o o h/@slddlmZddlmZddlmZddlmZddlm Z m Z ddl m Z ddl mZGdd d eZd S) )Basic)Tuple)Array_sympify)flatteniterable)as_int) defaultdictc@seZdZdZdZdZdZdZeddZ eddZ eddZ ed d Z ed d Z ed dZeddZeddZddZeddZddZdddZdddZdS)Prufera The Prufer correspondence is an algorithm that describes the bijection between labeled trees and the Prufer code. A Prufer code of a labeled tree is unique up to isomorphism and has a length of n - 2. Prufer sequences were first used by Heinz Prufer to give a proof of Cayley's formula. References ========== .. [1] https://mathworld.wolfram.com/LabeledTree.html NcCs*|jdur||jdd|j|_|jS)aHReturns Prufer sequence for the Prufer object. This sequence is found by removing the highest numbered vertex, recording the node it was attached to, and continuing until only two vertices remain. The Prufer sequence is the list of recorded nodes. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).prufer_repr [3, 3, 3, 4] >>> Prufer([1, 0, 0]).prufer_repr [1, 0, 0] See Also ======== to_prufer N) _prufer_repr to_prufer _tree_reprnodesselfrn/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/combinatorics/prufer.py prufer_repr s zPrufer.prufer_reprcCs&|jdur||jdd|_|jS)aReturns the tree representation of the Prufer object. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).tree_repr [[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]] >>> Prufer([1, 0, 0]).tree_repr [[1, 2], [0, 1], [0, 3], [0, 4]] See Also ======== to_tree N)rto_treer rrrr tree_repr;s zPrufer.tree_reprcCs|jS)a Returns the number of nodes in the tree. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).nodes 6 >>> Prufer([1, 0, 0]).nodes 5 )_nodesrrrrrRsz Prufer.nodescCs|jdur ||_|jS)aReturns the rank of the Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> p = Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]) >>> p.rank 778 >>> p.next(1).rank 779 >>> p.prev().rank 777 See Also ======== prufer_rank, next, prev, size N)_rank prufer_rankrrrrrankbs  z Prufer.rankcCs||jjdS)a6Return the number of possible trees of this Prufer object. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer([0]*4).size == Prufer([6]*4).size == 1296 True See Also ======== prufer_rank, rank, next, prev )prevrrrrrsize|sz Prufer.sizec Cstt}g}|D]}||dd7<||dd7<qt|dD]U}t|D] }||dkr5nq+d}|D]}||dkrG|d}n ||dkrQ|d}|durWnq:||||fD]}||d8<||st||qa||q%|S)aReturn the Prufer sequence for a tree given as a list of edges where ``n`` is the number of nodes in the tree. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> a.prufer_repr [0, 0] >>> Prufer.to_prufer([[0, 1], [0, 2], [0, 3]], 4) [0, 0] See Also ======== prufer_repr: returns Prufer sequence of a Prufer object. rrN)r intrangeappendpopremove) treendLedgeixyjrrrr s6         zPrufer.to_prufercsg}g}t|d}tdd|D] }|d7<q|D]*}t|D] }|dkr/nq%|d8<|d8<|t||gqfddt|DpXddg}|||S)aBReturn the tree (as a list of edges) of the given Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([0, 2], 4) >>> a.tree_repr [[0, 1], [0, 2], [2, 3]] >>> Prufer.to_tree([0, 2]) [[0, 1], [0, 2], [2, 3]] References ========== .. [1] https://hamberg.no/erlend/posts/2010-11-06-prufer-sequence-compact-tree-representation.html See Also ======== tree_repr: returns tree representation of a Prufer object. rcSsdS)Nrrrrrrsz Prufer.to_tree..rcsg|] }|dkr|qSrr.0r)r&rr sz"Prufer.to_tree..r)lenr r r!sorted)pruferr$lastr%pr)r,rr1rrs"     zPrufer.to_treec s~t}|dd|D]'}tt|dD]}|||d\}}||kr*||}}|||fqq g}t}d}|D],} || durPt| dn| d|dur_t| d|n| d}|t| q>tt|d|} | rfdd| D} t| dkrd| } t | dt | } t | dkrt |D]\}} fd d| D||<q|8}t ||dfS) aAReturn a list of edges and the number of nodes from the given runs that connect nodes in an integer-labelled tree. All node numbers will be shifted so that the minimum node is 0. It is not a problem if edges are repeated in the runs; only unique edges are returned. There is no assumption made about what the range of the node labels should be, but all nodes from the smallest through the largest must be present. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer.edges([1, 2, 3], [2, 4, 5]) # a T ([[0, 1], [1, 2], [1, 3], [3, 4]], 5) Duplicate edges are removed: >>> Prufer.edges([0, 1, 2, 3], [1, 4, 5], [1, 4, 6]) # a K ([[0, 1], [1, 2], [1, 4], [2, 3], [4, 5], [4, 6]], 7) rrrNcsg|]}|qSrrr/nminrrr2z Prufer.edges..Node %s is missing.Nodes %s are missing.csg|]}|qSrr)r0r%r8rrr2!r:) setr r3addupdateminmaxr!listr"r4 ValueError enumerate) runserr)abrvgotnmaxeimissingmsgrr8redgess<      z Prufer.edgescCs@d}d}t|jdddD]}|||j|7}||j9}q |S)a)Computes the rank of a Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> a.prufer_rank() 0 See Also ======== rank, next, prev, size rr)r rr)rrGr7r)rrrr%s  zPrufer.prufer_rankcsjt|t|}}ttt|dddD]}|||<|||}qtfddttDS)zFinds the unranked Prufer sequence. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> Prufer.unrank(0, 4) Prufer([0, 0]) rQrRcsg|]}|qSrrr/r'rrr2Nr:z!Prufer.unrank..)r r rr r r3)rrr%r)rrSrunrank=s   z Prufer.unrankc OsJ|dr t|dnt}|ftdd|ddD}tj|g|Ri|}t|dg}|drt|ddr|ddsFtdt|dkrQ|d}n6t t |d}t |d}|t|krt t ||}t|dkr}d| }t|dt|}t|d d |dD|_||_|S|d|_t|jd |_|S) aThe constructor for the Prufer object. Examples ======== >>> from sympy.combinatorics.prufer import Prufer A Prufer object can be constructed from a list of edges: >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> a.prufer_repr [0, 0] If the number of nodes is given, no checking of the nodes will be performed; it will be assumed that nodes 0 through n - 1 are present: >>> Prufer([[0, 1], [0, 2], [0, 3]], 4) Prufer([[0, 1], [0, 2], [0, 3]], 4) A Prufer object can be constructed from a Prufer sequence: >>> b = Prufer([1, 3]) >>> b.tree_repr [[0, 1], [1, 3], [2, 3]] rcss|]}t|VqdS)Nr)r0argrrr msz!Prufer.__new__..rNz-Prufer expects at least one edge in the tree.r;r<cSsg|]}t|qSr)rBr/rrrr2r:z"Prufer.__new__..r)rrtupler__new__rBrrCr3r=rrAr r"r4rrr ) clsargskw_argsarg0ret_objnnodesrrNrOrrrrXPs4          zPrufer.__new__rcCst|j||jS)aGenerates the Prufer sequence that is delta beyond the current one. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [0, 2], [0, 3]]) >>> b = a.next(1) # == a.next() >>> b.tree_repr [[0, 2], [0, 1], [1, 3]] >>> b.rank 1 See Also ======== prufer_rank, rank, prev, size r rTrrrdeltarrrnextsz Prufer.nextcCst|j||jS)aGenerates the Prufer sequence that is -delta before the current one. Examples ======== >>> from sympy.combinatorics.prufer import Prufer >>> a = Prufer([[0, 1], [1, 2], [2, 3], [1, 4]]) >>> a.rank 36 >>> b = a.prev() >>> b Prufer([1, 2, 0]) >>> b.rank 35 See Also ======== prufer_rank, rank, next, size r_r`rrrrsz Prufer.prevr.)__name__ __module__ __qualname____doc__r rrrpropertyrrrrr staticmethodr rrPr classmethodrTrXrbrrrrrr s8      2 , 5  7r N) sympy.corersympy.core.containersrsympy.tensor.arrayrsympy.core.sympifyrsympy.utilities.iterablesrrsympy.utilities.miscr collectionsr r rrrrs