結果

問題 No.2507 Yet Another Subgraph Counting
ユーザー suisen
提出日時 2023-08-31 22:08:43
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 1,406 ms / 2,000 ms
コード長 4,565 bytes
コンパイル時間 260 ms
コンパイル使用メモリ 81,920 KB
実行使用メモリ 114,404 KB
最終ジャッジ日時 2024-09-15 14:17:58
合計ジャッジ時間 13,992 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge6
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ファイルパターン 結果
other AC * 52
権限があれば一括ダウンロードができます

ソースコード

diff # 

from typing import List, Tuple

N_MAX = 13
popcount = [0] * (1 << N_MAX)
for S in range(1, 1 << N_MAX):
    popcount[S] = popcount[S & (S - 1)] + 1

def addeq_poly(f: List[int], g: List[int]):
    """
    f += g
    """
    for i, gi in enumerate(g):
        f[i] += gi

def subeq_poly(f: List[int], g: List[int]):
    """
    f -= g
    """
    for i, gi in enumerate(g):
        f[i] -= gi

def subset_zeta(f: List[int], n: int):
    """
    Inplace conversion from f to ζf. ζf is defined as follows:
        (ζf)(S) = Σ[T⊆S] f(T)
    """
    block = 1
    while block < 1 << n:
        offset = 0
        while offset < 1 << n:
            for p in range(offset, offset + block):
                f[p + block] += f[p]
            offset += 2 * block
        block <<= 1

def subset_zeta_poly(f: List[List[int]], n: int):
    """
    Inplace conversion from f to ζf. ζf is defined as follows:
        (ζf)(S) = Σ[T⊆S] f(T)
    """
    block = 1
    while block < 1 << n:
        offset = 0
        while offset < 1 << n:
            for p in range(offset, offset + block):
                addeq_poly(f[p + block], f[p])
            offset += 2 * block
        block <<= 1

def subset_mobius_poly(f: List[List[int]], n: int):
    """
    Inplace conversion from f to μf. μf is defined as follows:
        (μf)(S) = Σ[T⊆S] (-1)^(|S/T|) f(T)
    """
    block = 1
    while block < 1 << n:
        offset = 0
        while offset < 1 << n:
            for p in range(offset, offset + block):
                subeq_poly(f[p + block], f[p])
            offset += 2 * block
        block <<= 1

def mul_poly(f: List[int], g: List[int]):
    """
    Returns h = fg mod x^n, where f, g are polynomials with degree n-1 defined as follows:
        f(x) = Σ_i f[i] x^i,
        g(x) = Σ_i g[i] x^i.
    """
    n = len(f)
    h = [0] * n
    for i in range(n):
        for j in range(n - i):
            h[i + j] += f[i] * g[j]

    return h

def add_rank(f: List[int], n: int):
    """
    Add rank
    """
    return [[(i == popcount[S]) * f[S] for i in range(n + 1)] for S in range(1 << n)]

def remove_rank(rf: List[List[int]], n: int):
    """
    Remove rank
    """
    return [rf[S][popcount[S]] for S in range(1 << n)]

def subset_exp(f: List[int], n: int):
    """
    Subset exp of Σ[S⊆{0,1,...,n-1}] f(S) x^S
    """
    assert f[0] == 0
    rf = add_rank([1], 0)
    for i in range(n):
        rg = add_rank(f[1 << i: 1 << (i + 1)], i)
        subset_zeta_poly(rg, i)
        for S in range(1 << i):
            rf[S].append(0)
            rg[S].insert(0, 1)
            rh = mul_poly(rf[S], rg[S])
            rf.append(rh)
    subset_mobius_poly(rf, n)
    return remove_rank(rf, n)


def count_cycles(n: int, edges: List[Tuple[int, int]]):
    cycle = [0] * (1 << n)
    adj = [[] for _ in range(n)]
    for u, v in edges:
        adj[u].append(v)
        adj[v].append(u)

    cycle_dp = [[0] * n for _ in range(1 << n)]
    for v in range(n):
        cycle_dp[1 << v][v] = 1
    for s in range(1, 1 << n):
        start = 0
        while not ((s >> start) & 1):
            start += 1
        for cur in range(n):
            if cycle_dp[s][cur] == 0:
                continue
            for nxt in adj[cur]:
                if start == nxt:
                    cycle[s] += cycle_dp[s][cur]
                elif start < nxt and not ((s >> nxt) & 1):
                    cycle_dp[s | (1 << nxt)][nxt] += cycle_dp[s][cur]

    for s in range(1, 1 << n):
        if popcount[s] == 1:
            cycle[s] = 1
        elif popcount[s] == 2:
            cycle[s] = 0
        else:
            cycle[s] //= 2
    return cycle

n, m = map(int, input().split())

edges = []
for _ in range(m):
    u, v = map(int, input().split())
    u -= 1
    v -= 1
    edges.append((u, v))

# E[S] = # of edges connecting vertices in S
E = [0] * (1 << n)
for u, v in edges:
    E[(1 << u) | (1 << v)] += 1
subset_zeta(E, n)

cycle = count_cycles(n, edges)

f = [0] * (1 << n)
for C in range(1, 1 << n):
    if cycle[C] == 0:
        continue

    # max C
    t = C.bit_length() - 1
    # {0, ..., tX} - C
    S = ((1 << (t + 1)) - 1) ^ C
    k = popcount[S]

    bit_deposit = [0] * (1 << k)
    bit_deposit[0] = S
    for A in range(1, 1 << k):
        bit_deposit[A] = (bit_deposit[A - 1] - 1) & S
    bit_deposit.reverse()

    g = [f[bit_deposit[A]] * (E[bit_deposit[A] | C] - E[bit_deposit[A]] - E[C]) for A in range(1 << k)]

    for A, hA in enumerate(subset_exp(g, k)):
        f[bit_deposit[A] | C] += cycle[C] * hA

print(subset_exp(f, n)[-1])
0