codeforces：C. Set Construction【构造 + 入度观察】

题目截图

题目分析

• 题目要找n个集合
• 给出一个矩阵b
• 如果bij = 1，表示第i个集合为第j个集合的真子集
• bij = 0，表示不是真子集
• 寻找集合间的关系，g记录下一个更大的集合，smaller表示被本集合包含的集合的总个数
• 首先lst中记录的是没有更小孩子的最小集合，cur作为它们的独特标识
• 加入cur后，遍历与其关联的所有更大集合，并加入cur
• 同时对这些更大集合的smaller - 1，如果为0，则可以加入lst，因为它了无牵挂
• 这样一次遍历后，可以令cur + 1
• 因为要保持独特性，下一轮遍历的集合应该有它们独特的符号，从而可以区分

ac code

# -*- coding: utf-8 -*-
# @Time    : 2022/11/24 15:15
# @Author  : bridgekiller
# @FileName: 1761C.py
# @Software: PyCharm
# @Blog    ：bridge-killer.blog.csdn.net

import os
import sys
import math
import random
import threading
from copy import deepcopy
from io import BytesIO, IOBase
from types import GeneratorType
from functools import lru_cache, reduce
from bisect import bisect_left, bisect_right
from collections import Counter, defaultdict, deque
from itertools import accumulate, combinations, permutations
from heapq import nsmallest, nlargest, heapify, heappop, heappush
from typing import Generic, Iterable, Iterator, TypeVar, Union, List


def debug(func):
    def wrapper(*args, **kwargs):
        print('----------------')
        res = func(*args, **kwargs)
        print('----------------')
        return res

    return wrapper


def bootstrap(f, stack=[]):
    def wrappedfunc(*args, **kwargs):
        if stack:
            return f(*args, **kwargs)
        else:
            to = f(*args, **kwargs)
            while True:
                if type(to) is GeneratorType:
                    stack.append(to)
                    to = next(to)
                else:
                    stack.pop()
                    if not stack:
                        break
                    to = stack[-1].send(to)
            return to

    return wrappedfunc


class SegTree:
    '''
    支持增量更新，覆盖更新，序列更新，任意RMQ操作
    基于二叉树实现
    初始化：O(1)
    增量更新或覆盖更新的单次操作复杂度：O(log k)
    序列更新的单次复杂度：O(n)
    '''

    def __init__(self, f1, f2, l, r, v=0):
        '''
        初始化线段树[left,right)
        f1,f2示例：
        线段和:
        f1=lambda a,b:a+b
        f2=lambda a,n:a*n
        线段最大值:
        f1=lambda a,b:max(a,b)
        f2=lambda a,n:a
        线段最小值:
        f1=lambda a,b:min(a,b)
        f2=lambda a,n:a
        '''
        self.ans = f2(v, r - l)
        self.f1 = f1
        self.f2 = f2
        self.l = l  # left
        self.r = r  # right
        self.v = v  # init value
        self.lazy_tag = 0  # Lazy tag
        self.left = None  # SubTree(left,bottom)
        self.right = None  # SubTree(right,bottom)

    @property
    def mid_h(self):
        return (self.l + self.r) // 2

    def create_subtrees(self):
        midh = self.mid_h
        if not self.left and midh > self.l:
            self.left = SegTree(self.f1, self.f2, self.l, midh)
        if not self.right:
            self.right = SegTree(self.f1, self.f2, midh, self.r)

    def init_seg(self, M):
        '''
        将线段树的值初始化为矩阵Matrx
        输入保证Matrx与线段大小一致
        '''
        m0 = M[0]
        self.lazy_tag = 0
        for a in M:
            if a != m0:
                break
        else:
            self.v = m0
            self.ans = self.f2(m0, len(M))
            return self.ans
        self.v = '#'
        midh = self.mid_h
        self.create_subtrees()
        self.ans = self.f1(self.left.init_seg(M[:midh - self.l]), self.right.init_seg(M[midh - self.l:]))
        return self.ans

    def cover_seg(self, l, r, v):
        '''
        将线段[left,right)覆盖为val
        '''
        if self.v == v or l >= self.r or r <= self.l:
            return self.ans
        if l <= self.l and r >= self.r:
            self.v = v
            self.lazy_tag = 0
            self.ans = self.f2(v, self.r - self.l)
            return self.ans
        self.create_subtrees()
        if self.v != '#':
            if self.left:
                self.left.v = self.v
                self.left.ans = self.f2(self.v, self.left.r - self.left.l)
            if self.right:
                self.right.v = self.v
                self.right.ans = self.f2(self.v, self.right.r - self.right.l)
            self.v = '#'
        # push up
        self.ans = self.f1(self.left.cover_seg(l, r, v), self.right.cover_seg(l, r, v))
        return self.ans

    def inc_seg(self, l, r, v):
        '''
        将线段[left,right)增加val
        '''
        if v == 0 or l >= self.r or r <= self.l:
            return self.ans
        # self.ans = '?'
        if l <= self.l and r >= self.r:
            if self.v == '#':
                self.lazy_tag += v
            else:
                self.v += v
            self.ans += self.f2(v, self.r - self.l)
            return self.ans
        self.create_subtrees()
        if self.v != '#':
            self.left.v = self.v
            self.left.ans = self.f2(self.v, self.left.r - self.left.l)
            self.right.v = self.v
            self.right.ans = self.f2(self.v, self.right.r - self.right.l)
            self.v = '#'
        self.pushdown()
        self.ans = self.f1(self.left.inc_seg(l, r, v), self.right.inc_seg(l, r, v))
        return self.ans

    def inc_idx(self, idx, v):
        '''
        increase idx by val
        '''
        if v == 0 or idx >= self.r or idx < self.l:
            return self.ans
        if idx == self.l == self.r - 1:
            self.v += v
            self.ans += self.f2(v, 1)
            return self.ans
        self.create_subtrees()
        if self.v != '#':
            self.left.v = self.v
            self.left.ans = self.f2(self.v, self.left.r - self.left.l)
            self.right.v = self.v
            self.right.ans = self.f2(self.v, self.right.r - self.right.l)
            self.v = '#'
        self.pushdown()
        self.ans = self.f1(self.left.inc_idx(idx, v), self.right.inc_idx(idx, v))
        return self.ans

    def pushdown(self):
        if self.lazy_tag != 0:
            if self.left:
                if self.left.v != '#':
                    self.left.v += self.lazy_tag
                    self.left.lazy_tag = 0
                else:
                    self.left.lazy_tag += self.lazy_tag
                self.left.ans += self.f2(self.lazy_tag, self.left.r - self.left.l)
            if self.right:
                if self.right.v != '#':
                    self.right.v += self.lazy_tag
                    self.right.lazy_tag = 0
                else:
                    self.right.lazy_tag += self.lazy_tag
                self.right.ans += self.f2(self.lazy_tag, self.right.r - self.right.l)
            self.lazy_tag = 0

    def query(self, l, r):
        '''
        查询线段[right,bottom)的RMQ
        '''
        if l >= r: return 0
        if l <= self.l and r >= self.r:
            return self.ans
        if self.v != '#':
            return self.f2(self.v, min(self.r, r) - max(self.l, l))
        midh = self.mid_h
        anss = []
        if l < midh:
            anss.append(self.left.query(l, r))
        if r > midh:
            anss.append(self.right.query(l, r))
        return reduce(self.f1, anss)


class SortedList:
    def __init__(self, iterable=[], _load=200):
        """Initialize sorted list instance."""
        values = sorted(iterable)
        self._len = _len = len(values)
        self._load = _load
        self._lists = _lists = [values[i:i + _load] for i in range(0, _len, _load)]
        self._list_lens = [len(_list) for _list in _lists]
        self._mins = [_list[0] for _list in _lists]
        self._fen_tree = []
        self._rebuild = True

    def _fen_build(self):
        """Build a fenwick tree instance."""
        self._fen_tree[:] = self._list_lens
        _fen_tree = self._fen_tree
        for i in range(len(_fen_tree)):
            if i | i + 1 < len(_fen_tree):
                _fen_tree[i | i + 1] += _fen_tree[i]
        self._rebuild = False

    def _fen_update(self, index, value):
        """Update `fen_tree[index] += value`."""
        if not self._rebuild:
            _fen_tree = self._fen_tree
            while index < len(_fen_tree):
                _fen_tree[index] += value
                index |= index + 1

    def _fen_query(self, end):
        """Return `sum(_fen_tree[:end])`."""
        if self._rebuild:
            self._fen_build()

        _fen_tree = self._fen_tree
        x = 0
        while end:
            x += _fen_tree[end - 1]
            end &= end - 1
        return x

    def _fen_findkth(self, k):
        """Return a pair of (the largest `idx` such that `sum(_fen_tree[:idx]) <= k`, `k - sum(_fen_tree[:idx])`)."""
        _list_lens = self._list_lens
        if k < _list_lens[0]:
            return 0, k
        if k >= self._len - _list_lens[-1]:
            return len(_list_lens) - 1, k + _list_lens[-1] - self._len
        if self._rebuild:
            self._fen_build()

        _fen_tree = self._fen_tree
        idx = -1
        for d in reversed(range(len(_fen_tree).bit_length())):
            right_idx = idx + (1 << d)
            if right_idx < len(_fen_tree) and k >= _fen_tree[right_idx]:
                idx = right_idx
                k -= _fen_tree[idx]
        return idx + 1, k

    def _delete(self, pos, idx):
        """Delete value at the given `(pos, idx)`."""
        _lists = self._lists
        _mins = self._mins
        _list_lens = self._list_lens

        self._len -= 1
        self._fen_update(pos, -1)
        del _lists[pos][idx]
        _list_lens[pos] -= 1

        if _list_lens[pos]:
            _mins[pos] = _lists[pos][0]
        else:
            del _lists[pos]
            del _list_lens[pos]
            del _mins[pos]
            self._rebuild = True

    def _loc_left(self, value):
        """Return an index pair that corresponds to the first position of `value` in the sorted list."""
        if not self._len:
            return 0, 0

        _lists = self._lists
        _mins = self._mins

        lo, pos = -1, len(_lists) - 1
        while lo + 1 < pos:
            mi = (lo + pos) >> 1
            if value <= _mins[mi]:
                pos = mi
            else:
                lo = mi

        if pos and value <= _lists[pos - 1][-1]:
            pos -= 1

        _list = _lists[pos]
        lo, idx = -1, len(_list)
        while lo + 1 < idx:
            mi = (lo + idx) >> 1
            if value <= _list[mi]:
                idx = mi
            else:
                lo = mi

        return pos, idx

    def _loc_right(self, value):
        """Return an index pair that corresponds to the last position of `value` in the sorted list."""
        if not self._len:
            return 0, 0

        _lists = self._lists
        _mins = self._mins

        pos, hi = 0, len(_lists)
        while pos + 1 < hi:
            mi = (pos + hi) >> 1
            if value < _mins[mi]:
                hi = mi
            else:
                pos = mi

        _list = _lists[pos]
        lo, idx = -1, len(_list)
        while lo + 1 < idx:
            mi = (lo + idx) >> 1
            if value < _list[mi]:
                idx = mi
            else:
                lo = mi

        return pos, idx

    def add(self, value):
        """Add `value` to sorted list."""
        _load = self._load
        _lists = self._lists
        _mins = self._mins
        _list_lens = self._list_lens

        self._len += 1
        if _lists:
            pos, idx = self._loc_right(value)
            self._fen_update(pos, 1)
            _list = _lists[pos]
            _list.insert(idx, value)
            _list_lens[pos] += 1
            _mins[pos] = _list[0]
            if _load + _load < len(_list):
                _lists.insert(pos + 1, _list[_load:])
                _list_lens.insert(pos + 1, len(_list) - _load)
                _mins.insert(pos + 1, _list[_load])
                _list_lens[pos] = _load
                del _list[_load:]
                self._rebuild = True
        else:
            _lists.append([value])
            _mins.append(value)
            _list_lens.append(1)
            self._rebuild = True

    def discard(self, value):
        """Remove `value` from sorted list if it is a member."""
        _lists = self._lists
        if _lists:
            pos, idx = self._loc_right(value)
            if idx and _lists[pos][idx - 1] == value:
                self._delete(pos, idx - 1)

    def remove(self, value):
        """Remove `value` from sorted list; `value` must be a member."""
        _len = self._len
        self.discard(value)
        if _len == self._len:
            raise ValueError('{0!r} not in list'.format(value))

    def pop(self, index=-1):
        """Remove and return value at `index` in sorted list."""
        pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
        value = self._lists[pos][idx]
        self._delete(pos, idx)
        return value

    def bisect_left(self, value):
        """Return the first index to insert `value` in the sorted list."""
        pos, idx = self._loc_left(value)
        return self._fen_query(pos) + idx

    def bisect_right(self, value):
        """Return the last index to insert `value` in the sorted list."""
        pos, idx = self._loc_right(value)
        return self._fen_query(pos) + idx

    def count(self, value):
        """Return number of occurrences of `value` in the sorted list."""
        return self.bisect_right(value) - self.bisect_left(value)

    def __len__(self):
        """Return the size of the sorted list."""
        return self._len

    def __getitem__(self, index):
        """Lookup value at `index` in sorted list."""
        pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
        return self._lists[pos][idx]

    def __delitem__(self, index):
        """Remove value at `index` from sorted list."""
        pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
        self._delete(pos, idx)

    def __contains__(self, value):
        """Return true if `value` is an element of the sorted list."""
        _lists = self._lists
        if _lists:
            pos, idx = self._loc_left(value)
            return idx < len(_lists[pos]) and _lists[pos][idx] == value
        return False

    def __iter__(self):
        """Return an iterator over the sorted list."""
        return (value for _list in self._lists for value in _list)

    def __reversed__(self):
        """Return a reverse iterator over the sorted list."""
        return (value for _list in reversed(self._lists) for value in reversed(_list))

    def __repr__(self):
        """Return string representation of sorted list."""
        return 'SortedList({0})'.format(list(self))


T = TypeVar('T')


class SortedSet(Generic[T]):
    BUCKET_RATIO = 50
    REBUILD_RATIO = 170

    def _build(self, a=None) -> None:
        "Evenly divide `a` into buckets."
        if a is None: a = list(self)
        size = self.size = len(a)
        bucket_size = int(math.ceil(math.sqrt(size / self.BUCKET_RATIO)))
        self.a = [a[size * i // bucket_size: size * (i + 1) // bucket_size] for i in range(bucket_size)]

    def __init__(self, a: Iterable[T] = []) -> None:
        "Make a new SortedSet from iterable. / O(N) if sorted and unique / O(N log N)"
        a = list(a)
        if not all(a[i] < a[i + 1] for i in range(len(a) - 1)):
            a = sorted(set(a))
        self._build(a)

    def __iter__(self) -> Iterator[T]:
        for i in self.a:
            for j in i: yield j

    def __reversed__(self) -> Iterator[T]:
        for i in reversed(self.a):
            for j in reversed(i): yield j

    def __len__(self) -> int:
        return self.size

    def __repr__(self) -> str:
        return "SortedSet" + str(self.a)

    def __str__(self) -> str:
        s = str(list(self))
        return "{" + s[1: len(s) - 1] + "}"

    def _find_bucket(self, x: T) -> List[T]:
        "Find the bucket which should contain x. self must not be empty."
        for a in self.a:
            if x <= a[-1]: return a
        return a

    def __contains__(self, x: T) -> bool:
        if self.size == 0: return False
        a = self._find_bucket(x)
        i = bisect_left(a, x)
        return i != len(a) and a[i] == x

    def add(self, x: T) -> bool:
        "Add an element and return True if added. / O(√N)"
        if self.size == 0:
            self.a = [[x]]
            self.size = 1
            return True
        a = self._find_bucket(x)
        i = bisect_left(a, x)
        if i != len(a) and a[i] == x: return False
        a.insert(i, x)
        self.size += 1
        if len(a) > len(self.a) * self.REBUILD_RATIO:
            self._build()
        return True

    def discard(self, x: T) -> bool:
        "Remove an element and return True if removed. / O(√N)"
        if self.size == 0: return False
        a = self._find_bucket(x)
        i = bisect_left(a, x)
        if i == len(a) or a[i] != x: return False
        a.pop(i)
        self.size -= 1
        if len(a) == 0: self._build()
        return True

    def lt(self, x: T) -> Union[T, None]:
        "Find the largest element < x, or None if it doesn't exist."
        for a in reversed(self.a):
            if a[0] < x:
                return a[bisect_left(a, x) - 1]

    def le(self, x: T) -> Union[T, None]:
        "Find the largest element <= x, or None if it doesn't exist."
        for a in reversed(self.a):
            if a[0] <= x:
                return a[bisect_right(a, x) - 1]

    def gt(self, x: T) -> Union[T, None]:
        "Find the smallest element > x, or None if it doesn't exist."
        for a in self.a:
            if a[-1] > x:
                return a[bisect_right(a, x)]

    def ge(self, x: T) -> Union[T, None]:
        "Find the smallest element >= x, or None if it doesn't exist."
        for a in self.a:
            if a[-1] >= x:
                return a[bisect_left(a, x)]

    def __getitem__(self, x: int) -> T:
        "Return the x-th element, or IndexError if it doesn't exist."
        if x < 0: x += self.size
        if x < 0: raise IndexError
        for a in self.a:
            if x < len(a): return a[x]
            x -= len(a)
        raise IndexError

    def index(self, x: T) -> int:
        "Count the number of elements < x."
        ans = 0
        for a in self.a:
            if a[-1] >= x:
                return ans + bisect_left(a, x)
            ans += len(a)
        return ans

    def index_right(self, x: T) -> int:
        "Count the number of elements <= x."
        ans = 0
        for a in self.a:
            if a[-1] > x:
                return ans + bisect_right(a, x)
            ans += len(a)
        return ans


BUFSIZE = 4096


class FastIO(IOBase):
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)


class IOWrapper(IOBase):
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")


sys.stdin = IOWrapper(sys.stdin)
sys.stdout = IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")


def I():
    return input()


def II():
    return int(input())


def MI():
    return map(int, input().split())


def LI():
    return list(input().split())


def LII():
    return list(map(int, input().split()))


def GMI():
    return map(lambda x: int(x) - 1, input().split())


def LGMI():
    return list(map(lambda x: int(x) - 1, input().split()))


def solve():
    n = II()
    b = []
    for _ in range(n):
        row = I()
        b.append(row)
    g = defaultdict(list)
    smallers = [0] * n # smaller[i]表示被第i个节点包含的总节点个数
    for i in range(n):
        for j in range(n):
            if b[i][j] == '1': # 记录下一个位置，以及累加比下一个位置小的个数
                g[i].append(j)
                smallers[j] += 1
    # 记录最小的set
    ans = [set() for _ in range(n)]
    lst = [i for i in range(n) if smallers[i] == 0]
    cur = 1 # 独特的元素
    while lst:
        top = lst.pop()
        ans[top].add(cur) # 独特的元素
        # 给它的所有父亲加上独特的元素
        for fa in g[top]:
            ans[fa].add(cur)
            smallers[fa] -= 1 # 可以去掉一个牵绊
            if smallers[fa] == 0: # 了无牵绊，可以加入
                lst.append(fa)
        cur += 1
    for i in range(n):
        print(len(ans[i]), *ans[i])



if __name__ == '__main__':
    for _ in range(II()):
        solve()

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总结

• 这是一道图论入度的构造题
• 难点是如何保证某些bij = 0
• 这就需要进行分组，然后引入独特性cur