decrement to [3, 3,
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1]. A singleton part is never decremented -- [1, 0, 0] is not
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decremented to [0, 3, 1]. Instead, the decrement operator needs
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to fail for this case. In Knuth's pseudocode, the decrement
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operator is step m5.
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Spread unallocated multiplicity - Once a part has been decremented,
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it cannot be the rightmost part in the partition. There is some
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multiplicity that has not been allocated, and new parts must be
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created above it in the stack to use up this multiplicity. To
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maintain the invariant that the parts on the stack are in
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decreasing order, these new parts must be less than or equal to
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the decremented part.
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For example, if the multiset is [5, 3, 1], and its most
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significant part has just been decremented to [5, 3, 0], the
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spread operation will add a new part so that the stack becomes
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[[5, 3, 0], [0, 0, 1]]. If the most significant part (for the
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same multiset) has been decremented to [2, 0, 0] the stack becomes
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[[2, 0, 0], [2, 0, 0], [1, 3, 1]]. In the pseudocode, the spread
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operation for one part is step m2. The complete spread operation
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is a loop of steps m2 and m3.
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In order to facilitate the spread operation, Knuth stores, for each
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component of each part, not just the multiplicity of that component
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in the part, but also the total multiplicity available for this
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component in this part or any lesser part above it on the stack.
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One added twist is that Knuth does not represent the part vectors as
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arrays. Instead, he uses a sparse representation, in which a
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component of a part is represented as a component number (c), plus
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the multiplicity of the component in that part (v) as well as the
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total multiplicity available for that component (u). This saves
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time that would be spent skipping over zeros.
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"""
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class PartComponent:
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"""Internal class used in support of the multiset partitions
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enumerators and the associated visitor functions.
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Represents one component of one part of the current partition.
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A stack of these, plus an auxiliary frame array, f, represents a
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partition of the multiset.
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Knuth's pseudocode makes c, u, and v separate arrays.
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"""
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__slots__ = ('c', 'u', 'v')
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def __init__(self):
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self.c = 0 # Component number
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self.u = 0 # The as yet unpartitioned amount in component c
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# *before* it is allocated by this triple
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self.v = 0 # Amount of c component in the current part
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# (v<=u). An invariant of the representation is
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# that the next higher triple for this component
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# (if there is one) will have a value of u-v in
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# its u attribute.
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def __repr__(self):
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"for debug/algorithm animation purposes"
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return 'c:%d u:%d v:%d' % (self.c, self.u, self.v)
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def __eq__(self, other):
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"""Define value oriented equality, which is useful for testers"""
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return (isinstance(other, self.__class__) and
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self.c == other.c and
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self.u == other.u and
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self.v == other.v)
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def __ne__(self, other):
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"""Defined for consistency with __eq__"""
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return not self == other
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# This function tries to be a faithful implementation of algorithm
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# 7.1.2.5M in Volume 4A, Combinatoral Algorithms, Part 1, of The Art
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# of Computer Programming, by Donald Knuth. This includes using
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# (mostly) the same variable names, etc. This makes for rather
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# low-level Python.
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# Changes from Knuth's pseudocode include
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# - use PartComponent struct/object instead of 3 arrays
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# - make the function a generator
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# - map (with some difficulty) the GOTOs to Python control structures.
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# - Knuth uses 1-based numbering for components, this code is 0-based
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# - renamed variable l to lpart.
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# - flag variable x takes on values True/False instead of 1/0
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#
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def multiset_partitions_taocp(multiplicities):
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"""Enumerates partitions of a multiset.
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Parameters
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==========
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multiplicities
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list of integer multiplicities of the components of the multiset.
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Yields
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======
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state
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Internal data structure which encodes a particular partition.
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This output is then usually processed by a visitor function
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which combines the information from this data structure with
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the components themselves to produce an actual partition.
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Unless they wish to create their own visitor function, users will
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have little need to look inside this data structure. But, for
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reference, it is a 3-element list with components:
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f
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is a frame array, which is used to divide pstack into parts.
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lpart
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points to the base of the topmost part.
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pstack
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is an array of PartComponent objects.
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The ``state`` output offers a peek into the internal data
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structures of the enumeration function. The client should
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treat this as read-only; any modification of the data
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structure will cause unpredictable (and almost certainly
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incorrect) results. Also, the components of ``state`` are
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modified in place at each iteration. Hence, the visitor must
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be called at each loop iteration. Accumulating the ``state``
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instances and processing them later will not work.
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Examples
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========
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>>> from sympy.utilities.enumerative import list_visitor
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>>> from sympy.utilities.enumerative import multiset_partitions_taocp
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>>> # variables components and multiplicities represent the multiset 'abb'
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>>> components = 'ab'
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>>> multiplicities = [1, 2]
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>>> states = multiset_partitions_taocp(multiplicities)
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>>> list(list_visitor(state, components) for state in states)
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[[['a', 'b', 'b']],
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[['a', 'b'], ['b']],
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[['a'], ['b', 'b']],
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[['a'], ['b'], ['b']]]
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See Also
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========
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sympy.utilities.iterables.multiset_partitions: Takes a multiset
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as input and directly yields multiset partitions. It
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dispatches to a number of functions, including this one, for
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implementation. Most users will find it more convenient to
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use than multiset_partitions_taocp.
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"""
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# Important variables.
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# m is the number of components, i.e., number of distinct elements
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m = len(multiplicities)
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# n is the cardinality, total number of elements whether or not distinct
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n = sum(multiplicities)
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# The main data structure, f segments pstack into parts. See
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# list_visitor() for example code indicating how this internal
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# state corresponds to a partition.
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# Note: allocation of space for stack is conservative. Knuth's
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# exercise 7.2.1.5.68 gives some indication of how to tighten this
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# bound, but this is not implemented.
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pstack = [PartComponent() for i in range(n * m + 1)]
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f = [0] * (n + 1)
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# Step M1 in Knuth (Initialize)
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# Initial state - entire multiset in one part.
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for j in range(m):
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ps = pstack[j]
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ps.c = j
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ps.u = multiplicities[j]
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ps.v = multiplicities[j]
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# Other variables
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f[0] = 0
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a = 0
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lpart = 0
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f[1] = m
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b = m # in general, current stack frame is from a to b - 1
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while True:
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while True:
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# Step M2 (Subtract v from u)
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j = a
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k = b
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x = False
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while j < b:
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pstack[k].u = pstack[j].u - pstack[j].v
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if pstack[k].u == 0:
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x = True
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elif not x:
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pstack[k].c = pstack[j].c
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pstack[k].v = min(pstack[j].v, pstack[k].u)
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x = pstack[k].u < pstack[j].v
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k = k + 1
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else: # x is True
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pstack[k].c = pstack[j].c
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pstack[k].v = pstack[k].u
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k = k + 1
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j = j + 1
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# Note: x is True iff v has changed
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# Step M3 (Push if nonzero.)
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if k > b:
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a = b
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b = k
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lpart = lpart + 1
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f[lpart + 1] = b
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# Return to M2
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else:
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break # Continue to M4
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# M4 Visit a partition
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state = [f, lpart, pstack]
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yield state
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# M5 (Decrease v)
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while True:
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j = b-1
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while (pstack[j].v == 0):
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j = j - 1
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if j == a and pstack[j].v == 1:
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# M6 (Backtrack)
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if lpart == 0:
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return
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lpart = lpart - 1
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b = a
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a = f[lpart]
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# Return to M5
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else:
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pstack[j].v = pstack[j].v - 1
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for k in range(j + 1, b):
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pstack[k].v = pstack[k].u
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break # GOTO M2
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# --------------- Visitor functions for multiset partitions ---------------
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# A visitor takes the partition state generated by
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# multiset_partitions_taocp or other enumerator, and produces useful
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# output (such as the actual partition).
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def factoring_visitor(state, primes):
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"""Use with multiset_partitions_taocp to enumerate the ways a
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number can be expressed as a product of factors. For this usage,
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the exponents of the prime factors of a number are arguments to
|
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the partition enumerator, while the corresponding prime factors
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are input here.
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Examples
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========
|
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To enumerate the factorings of a number we can think of the elements of the
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partition as being the prime factors and the multiplicities as being their
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exponents.
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>>> from sympy.utilities.enumerative import factoring_visitor
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>>> from sympy.utilities.enumerative import multiset_partitions_taocp
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>>> from sympy import factorint
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>>> primes, multiplicities = zip(*factorint(24).items())
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>>> primes
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(2, 3)
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>>> multiplicities
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(3, 1)
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>>> states = multiset_partitions_taocp(multiplicities)
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>>> list(factoring_visitor(state, primes) for state in states)
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[[24], [8, 3], [12, 2], [4, 6], [4, 2, 3], [6, 2, 2], [2, 2, 2, 3]]
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"""
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f, lpart, pstack = state
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factoring = []
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for i in range(lpart + 1):
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factor = 1
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for ps in pstack[f[i]: f[i + 1]]:
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if ps.v > 0:
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factor *= primes[ps.c] ** ps.v
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factoring.append(factor)
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return factoring
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def list_visitor(state, components):
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"""Return a list of lists to represent the partition.
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Examples
|
||
|
|
========
|
||
|
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|
||
|
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>>> from sympy.utilities.enumerative import list_visitor
|
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|
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>>> from sympy.utilities.enumerative import multiset_partitions_taocp
|
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|
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>>> states = multiset_partitions_taocp([1, 2, 1])
|
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>>> s = next(states)
|
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>>> list_visitor(s, 'abc') # for multiset 'a b b c'
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[['a', 'b', 'b', 'c']]
|
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>>> s = next(states)
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>>> list_visitor(s, [1, 2, 3]) # for multiset '1 2 2 3
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[[1, 2, 2], [3]]
|
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|
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"""
|
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|
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f, lpart, pstack = state
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partition = []
|
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for i in range(lpart+1):
|
||
|
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part = []
|
||
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for ps in pstack[f[i]:f[i+1]]:
|
||
|
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if ps.v > 0:
|
||
|
|
part.extend([components[ps.c]] * ps.v)
|
||
|
|
partition.append(part)
|
||
|
|
|
||
|
|
return partition
|
||
|
|
|
||
|
|
|
||
|
|
class MultisetPartitionTraverser():
|
||
|
|
"""
|
||
|
|
Has methods to ``enumerate`` and ``count`` the partitions of a multiset.
|
||
|
|
|
||
|
|
This implements a refactored and extended version of Knuth's algorithm
|
||
|
|
7.1.2.5M [AOCP]_."
|
||
|
|
|
||
|
|
The enumeration methods of this class are generators and return
|
||
|
|
data structures which can be interpreted by the same visitor
|
||
|
|
functions used for the output of ``multiset_partitions_taocp``.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.utilities.enumerative import MultisetPartitionTraverser
|
||
|
|
>>> m = MultisetPartitionTraverser()
|
||
|
|
>>> m.count_partitions([4,4,4,2])
|
||
|
|
127750
|
||
|
|
>>> m.count_partitions([3,3,3])
|
||
|
|
686
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
multiset_partitions_taocp
|
||
|
|
sympy.utilities.iterables.multiset_partitions
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [AOCP] Algorithm 7.1.2.5M in Volume 4A, Combinatoral Algorithms,
|
||
|
|
Part 1, of The Art of Computer Programming, by Donald Knuth.
|
||
|
|
|
||
|
|
.. [Factorisatio] On a Problem of Oppenheim concerning
|
||
|
|
"Factorisatio Numerorum" E. R. Canfield, Paul Erdos, Carl
|
||
|
|
Pomerance, JOURNAL OF NUMBER THEORY, Vol. 17, No. 1. August
|
||
|
|
1983. See section 7 for a description of an algorithm
|
||
|
|
similar to Knuth's.
|
||
|
|
|
||
|
|
.. [Yorgey] Generating Multiset Partitions, Brent Yorgey, The
|
||
|
|
Monad.Reader, Issue 8, September 2007.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __init__(self):
|
||
|
|
self.debug = False
|
||
|
|
# TRACING variables. These are useful for gathering
|
||
|
|
# statistics on the algorithm itself, but have no particular
|
||
|
|
# benefit to a user of the code.
|
||
|
|
self.k1 = 0
|
||
|
|
self.k2 = 0
|
||
|
|
self.p1 = 0
|
||
|
|
|
||
|
|
def db_trace(self, msg):
|
||
|
|
"""Useful for understanding/debugging the algorithms. Not
|
||
|
|
generally activated in end-user code."""
|
||
|
|
if self.debug:
|
||
|
|
# XXX: animation_visitor is undefined... Clearly this does not
|
||
|
|
# work and was not tested. Previous code in comments below.
|
||
|
|
raise RuntimeError
|
||
|
|
#letters = 'abcdefghijklmnopqrstuvwxyz'
|
||
|
|
#state = [self.f, self.lpart, self.pstack]
|
||
|
|
#print("DBG:", msg,
|
||
|
|
# ["".join(part) for part in list_visitor(state, letters)],
|
||
|
|
# animation_visitor(state))
|
||
|
|
|
||
|
|
#
|
||
|
|
# Helper methods for enumeration
|
||
|
|
#
|
||
|
|
def _initialize_enumeration(self, multiplicities):
|
||
|
|
"""Allocates and initializes the partition stack.
|
||
|
|
|
||
|
|
This is called from the enumeration/counting routines, so
|
||
|
|
there is no need to call it separately."""
|
||
|
|
|
||
|
|
num_components = len(multiplicities)
|
||
|
|
# cardinality is the total number of elements, whether or not distinct
|
||
|
|
cardinality = sum(multiplicities)
|
||
|
|
|
||
|
|
# pstack is the partition stack, which is segmented by
|
||
|
|
# f into parts.
|
||
|
|
self.pstack = [PartComponent() for i in
|
||
|
|
range(num_components * cardinality + 1)]
|
||
|
|
self.f = [0] * (cardinality + 1)
|
||
|
|
|
||
|
|
# Initial state - entire multiset in one part.
|
||
|
|
for j in range(num_components):
|
||
|
|
ps = self.pstack[j]
|
||
|
|
ps.c = j
|
||
|
|
ps.u = multiplicities[j]
|
||
|
|
ps.v = multiplicities[j]
|
||
|
|
|
||
|
|
self.f[0] = 0
|
||
|
|
self.f[1] = num_components
|
||
|
|
self.lpart = 0
|
||
|
|
|
||
|
|
# The decrement_part() method corresponds to step M5 in Knuth's
|
||
|
|
# algorithm. This is the base version for enum_all(). Modified
|
||
|
|
# versions of this method are needed if we want to restrict
|
||
|
|
# sizes of the partitions produced.
|
||
|
|
def decrement_part(self, part):
|
||
|
|
"""Decrements part (a subrange of pstack), if possible, returning
|
||
|
|
True iff the part was successfully decremented.
|
||
|
|
|
||
|
|
If you think of the v values in the part as a multi-digit
|
||
|
|
integer (least significant digit on the right) this is
|
||
|
|
basically decrementing that integer, but with the extra
|
||
|
|
constraint that the leftmost digit cannot be decremented to 0.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
part
|
||
|
|
The part, represented as a list of PartComponent objects,
|
||
|
|
which is to be decremented.
|
||
|
|
|
||
|
|
"""
|
||
|
|
plen = len(part)
|
||
|
|
for j in range(plen - 1, -1, -1):
|
||
|
|
if j == 0 and part[j].v > 1 or j > 0 and part[j].v > 0:
|
||
|
|
# found val to decrement
|
||
|
|
part[j].v -= 1
|
||
|
|
# Reset trailing parts back to maximum
|
||
|
|
for k in range(j + 1, plen):
|
||
|
|
part[k].v = part[k].u
|
||
|
|
return True
|
||
|
|
return False
|
||
|
|
|
||
|
|
# Version to allow number of parts to be bounded from above.
|
||
|
|
# Corresponds to (a modified) step M5.
|
||
|
|
def decrement_part_small(self, part, ub):
|
||
|
|
"""Decrements part (a subrange of pstack), if possible, returning
|
||
|
|
True iff the part was successfully decremented.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
part
|
||
|
|
part to be decremented (topmost part on the stack)
|
||
|
|
|
||
|
|
ub
|
||
|
|
the maximum number of parts allowed in a partition
|
||
|
|
returned by the calling traversal.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
The goal of this modification of the ordinary decrement method
|
||
|
|
is to fail (meaning that the subtree rooted at this part is to
|
||
|
|
be skipped) when it can be proved that this part can only have
|
||
|
|
child partitions which are larger than allowed by ``ub``. If a
|
||
|
|
decision is made to fail, it must be accurate, otherwise the
|
||
|
|
enumeration will miss some partitions. But, it is OK not to
|
||
|
|
capture all the possible failures -- if a part is passed that
|
||
|
|
shouldn't be, the resulting too-large partitions are filtered
|
||
|
|
by the enumeration one level up. However, as is usual in
|
||
|
|
constrained enumerations, failing early is advantageous.
|
||
|
|
|
||
|
|
The tests used by this method catch the most common cases,
|
||
|
|
although this implementation is by no means the last word on
|
||
|
|
this problem. The tests include:
|
||
|
|
|
||
|
|
1) ``lpart`` must be less than ``ub`` by at least 2. This is because
|
||
|
|
once a part has been decremented, the partition
|
||
|
|
will gain at least one child in the spread step.
|
||
|
|
|
||
|
|
2) If the leading component of the part is about to be
|
||
|
|
decremented, check for how many parts will be added in
|
||
|
|
order to use up the unallocated multiplicity in that
|
||
|
|
leading component, and fail if this number is greater than
|
||
|
|
allowed by ``ub``. (See code for the exact expression.) This
|
||
|
|
test is given in the answer to Knuth's problem 7.2.1.5.69.
|
||
|
|
|
||
|
|
3) If there is *exactly* enough room to expand the leading
|
||
|
|
component by the above test, check the next component (if
|
||
|
|
it exists) once decrementing has finished. If this has
|
||
|
|
``v == 0``, this next component will push the expansion over the
|
||
|
|
limit by 1, so fail.
|
||
|
|
"""
|
||
|
|
if self.lpart >= ub - 1:
|
||
|
|
self.p1 += 1 # increment to keep track of usefulness of tests
|
||
|
|
return False
|
||
|
|
plen = len(part)
|
||
|
|
for j in range(plen - 1, -1, -1):
|
||
|
|
# Knuth's mod, (answer to problem 7.2.1.5.69)
|
||
|
|
if j == 0 and (part[0].v - 1)*(ub - self.lpart) < part[0].u:
|
||
|
|
self.k1 += 1
|
||
|
|
return False
|
||
|
|
|
||
|
|
if j == 0 and part[j].v > 1 or j > 0 and part[j].v > 0:
|
||
|
|
# found val to decrement
|
||
|
|
part[j].v -= 1
|
||
|
|
# Reset trailing parts back to maximum
|
||
|
|
for k in range(j + 1, plen):
|
||
|
|
part[k].v = part[k].u
|
||
|
|
|
||
|
|
# Have now decremented part, but are we doomed to
|
||
|
|
# failure when it is expanded? Check one oddball case
|
||
|
|
# that turns out to be surprisingly common - exactly
|
||
|
|
# enough room to expand the leading component, but no
|
||
|
|
# room for the second component, which has v=0.
|
||
|
|
if (plen > 1 and part[1].v == 0 and
|
||
|
|
(part[0].u - part[0].v) ==
|
||
|
|
((ub - self.lpart - 1) * part[0].v)):
|
||
|
|
self.k2 += 1
|
||
|
|
self.db_trace("Decrement fails test 3")
|
||
|
|
return False
|
||
|
|
return True
|
||
|
|
return False
|
||
|
|
|
||
|
|
def decrement_part_large(self, part, amt, lb):
|
||
|
|
"""Decrements part, while respecting size constraint.
|
||
|
|
|
||
|
|
A part can have no children which are of sufficient size (as
|
||
|
|
indicated by ``lb``) unless that part has sufficient
|
||
|
|
unallocated multiplicity. When enforcing the size constraint,
|
||
|
|
this method will decrement the part (if necessary) by an
|
||
|
|
amount needed to ensure sufficient unallocated multiplicity.
|
||
|
|
|
||
|
|
Returns True iff the part was successfully decremented.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
part
|
||
|
|
part to be decremented (topmost part on the stack)
|
||
|
|
|
||
|
|
amt
|
||
|
|
Can only take values 0 or 1. A value of 1 means that the
|
||
|
|
part must be decremented, and then the size constraint is
|
||
|
|
enforced. A value of 0 means just to enforce the ``lb``
|
||
|
|
size constraint.
|
||
|
|
|
||
|
|
lb
|
||
|
|
The partitions produced by the calling enumeration must
|
||
|
|
have more parts than this value.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if amt == 1:
|
||
|
|
# In this case we always need to increment, *before*
|
||
|
|
# enforcing the "sufficient unallocated multiplicity"
|
||
|
|
# constraint. Easiest for this is just to call the
|
||
|
|
# regular decrement method.
|
||
|
|
if not self.decrement_part(part):
|
||
|
|
return False
|
||
|
|
|
||
|
|
# Next, perform any needed additional decrementing to respect
|
||
|
|
# "sufficient unallocated multiplicity" (or fail if this is
|
||
|
|
# not possible).
|
||
|
|
min_unalloc = lb - self.lpart
|
||
|
|
if min_unalloc <= 0:
|
||
|
|
return True
|
||
|
|
total_mult = sum(pc.u for pc in part)
|
||
|
|
total_alloc = sum(pc.v for pc in part)
|
||
|
|
if total_mult <= min_unalloc:
|
||
|
|
return False
|
||
|
|
|
||
|
|
deficit = min_unalloc - (total_mult - total_alloc)
|
||
|
|
if deficit <= 0:
|
||
|
|
return True
|
||
|
|
|
||
|
|
for i in range(len(part) - 1, -1, -1):
|
||
|
|
if i == 0:
|
||
|
|
if part[0].v > deficit:
|
||
|
|
part[0].v -= deficit
|
||
|
|
return True
|
||
|
|
else:
|
||
|
|
return False # This shouldn't happen, due to above check
|
||
|
|
else:
|
||
|
|
if part[i].v >= deficit:
|
||
|
|
part[i].v -= deficit
|
||
|
|
return True
|
||
|
|
else:
|
||
|
|
deficit -= part[i].v
|
||
|
|
part[i].v = 0
|
||
|
|
|
||
|
|
def decrement_part_range(self, part, lb, ub):
|
||
|
|
"""Decrements part (a subrange of pstack), if possible, returning
|
||
|
|
True iff the part was successfully decremented.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
part
|
||
|
|
part to be decremented (topmost part on the stack)
|
||
|
|
|
||
|
|
ub
|
||
|
|
the maximum number of parts allowed in a partition
|
||
|
|
returned by the calling traversal.
|
||
|
|
|
||
|
|
lb
|
||
|
|
The partitions produced by the calling enumeration must
|
||
|
|
have more parts than this value.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
Combines the constraints of _small and _large decrement
|
||
|
|
methods. If returns success, part has been decremented at
|
||
|
|
least once, but perhaps by quite a bit more if needed to meet
|
||
|
|
the lb constraint.
|
||
|
|
"""
|
||
|
|
|
||
|
|
# Constraint in the range case is just enforcing both the
|
||
|
|
# constraints from _small and _large cases. Note the 0 as the
|
||
|
|
# second argument to the _large call -- this is the signal to
|
||
|
|
# decrement only as needed to for constraint enforcement. The
|
||
|
|
# short circuiting and left-to-right order of the 'and'
|
||
|
|
# operator is important for this to work correctly.
|
||
|
|
return self.decrement_part_small(part, ub) and \
|
||
|
|
self.decrement_part_large(part, 0, lb)
|
||
|
|
|
||
|
|
def spread_part_multiplicity(self):
|
||
|
|
"""Returns True if a new part has been created, and
|
||
|
|
adjusts pstack, f and lpart as needed.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
Spreads unallocated multiplicity from the current top part
|
||
|
|
into a new part created above the current on the stack. This
|
||
|
|
new part is constrained to be less than or equal to the old in
|
||
|
|
terms of the part ordering.
|
||
|
|
|
||
|
|
This call does nothing (and returns False) if the current top
|
||
|
|
part has no unallocated multiplicity.
|
||
|
|
|
||
|
|
"""
|
||
|
|
j = self.f[self.lpart] # base of current top part
|
||
|
|
k = self.f[self.lpart + 1] # ub of current; potential base of next
|
||
|
|
base = k # save for later comparison
|
||
|
|
|
||
|
|
changed = False # Set to true when the new part (so far) is
|
||
|
|
# strictly less than (as opposed to less than
|
||
|
|
# or equal) to the old.
|
||
|
|
for j in range(self.f[self.lpart], self.f[self.lpart + 1]):
|
||
|
|
self.pstack[k].u = self.pstack[j].u - self.pstack[j].v
|
||
|
|
if self.pstack[k].u == 0:
|
||
|
|
changed = True
|
||
|
|
else:
|
||
|
|
self.pstack[k].c = self.pstack[j].c
|
||
|
|
if changed: # Put all available multiplicity in this part
|
||
|
|
self.pstack[k].v = self.pstack[k].u
|
||
|
|
else: # Still maintaining ordering constraint
|
||
|
|
if self.pstack[k].u < self.pstack[j].v:
|
||
|
|
self.pstack[k].v = self.pstack[k].u
|
||
|
|
changed = True
|
||
|
|
else:
|
||
|
|
self.pstack[k].v = self.pstack[j].v
|
||
|
|
k = k + 1
|
||
|
|
if k > base:
|
||
|
|
# Adjust for the new part on stack
|
||
|
|
self.lpart = self.lpart + 1
|
||
|
|
self.f[self.lpart + 1] = k
|
||
|
|
return True
|
||
|
|
return False
|
||
|
|
|
||
|
|
def top_part(self):
|
||
|
|
"""Return current top part on the stack, as a slice of pstack.
|
||
|
|
|
||
|
|
"""
|
||
|
|
return self.pstack[self.f[self.lpart]:self.f[self.lpart + 1]]
|
||
|
|
|
||
|
|
# Same interface and functionality as multiset_partitions_taocp(),
|
||
|
|
# but some might find this refactored version easier to follow.
|
||
|
|
def enum_all(self, multiplicities):
|
||
|
|
"""Enumerate the partitions of a multiset.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.utilities.enumerative import list_visitor
|
||
|
|
>>> from sympy.utilities.enumerative import MultisetPartitionTraverser
|
||
|
|
>>> m = MultisetPartitionTraverser()
|
||
|
|
>>> states = m.enum_all([2,2])
|
||
|
|
>>> list(list_visitor(state, 'ab') for state in states)
|
||
|
|
[[['a', 'a', 'b', 'b']],
|
||
|
|
[['a', 'a', 'b'], ['b']],
|
||
|
|
[['a', 'a'], ['b', 'b']],
|
||
|
|
[['a', 'a'], ['b'], ['b']],
|
||
|
|
[['a', 'b', 'b'], ['a']],
|
||
|
|
[['a', 'b'], ['a', 'b']],
|
||
|
|
[['a', 'b'], ['a'], ['b']],
|
||
|
|
[['a'], ['a'], ['b', 'b']],
|
||
|
|
[['a'], ['a'], ['b'], ['b']]]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
multiset_partitions_taocp():
|
||
|
|
which provides the same result as this method, but is
|
||
|
|
about twice as fast. Hence, enum_all is primarily useful
|
||
|
|
for testing. Also see the function for a discussion of
|
||
|
|
states and visitors.
|
||
|
|
|
||
|
|
"""
|
||
|
|
self._initialize_enumeration(multiplicities)
|
||
|
|
while True:
|
||
|
|
while self.spread_part_multiplicity():
|
||
|
|
pass
|
||
|
|
|
||
|
|
# M4 Visit a partition
|
||
|
|
state = [self.f, self.lpart, self.pstack]
|
||
|
|
yield state
|
||
|
|
|
||
|
|
# M5 (Decrease v)
|
||
|
|
while not self.decrement_part(self.top_part()):
|
||
|
|
# M6 (Backtrack)
|
||
|
|
if self.lpart == 0:
|
||
|
|
return
|
||
|
|
self.lpart -= 1
|
||
|
|
|
||
|
|
def enum_small(self, multiplicities, ub):
|
||
|
|
"""Enumerate multiset partitions with no more than ``ub`` parts.
|
||
|
|
|
||
|
|
Equivalent to enum_range(multiplicities, 0, ub)
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
multiplicities
|
||
|
|
list of multiplicities of the components of the multiset.
|
||
|
|
|
||
|
|
ub
|
||
|
|
Maximum number of parts
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.utilities.enumerative import list_visitor
|
||
|
|
>>> from sympy.utilities.enumerative import MultisetPartitionTraverser
|
||
|
|
>>> m = MultisetPartitionTraverser()
|
||
|
|
>>> states = m.enum_small([2,2], 2)
|
||
|
|
>>> list(list_visitor(state, 'ab') for state in states)
|
||
|
|
[[['a', 'a', 'b', 'b']],
|
||
|
|
[['a', 'a', 'b'], ['b']],
|
||
|
|
[['a', 'a'], ['b', 'b']],
|
||
|
|
[['a', 'b', 'b'], ['a']],
|
||
|
|
[['a', 'b'], ['a', 'b']]]
|
||
|
|
|
||
|
|
The implementation is based, in part, on the answer given to
|
||
|
|
exercise 69, in Knuth [AOCP]_.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
enum_all, enum_large, enum_range
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
# Keep track of iterations which do not yield a partition.
|
||
|
|
# Clearly, we would like to keep this number small.
|
||
|
|
self.discarded = 0
|
||
|
|
if ub <= 0:
|
||
|
|
return
|
||
|
|
self._initialize_enumeration(multiplicities)
|
||
|
|
while True:
|
||
|
|
good_partition = True
|
||
|
|
while self.spread_part_multiplicity():
|
||
|
|
self.db_trace("spread 1")
|
||
|
|
if self.lpart >= ub:
|
||
|
|
self.discarded += 1
|
||
|
|
good_partition = False
|
||
|
|
self.db_trace(" Discarding")
|
||
|
|
self.lpart = ub - 2
|
||
|
|
break
|
||
|
|
|
||
|
|
# M4 Visit a partition
|
||
|
|
if good_partition:
|
||
|
|
state = [self.f, self.lpart, self.pstack]
|
||
|
|
yield state
|
||
|
|
|
||
|
|
# M5 (Decrease v)
|
||
|
|
while not self.decrement_part_small(self.top_part(), ub):
|
||
|
|
self.db_trace("Failed decrement, going to backtrack")
|
||
|
|
# M6 (Backtrack)
|
||
|
|
if self.lpart == 0:
|
||
|
|
return
|
||
|
|
self.lpart -= 1
|
||
|
|
self.db_trace("Backtracked to")
|
||
|
|
self.db_trace("decrement ok, about to expand")
|
||
|
|
|
||
|
|
def enum_large(self, multiplicities, lb):
|
||
|
|
"""Enumerate the partitions of a multiset with lb < num(parts)
|
||
|
|
|
||
|
|
Equivalent to enum_range(multiplicities, lb, sum(multiplicities))
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
multiplicities
|
||
|
|
list of multiplicities of the components of the multiset.
|
||
|
|
|
||
|
|
lb
|
||
|
|
Number of parts in the partition must be greater than
|
||
|
|
this lower bound.
|
||
|
|
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.utilities.enumerative import list_visitor
|
||
|
|
>>> from sympy.utilities.enumerative import MultisetPartitionTraverser
|
||
|
|
>>> m = MultisetPartitionTraverser()
|
||
|
|
>>> states = m.enum_large([2,2], 2)
|
||
|
|
>>> list(list_visitor(state, 'ab') for state in states)
|
||
|
|
[[['a', 'a'], ['b'], ['b']],
|
||
|
|
[['a', 'b'], ['a'], ['b']],
|
||
|
|
[['a'], ['a'], ['b', 'b']],
|
||
|
|
[['a'], ['a'], ['b'], ['b']]]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
enum_all, enum_small, enum_range
|
||
|
|
|
||
|
|
"""
|
||
|
|
self.discarded = 0
|
||
|
|
if lb >= sum(multiplicities):
|
||
|
|
return
|
||
|
|
self._initialize_enumeration(multiplicities)
|
||
|
|
self.decrement_part_large(self.top_part(), 0, lb)
|
||
|
|
while True:
|
||
|
|
good_partition = True
|
||
|
|
while self.spread_part_multiplicity():
|
||
|
|
if not self.decrement_part_large(self.top_part(), 0, lb):
|
||
|
|
# Failure here should be rare/impossible
|
||
|
|
self.discarded += 1
|
||
|
|
good_partition = False
|
||
|
|
break
|
||
|
|
|
||
|
|
# M4 Visit a partition
|
||
|
|
if good_partition:
|
||
|
|
state = [self.f, self.lpart, self.pstack]
|
||
|
|
yield state
|
||
|
|
|
||
|
|
# M5 (Decrease v)
|
||
|
|
while not self.decrement_part_large(self.top_part(), 1, lb):
|
||
|
|
# M6 (Backtrack)
|
||
|
|
if self.lpart == 0:
|
||
|
|
return
|
||
|
|
self.lpart -= 1
|
||
|
|
|
||
|
|
def enum_range(self, multiplicities, lb, ub):
|
||
|
|
|
||
|
|
"""Enumerate the partitions of a multiset with
|
||
|
|
``lb < num(parts) <= ub``.
|
||
|
|
|
||
|
|
In particular, if partitions with exactly ``k`` parts are
|
||
|
|
desired, call with ``(multiplicities, k - 1, k)``. This
|
||
|
|
method generalizes enum_all, enum_small, and enum_large.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.utilities.enumerative import list_visitor
|
||
|
|
>>> from sympy.utilities.enumerative import MultisetPartitionTraverser
|
||
|
|
>>> m = MultisetPartitionTraverser()
|
||
|
|
>>> states = m.enum_range([2,2], 1, 2)
|
||
|
|
>>> list(list_visitor(state, 'ab') for state in states)
|
||
|
|
[[['a', 'a', 'b'], ['b']],
|
||
|
|
[['a', 'a'], ['b', 'b']],
|
||
|
|
[['a', 'b', 'b'], ['a']],
|
||
|
|
[['a', 'b'], ['a', 'b']]]
|
||
|
|
|
||
|
|
"""
|
||
|
|
# combine the constraints of the _large and _small
|
||
|
|
# enumerations.
|
||
|
|
self.discarded = 0
|
||
|
|
if ub <= 0 or lb >= sum(multiplicities):
|
||
|
|
return
|
||
|
|
self._initialize_enumeration(multiplicities)
|
||
|
|
self.decrement_part_large(self.top_part(), 0, lb)
|
||
|
|
while True:
|
||
|
|
good_partition = True
|
||
|
|
while self.spread_part_multiplicity():
|
||
|
|
self.db_trace("spread 1")
|
||
|
|
if not self.decrement_part_large(self.top_part(), 0, lb):
|
||
|
|
# Failure here - possible in range case?
|
||
|
|
self.db_trace(" Discarding (large cons)")
|
||
|
|
self.discarded += 1
|
||
|
|
good_partition = False
|
||
|
|
break
|
||
|
|
elif self.lpart >= ub:
|
||
|
|
self.discarded += 1
|
||
|
|
good_partition = False
|
||
|
|
self.db_trace(" Discarding small cons")
|
||
|
|
self.lpart = ub - 2
|
||
|
|
break
|
||
|
|
|
||
|
|
# M4 Visit a partition
|
||
|
|
if good_partition:
|
||
|
|
state = [self.f, self.lpart, self.pstack]
|
||
|
|
yield state
|
||
|
|
|
||
|
|
# M5 (Decrease v)
|
||
|
|
while not self.decrement_part_range(self.top_part(), lb, ub):
|
||
|
|
self.db_trace("Failed decrement, going to backtrack")
|
||
|
|
# M6 (Backtrack)
|
||
|
|
if self.lpart == 0:
|
||
|
|
return
|
||
|
|
self.lpart -= 1
|
||
|
|
self.db_trace("Backtracked to")
|
||
|
|
self.db_trace("decrement ok, about to expand")
|
||
|
|
|
||
|
|
def count_partitions_slow(self, multiplicities):
|
||
|
|
"""Returns the number of partitions of a multiset whose elements
|
||
|
|
have the multiplicities given in ``multiplicities``.
|
||
|
|
|
||
|
|
Primarily for comparison purposes. It follows the same path as
|
||
|
|
enumerate, and counts, rather than generates, the partitions.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
count_partitions
|
||
|
|
Has the same calling interface, but is much faster.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# number of partitions so far in the enumeration
|
||
|
|
self.pcount = 0
|
||
|
|
self._initialize_enumeration(multiplicities)
|
||
|
|
while True:
|
||
|
|
while self.spread_part_multiplicity():
|
||
|
|
pass
|
||
|
|
|
||
|
|
# M4 Visit (count) a partition
|
||
|
|
self.pcount += 1
|
||
|
|
|
||
|
|
# M5 (Decrease v)
|
||
|
|
while not self.decrement_part(self.top_part()):
|
||
|
|
# M6 (Backtrack)
|
||
|
|
if self.lpart == 0:
|
||
|
|
return self.pcount
|
||
|
|
self.lpart -= 1
|
||
|
|
|
||
|
|
def count_partitions(self, multiplicities):
|
||
|
|
"""Returns the number of partitions of a multiset whose components
|
||
|
|
have the multiplicities given in ``multiplicities``.
|
||
|
|
|
||
|
|
For larger counts, this method is much faster than calling one
|
||
|
|
of the enumerators and counting the result. Uses dynamic
|
||
|
|
programming to cut down on the number of nodes actually
|
||
|
|
explored. The dictionary used in order to accelerate the
|
||
|
|
counting process is stored in the ``MultisetPartitionTraverser``
|
||
|
|
object and persists across calls. If the user does not
|
||
|
|
expect to call ``count_partitions`` for any additional
|
||
|
|
multisets, the object should be cleared to save memory. On
|
||
|
|
the other hand, the cache built up from one count run can
|
||
|
|
significantly speed up subsequent calls to ``count_partitions``,
|
||
|
|
so it may be advantageous not to clear the object.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.utilities.enumerative import MultisetPartitionTraverser
|
||
|
|
>>> m = MultisetPartitionTraverser()
|
||
|
|
>>> m.count_partitions([9,8,2])
|
||
|
|
288716
|
||
|
|
>>> m.count_partitions([2,2])
|
||
|
|
9
|
||
|
|
>>> del m
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
If one looks at the workings of Knuth's algorithm M [AOCP]_, it
|
||
|
|
can be viewed as a traversal of a binary tree of parts. A
|
||
|
|
part has (up to) two children, the left child resulting from
|
||
|
|
the spread operation, and the right child from the decrement
|
||
|
|
operation. The ordinary enumeration of multiset partitions is
|
||
|
|
an in-order traversal of this tree, and with the partitions
|
||
|
|
corresponding to paths from the root to the leaves. The
|
||
|
|
mapping from paths to partitions is a little complicated,
|
||
|
|
since the partition would contain only those parts which are
|
||
|
|
leaves or the parents of a spread link, not those which are
|
||
|
|
parents of a decrement link.
|
||
|
|
|
||
|
|
For counting purposes, it is sufficient to count leaves, and
|
||
|
|
this can be done with a recursive in-order traversal. The
|
||
|
|
number of leaves of a subtree rooted at a particular part is a
|
||
|
|
function only of that part itself, so memoizing has the
|
||
|
|
potential to speed up the counting dramatically.
|
||
|
|
|
||
|
|
This method follows a computational approach which is similar
|
||
|
|
to the hypothetical memoized recursive function, but with two
|
||
|
|
differences:
|
||
|
|
|
||
|
|
1) This method is iterative, borrowing its structure from the
|
||
|
|
other enumerations and maintaining an explicit stack of
|
||
|
|
parts which are in the process of being counted. (There
|
||
|
|
may be multisets which can be counted reasonably quickly by
|
||
|
|
this implementation, but which would overflow the default
|
||
|
|
Python recursion limit with a recursive implementation.)
|
||
|
|
|
||
|
|
2) Instead of using the part data structure directly, a more
|
||
|
|
compact key is constructed. This saves space, but more
|
||
|
|
importantly coalesces some parts which would remain
|
||
|
|
separate with physical keys.
|
||
|
|
|
||
|
|
Unlike the enumeration functions, there is currently no _range
|
||
|
|
version of count_partitions. If someone wants to stretch
|
||
|
|
their brain, it should be possible to construct one by
|
||
|
|
memoizing with a histogram of counts rather than a single
|
||
|
|
count, and combining the histograms.
|
||
|
|
"""
|
||
|
|
# number of partitions so far in the enumeration
|
||
|
|
self.pcount = 0
|
||
|
|
# dp_stack is list of lists of (part_key, start_count) pairs
|
||
|
|
self.dp_stack = []
|
||
|
|
|
||
|
|
# dp_map is map part_key-> count, where count represents the
|
||
|
|
# number of multiset which are descendants of a part with this
|
||
|
|
# key, **or any of its decrements**
|
||
|
|
|
||
|
|
# Thus, when we find a part in the map, we add its count
|
||
|
|
# value to the running total, cut off the enumeration, and
|
||
|
|
# backtrack
|
||
|
|
|
||
|
|
if not hasattr(self, 'dp_map'):
|
||
|
|
self.dp_map = {}
|
||
|
|
|
||
|
|
self._initialize_enumeration(multiplicities)
|
||
|
|
pkey = part_key(self.top_part())
|
||
|
|
self.dp_stack.append([(pkey, 0), ])
|
||
|
|
while True:
|
||
|
|
while self.spread_part_multiplicity():
|
||
|
|
pkey = part_key(self.top_part())
|
||
|
|
if pkey in self.dp_map:
|
||
|
|
# Already have a cached value for the count of the
|
||
|
|
# subtree rooted at this part. Add it to the
|
||
|
|
# running counter, and break out of the spread
|
||
|
|
# loop. The -1 below is to compensate for the
|
||
|
|
# leaf that this code path would otherwise find,
|
||
|
|
# and which gets incremented for below.
|
||
|
|
|
||
|
|
self.pcount += (self.dp_map[pkey] - 1)
|
||
|
|
self.lpart -= 1
|
||
|
|
break
|
||
|
|
else:
|
||
|
|
self.dp_stack.append([(pkey, self.pcount), ])
|
||
|
|
|
||
|
|
# M4 count a leaf partition
|
||
|
|
self.pcount += 1
|
||
|
|
|
||
|
|
# M5 (Decrease v)
|
||
|
|
while not self.decrement_part(self.top_part()):
|
||
|
|
# M6 (Backtrack)
|
||
|
|
for key, oldcount in self.dp_stack.pop():
|
||
|
|
self.dp_map[key] = self.pcount - oldcount
|
||
|
|
if self.lpart == 0:
|
||
|
|
return self.pcount
|
||
|
|
self.lpart -= 1
|
||
|
|
|
||
|
|
# At this point have successfully decremented the part on
|
||
|
|
# the stack and it does not appear in the cache. It needs
|
||
|
|
# to be added to the list at the top of dp_stack
|
||
|
|
pkey = part_key(self.top_part())
|
||
|
|
self.dp_stack[-1].append((pkey, self.pcount),)
|
||
|
|
|
||
|
|
|
||
|
|
def part_key(part):
|
||
|
|
"""Helper for MultisetPartitionTraverser.count_partitions that
|
||
|
|
creates a key for ``part``, that only includes information which can
|
||
|
|
affect the count for that part. (Any irrelevant information just
|
||
|
|
reduces the effectiveness of dynamic programming.)
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
This member function is a candidate for future exploration. There
|
||
|
|
are likely symmetries that can be exploited to coalesce some
|
||
|
|
``part_key`` values, and thereby save space and improve
|
||
|
|
performance.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# The component number is irrelevant for counting partitions, so
|
||
|
|
# leave it out of the memo key.
|
||
|
|
rval = []
|
||
|
|
for ps in part:
|
||
|
|
rval.append(ps.u)
|
||
|
|
rval.append(ps.v)
|
||
|
|
return tuple(rval)
|