U 9v f4@sbdZddlZddlZddddddgZdd dZd d Zd dZddZddZ ddZ ddZ dS)zTest sequences for graphiness. N is_graphicalis_multigraphicalis_pseudographicalis_digraphical%is_valid_degree_sequence_erdos_gallai%is_valid_degree_sequence_havel_hakimiegcCs>|dkrtt|}n$|dkr,tt|}nd}t||S)uReturns True if sequence is a valid degree sequence. A degree sequence is valid if some graph can realize it. Parameters ---------- sequence : list or iterable container A sequence of integer node degrees method : "eg" | "hh" (default: 'eg') The method used to validate the degree sequence. "eg" corresponds to the Erdős-Gallai algorithm [EG1960]_, [choudum1986]_, and "hh" to the Havel-Hakimi algorithm [havel1955]_, [hakimi1962]_, [CL1996]_. Returns ------- valid : bool True if the sequence is a valid degree sequence and False if not. Examples -------- >>> G = nx.path_graph(4) >>> sequence = (d for n, d in G.degree()) >>> nx.is_graphical(sequence) True References ---------- .. [EG1960] Erdős and Gallai, Mat. Lapok 11 264, 1960. .. [choudum1986] S.A. Choudum. "A simple proof of the Erdős-Gallai theorem on graph sequences." Bulletin of the Australian Mathematical Society, 33, pp 67-70, 1986. https://doi.org/10.1017/S0004972700002872 .. [havel1955] Havel, V. "A Remark on the Existence of Finite Graphs" Casopis Pest. Mat. 80, 477-480, 1955. .. [hakimi1962] Hakimi, S. "On the Realizability of a Set of Integers as Degrees of the Vertices of a Graph." SIAM J. Appl. Math. 10, 496-506, 1962. .. [CL1996] G. Chartrand and L. Lesniak, "Graphs and Digraphs", Chapman and Hall/CRC, 1996. rhhz`method` must be 'eg' or 'hh')rlistrnxZNetworkXException)sequencemethodZvalidmsgrA/tmp/pip-unpacked-wheel-_lngutwb/networkx/algorithms/graphical.pyrs* cCstj|}t|}dg|}d|ddf\}}}}|D]\}|dksJ||krRtjq6|dkr6t||t|||||df\}}}}||d7<q6|ds|||dkrtj|||||fS)Nr)r utilsmake_list_of_intslenNetworkXUnfeasiblemaxmin) deg_sequencepnum_degsdmaxdmindsumndrrr_basic_graphical_testsEs  (r!c CsRzt|\}}}}}Wntjk r.YdSX|dks\d||||d||dkr`dSdg|d}|dkrN||dkr|d8}qx||dkrdS||d|d||<}d}|}t|D]P} ||dkr|d8}q||d|d||<}|dkr|d||<|d7}qt|D](} || } || d|d|| <}q"qndS)aReturns True if deg_sequence can be realized by a simple graph. The validation proceeds using the Havel-Hakimi theorem [havel1955]_, [hakimi1962]_, [CL1996]_. Worst-case run time is $O(s)$ where $s$ is the sum of the sequence. Parameters ---------- deg_sequence : list A list of integers where each element specifies the degree of a node in a graph. Returns ------- valid : bool True if deg_sequence is graphical and False if not. Notes ----- The ZZ condition says that for the sequence d if .. math:: |d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)} then d is graphical. This was shown in Theorem 6 in [1]_. References ---------- .. [1] I.E. Zverovich and V.E. Zverovich. "Contributions to the theory of graphic sequences", Discrete Mathematics, 105, pp. 292-303 (1992). .. [havel1955] Havel, V. "A Remark on the Existence of Finite Graphs" Casopis Pest. Mat. 80, 477-480, 1955. .. [hakimi1962] Hakimi, S. "On the Realizability of a Set of Integers as Degrees of the Vertices of a Graph." SIAM J. Appl. Math. 10, 496-506, 1962. .. [CL1996] G. Chartrand and L. Lesniak, "Graphs and Digraphs", Chapman and Hall/CRC, 1996. FrrTr!r rrange) rrrrrrmodstubsmslenkistubrrrrYs4&,           c Cs,zt|\}}}}}Wntjk r.YdSX|dks\d||||d||dkr`dSd\}}}} t||ddD]} | |dkrdS|| dkr||| } | || kr| |} || | 7}t| D],} |||| 7}| || ||| 7} q|| 7}|||d||| kr|dSq|dS)uTReturns True if deg_sequence can be realized by a simple graph. The validation is done using the Erdős-Gallai theorem [EG1960]_. Parameters ---------- deg_sequence : list A list of integers Returns ------- valid : bool True if deg_sequence is graphical and False if not. Notes ----- This implementation uses an equivalent form of the Erdős-Gallai criterion. Worst-case run time is $O(n)$ where $n$ is the length of the sequence. Specifically, a sequence d is graphical if and only if the sum of the sequence is even and for all strong indices k in the sequence, .. math:: \sum_{i=1}^{k} d_i \leq k(k-1) + \sum_{j=k+1}^{n} \min(d_i,k) = k(n-1) - ( k \sum_{j=0}^{k-1} n_j - \sum_{j=0}^{k-1} j n_j ) A strong index k is any index where d_k >= k and the value n_j is the number of occurrences of j in d. The maximal strong index is called the Durfee index. This particular rearrangement comes from the proof of Theorem 3 in [2]_. The ZZ condition says that for the sequence d if .. math:: |d| >= \frac{(\max(d) + \min(d) + 1)^2}{4*\min(d)} then d is graphical. This was shown in Theorem 6 in [2]_. References ---------- .. [1] A. Tripathi and S. Vijay. "A note on a theorem of Erdős & Gallai", Discrete Mathematics, 265, pp. 417-420 (2003). .. [2] I.E. Zverovich and V.E. Zverovich. "Contributions to the theory of graphic sequences", Discrete Mathematics, 105, pp. 292-303 (1992). .. [EG1960] Erdős and Gallai, Mat. Lapok 11 264, 1960. Frr"rT)rrrrr#) rrrrrrr'Zsum_degZsum_njZsum_jnjZdkZrun_sizevrrrrs,2,      cCszztj|}Wntjk r(YdSXd\}}|D]&}|dkrHdS||t||}}q6|dsr|d|krvdSdS)a,Returns True if some multigraph can realize the sequence. Parameters ---------- sequence : list A list of integers Returns ------- valid : bool True if deg_sequence is a multigraphic degree sequence and False if not. Notes ----- The worst-case run time is $O(n)$ where $n$ is the length of the sequence. References ---------- .. [1] S. L. Hakimi. "On the realizability of a set of integers as degrees of the vertices of a linear graph", J. SIAM, 10, pp. 496-506 (1962). FrrrrT)r rr NetworkXErrorr)r rrrr rrrrscCsFztj|}Wntjk r(YdSXt|ddkoDt|dkS)aReturns True if some pseudograph can realize the sequence. Every nonnegative integer sequence with an even sum is pseudographical (see [1]_). Parameters ---------- sequence : list or iterable container A sequence of integer node degrees Returns ------- valid : bool True if the sequence is a pseudographic degree sequence and False if not. Notes ----- The worst-case run time is $O(n)$ where n is the length of the sequence. References ---------- .. [1] F. Boesch and F. Harary. "Line removal algorithms for graphs and their degree lists", IEEE Trans. Circuits and Systems, CAS-23(12), pp. 778-782 (1976). Frr)r rrr-sumr)r rrrrrs cCstztj|}tj|}Wntjk r4YdSXddt|t|f\}}}}t||}d} |dkrldSgg} } t|D]} d\} }| |kr|| }| |kr|| } | dks|dkrdS|| ||t| | }}} | dkr| d|d| fq~|dkr~| d|q~||kr&dSt | t | dg| d}| rpt | \}}|d9}|t| t| kr~dSd}t|D]}| r| r| dd| dkrt | }d}nt | \}}|dkrdS|ddks|dkr|d|f||<|d7}qt|D]:}||}|ddkrBt | |nt | |dq|dkrHt | |qHdS)a4Returns True if some directed graph can realize the in- and out-degree sequences. Parameters ---------- in_sequence : list or iterable container A sequence of integer node in-degrees out_sequence : list or iterable container A sequence of integer node out-degrees Returns ------- valid : bool True if in and out-sequences are digraphic False if not. Notes ----- This algorithm is from Kleitman and Wang [1]_. The worst case runtime is $O(s \times \log n)$ where $s$ and $n$ are the sum and length of the sequences respectively. References ---------- .. [1] D.J. Kleitman and D.L. Wang Algorithms for Constructing Graphs and Digraphs with Given Valences and Factors, Discrete Mathematics, 6(1), pp. 79-88 (1973) FrTr,r*r) r rrr-rrr$appendheapqheapifyheappopheappush)Z in_sequenceZ out_sequenceZin_deg_sequenceZout_deg_sequenceZsuminZsumoutZninZnoutZmaxnZmaxinZstubheapZzeroheaprZin_degZout_degr%ZfreeoutZfreeinr&r(ZstuboutZstubinr)rrrr8sj         "     )r) __doc__r0Znetworkxr __all__rr!rrrrrrrrrs  4LM%!