"""Signed matching polynomials of connected graphs -- numberdb.org/T239

For every connected simple graph G on at most seven vertices, named by its
canonical graph6 string, this stores

    mu(G, x) = sum_k (-1)^k m_k x^(n - 2k),

where n is the number of vertices and m_k is the number of k-edge matchings
of G.

Run it with SageMath:

    $ sage -pip install numberdb          # once
    $ sage -python generate.py            # check the table against this code
    $ sage -python generate.py --publish  # fill the draft, with NUMBERDB_API_KEY set

The generator counts matchings directly from edge subsets. This keeps the
arithmetic in ZZ and gives an independent path to compare with Sage's
matching_polynomial().
"""

import os
import sys
from functools import lru_cache
from itertools import combinations

import numberdb.sage as numberdb
from sage.graphs.graph import Graph
from sage.graphs.graph_generators import graphs
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing


T239 = "T239"
MAX_VERTICES = 7

_R = PolynomialRing(ZZ, "x")
_x = _R.gen()

NAMES = {
    "@": "complete graph $K_1$",
    "A_": "complete graph $K_2$",
    "BW": "path $P_3$",
    "Bw": "complete graph $K_3$",
    "CF": "star $K_{1,3}$",
    "CL": "path $P_4$",
    "C]": "cycle $C_4$",
    "C^": "diamond graph",
    "C~": "complete graph $K_4$",
    "D?{": "star $K_{1,4}$",
    "DBg": "path $P_5$",
    "DBk": "bull graph",
    "DB{": "dart graph",
    "DFw": "complete bipartite graph $K_{2,3}$",
    "DK{": "butterfly graph",
    "DLo": "cycle $C_5$",
    "DN{": "house X graph",
    "D]{": "wheel $W_5$",
    "Dbk": "house graph",
    "D~{": "complete graph $K_5$",
    "E?Bw": "star $K_{1,5}$",
    "E?~o": "complete bipartite graph $K_{2,4}$",
    "E@YO": "path $P_6$",
    "EFz_": "complete bipartite graph $K_{3,3}$",
    "EIe_": "cycle $C_6$",
    "ELrw": "wheel $W_6$",
    "E~~w": "complete graph $K_6$",
    "F??Fw": "star $K_{1,6}$",
    "F?B~o": "complete bipartite graph $K_{2,5}$",
    "F?~v_": "complete bipartite graph $K_{3,4}$",
    "F@HSO": "path $P_7$",
    "FHQSO": "cycle $C_7$",
    "FIefw": "wheel $W_7$",
    "FjaHw": "Moser spindle",
    "F~~~w": "complete graph $K_7$",
}


def _key_from_stdin():
    if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
        return
    token = sys.stdin.read().strip()
    if "=" in token and token.split("=", 1)[0].isupper():
        token = token.split("=", 1)[1].strip().strip("'\"")
    if token:
        os.environ["NUMBERDB_API_KEY"] = token


@lru_cache(maxsize=None)
def connected_graphs(max_vertices=MAX_VERTICES):
    out = []
    for n in range(1, max_vertices + 1):
        for graph in graphs.nauty_geng("%d -c" % n):
            canonical = graph.canonical_label(algorithm="sage")
            out.append((canonical.graph6_string(), canonical))
    return tuple(out)


@lru_cache(maxsize=None)
def graph_by_key(max_vertices=MAX_VERTICES):
    return {key: graph for key, graph in connected_graphs(max_vertices)}


def graph_from_key(key):
    return Graph(str(key))


@lru_cache(maxsize=None)
def matching_counts(key):
    graph = graph_by_key().get(key)
    if graph is None:
        graph = graph_from_key(key)
    edges = list(graph.edges(labels=False))
    limit = graph.num_verts() // 2
    counts = [ZZ(0)] * (limit + 1)
    counts[0] = ZZ(1)
    for size in range(1, limit + 1):
        total = ZZ(0)
        for chosen in combinations(edges, size):
            used = []
            for edge in chosen:
                used.extend(edge)
            if len(set(used)) == 2 * size:
                total += 1
        counts[size] = total
    return tuple(counts)


@lru_cache(maxsize=None)
def signed_matching_polynomial(key):
    counts = matching_counts(key)
    graph = graph_by_key().get(key)
    if graph is None:
        graph = graph_from_key(key)
    n = graph.num_verts()
    return sum(
        _R((-1) ** k) * _R(count) * _x ** (n - 2 * k)
        for k, count in enumerate(counts)
    )


def entry_comment(key):
    if key in NAMES:
        return "This is the %s." % NAMES[key]
    return ""


class SignedMatchingPolynomialsOfConnectedGraphs(numberdb.Generator):

    table = os.environ.get("NUMBERDB_TABLE", T239)
    parameters = ("g",)
    type = "Z[]"
    rigour = "exact"

    def enumerate(self, max_vertices=MAX_VERTICES):
        for key, _graph in connected_graphs(max_vertices):
            yield {"g": key}

    def value(self, params, digits):
        del digits
        key = params["g"]
        entry = {"number": signed_matching_polynomial(key)}
        comment = entry_comment(key)
        if comment:
            entry["comment"] = comment
        return entry


if __name__ == "__main__":
    _key_from_stdin()
    generator = SignedMatchingPolynomialsOfConnectedGraphs()
    if os.environ.get("NUMBERDB_PUBLISH") == "1" or "--publish" in sys.argv:
        print(generator.publish(
            message="signed matching polynomials of connected graphs",
            assisted_by=os.environ.get("NUMBERDB_ASSISTED_BY", "")))
    else:
        report = generator.verify(sample=None)
        print(report)
        sys.exit(0 if report.ok else 1)