Two Types of Spells (Codeforces 1398E)
Tags: Greedy, binary search

Polycarp plays a computer game (yet again). In this game, he fights monsters using magic spells.There are two types of spells: fire spell of power $$$x$$$ deals $$$x$$$ damage to the monster, and lightning spell of power $$$y$$$ deals $$$y$$$ damage to the monster and doubles the damage of the next spell Polycarp casts. Each spell can be cast only once per battle, but Polycarp can cast them in any order.For example, suppose that Polycarp knows three spells: a fire spell of power $$$5$$$, a lightning spell of power $$$1$$$, and a lightning spell of power $$$8$$$. There are $$$6$$$ ways to choose the order in which he casts the spells:  first, second, third. This order deals $$$5 + 1 + 2 \cdot 8 = 22$$$ damage;  first, third, second. This order deals $$$5 + 8 + 2 \cdot 1 = 15$$$ damage;  second, first, third. This order deals $$$1 + 2 \cdot 5 + 8 = 19$$$ damage;  second, third, first. This order deals $$$1 + 2 \cdot 8 + 2 \cdot 5 = 27$$$ damage;  third, first, second. This order deals $$$8 + 2 \cdot 5 + 1 = 19$$$ damage;  third, second, first. This order deals $$$8 + 2 \cdot 1 + 2 \cdot 5 = 20$$$ damage. Initially, Polycarp knows $$$0$$$ spells. His spell set changes $$$n$$$ times, each time he either learns a new spell or forgets an already known one. After each change, calculate the maximum possible damage Polycarp may deal using the spells he knows.

Input: The first line contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of changes to the spell set.Each of the next $$$n$$$ lines contains two integers $$$tp$$$ and $$$d$$$ ($$$0 \le tp_i \le 1$$$; $$$-10^9 \le d \le 10^9$$$; $$$d_i \neq 0$$$) — the description of the change. If $$$tp_i$$$ if equal to $$$0$$$, then Polycarp learns (or forgets) a fire spell, otherwise he learns (or forgets) a lightning spell.If $$$d_i > 0$$$, then Polycarp learns a spell of power $$$d_i$$$. Otherwise, Polycarp forgets a spell with power $$$-d_i$$$, and it is guaranteed that he knew that spell before the change.It is guaranteed that the powers of all spells Polycarp knows after each change are different (Polycarp never knows two spells with the same power).

Output: After each change, print the maximum damage Polycarp can deal with his current set of spells.


Sample Inputs:
6
1 5
0 10
1 -5
0 5
1 11
0 -10

Sample Outputs:
5
25
10
15
36
21

Solution (Python):
import os
import sys
from io import BytesIO, IOBase
BUFSIZE = 8192

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, sys.stdout) = (IOWrapper(sys.stdin), IOWrapper(sys.stdout))
input = lambda : sys.stdin.readline()

def RL():
    return map(int, sys.stdin.readline().split())

def RLL():
    return list(map(int, sys.stdin.readline().split()))

def N():
    return int(input())

def print_list(l):
    print(' '.join(map(str, l)))

class SortedList:

    def __init__(self, iterable=[], _load=200):
        """"""
        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):
        """"""
        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):
        """"""
        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):
        """"""
        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):
        """"""
        _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):
        """"""
        _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):
        """"""
        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):
        """"""
        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):
        """"""
        _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):
        """"""
        _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):
        """"""
        _len = self._len
        self.discard(value)
        if _len == self._len:
            raise ValueError('{0!r} not in list'.format(value))

    def pop(self, index=-1):
        """"""
        (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):
        """"""
        (pos, idx) = self._loc_left(value)
        return self._fen_query(pos) + idx

    def bisect_right(self, value):
        """"""
        (pos, idx) = self._loc_right(value)
        return self._fen_query(pos) + idx

    def count(self, value):
        """"""
        return self.bisect_right(value) - self.bisect_left(value)

    def __len__(self):
        """"""
        return self._len

    def __getitem__(self, index):
        """"""
        (pos, idx) = self._fen_findkth(self._len + index if index < 0 else index)
        return self._lists[pos][idx]

    def __delitem__(self, index):
        """"""
        (pos, idx) = self._fen_findkth(self._len + index if index < 0 else index)
        self._delete(pos, idx)

    def __contains__(self, value):
        """"""
        _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 (value for _list in self._lists for value in _list)

    def __reversed__(self):
        """"""
        return (value for _list in reversed(self._lists) for value in reversed(_list))

    def __repr__(self):
        """"""
        return 'SortedList({0})'.format(list(self))
spell = SortedList([0])
lightning = SortedList()
(s, k) = (0, -1)
for _ in range(N()):
    (t, power) = RL()
    s += power
    if t == 1:
        if lightning:
            spell.add(lightning[0])
            if lightning[0] > spell[k]:
                s += lightning[0] - spell[k]
        if power > 0:
            lightning.add(power)
            spell.add(power)
            if power > spell[k]:
                s += power - spell[k]
        else:
            lightning.discard(-power)
            if -power > spell[k]:
                s -= -power - spell[k]
            spell.discard(-power)
        if lightning:
            if lightning[0] > spell[k]:
                s -= lightning[0] - spell[k]
            spell.discard(lightning[0])
    elif power > 0:
        spell.add(power)
        if power > spell[k]:
            s += power - spell[k]
    else:
        if -power > spell[k]:
            s -= -power - spell[k]
        spell.discard(-power)
    k = max(k, -len(spell))
    if -k > len(lightning) + 1:
        k += 1
        s -= spell[k]
    elif -k < len(lightning) + 1 and spell[k] > 0:
        s += spell[k]
        k -= 1
    print(s)

Solution (Python):
import bisect
import io, os
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline

class TreeSet(object):
    """"""

    def __init__(self, elements):
        self._treeset = []
        self.addAll(elements)

    def addAll(self, elements):
        for element in elements:
            if element in self:
                continue
            self.add(element)

    def add(self, element):
        if element not in self:
            bisect.insort(self._treeset, element)

    def ceiling(self, e):
        index = bisect.bisect_right(self._treeset, e)
        if self[index - 1] == e:
            return e
        return self._treeset[bisect.bisect_right(self._treeset, e)]

    def floor(self, e):
        index = bisect.bisect_left(self._treeset, e)
        if self[index] == e:
            return e
        else:
            return self._treeset[bisect.bisect_left(self._treeset, e) - 1]

    def __getitem__(self, num):
        return self._treeset[num]

    def __len__(self):
        return len(self._treeset)

    def clear(self):
        """"""
        self._treeset = []

    def clone(self):
        """"""
        return TreeSet(self._treeset)

    def remove(self, element):
        """"""
        try:
            self._treeset.remove(element)
        except ValueError:
            return False
        return True

    def __iter__(self):
        """"""
        for element in self._treeset:
            yield element

    def pop(self, index):
        return self._treeset.pop(index)

    def __str__(self):
        return str(self._treeset)

    def __eq__(self, target):
        if isinstance(target, TreeSet):
            return self._treeset == target.treeset
        elif isinstance(target, list):
            return self._treeset == target

    def __contains__(self, e):
        """"""
        try:
            return e == self._treeset[bisect.bisect_left(self._treeset, e)]
        except:
            return False
if __name__ == '__main__':
    n = int(input())
    MAX_VALUE = int(1000000000.0 + 1)
    ts = TreeSet([-MAX_VALUE, MAX_VALUE])
    info = {}
    cntL = 0
    cntL2 = 0
    min2 = MAX_VALUE
    sum1 = 0
    sum2 = 0
    for _ in range(n):
        (type, power) = list(map(int, input().split()))
        if power > 0:
            ts.add(power)
            info[power] = type
            if min2 == ts[len(ts) - cntL - 1]:
                sum1 += power
                if type == 1:
                    cntL += 1
                    min2 = ts[len(ts) - cntL - 1]
                    sum2 += min2
                    sum1 -= min2
                    cntL2 += info[min2]
            else:
                sum2 += power
                cntL2 += info[power]
                if type == 0:
                    sum2 -= min2
                    sum1 += min2
                    cntL2 -= info.get(min2, 0)
                else:
                    cntL += 1
        else:
            power = abs(power)
            ts.remove(power)
            if min2 == ts[len(ts) - cntL - 1]:
                sum1 -= power
                if type == 1:
                    sum2 -= min2
                    sum1 += min2
                    cntL2 -= info.get(min2, 0)
                    cntL -= 1
            else:
                sum2 -= power
                cntL2 -= info[power]
                if type == 0:
                    min2 = ts[len(ts) - cntL - 1]
                    sum2 += min2
                    sum1 -= min2
                    cntL2 += info[min2]
                else:
                    cntL -= 1
        min2 = ts[len(ts) - cntL - 1]
        if cntL2 < cntL or cntL == 0:
            ans = 2 * sum2 + sum1
        else:
            if len(ts) - cntL - 2 > 0:
                max1 = ts[len(ts) - cntL - 2]
            else:
                max1 = 0
            ans = 2 * sum2 - min2 + sum1 + max1
        print(ans)

Solution (Python):
import os
import sys
from io import BytesIO, IOBase
BUFSIZE = 8192

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, sys.stdout) = (IOWrapper(sys.stdin), IOWrapper(sys.stdout))
input = lambda : sys.stdin.readline()

def RL():
    return map(int, sys.stdin.readline().split())

def RLL():
    return list(map(int, sys.stdin.readline().split()))

def N():
    return int(input())

def print_list(l):
    print(' '.join(map(str, l)))

class SortedList:

    def __init__(self, iterable=[], _load=200):
        """"""
        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):
        """"""
        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):
        """"""
        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):
        """"""
        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):
        """"""
        _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):
        """"""
        _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):
        """"""
        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):
        """"""
        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):
        """"""
        _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):
        """"""
        _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):
        """"""
        _len = self._len
        self.discard(value)
        if _len == self._len:
            raise ValueError('{0!r} not in list'.format(value))

    def pop(self, index=-1):
        """"""
        (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):
        """"""
        (pos, idx) = self._loc_left(value)
        return self._fen_query(pos) + idx

    def bisect_right(self, value):
        """"""
        (pos, idx) = self._loc_right(value)
        return self._fen_query(pos) + idx

    def count(self, value):
        """"""
        return self.bisect_right(value) - self.bisect_left(value)

    def __len__(self):
        """"""
        return self._len

    def __getitem__(self, index):
        """"""
        (pos, idx) = self._fen_findkth(self._len + index if index < 0 else index)
        return self._lists[pos][idx]

    def __delitem__(self, index):
        """"""
        (pos, idx) = self._fen_findkth(self._len + index if index < 0 else index)
        self._delete(pos, idx)

    def __contains__(self, value):
        """"""
        _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 (value for _list in self._lists for value in _list)

    def __reversed__(self):
        """"""
        return (value for _list in reversed(self._lists) for value in reversed(_list))

    def __repr__(self):
        """"""
        return 'SortedList({0})'.format(list(self))
spell = SortedList([0])
lightning = SortedList()
data = [map(int, line.split()) for line in sys.stdin.readlines()]
n = data[0].__next__()
(s, k) = (0, -1)
for i in range(1, n + 1):
    (t, power) = data[i]
    s += power
    if t == 1:
        if lightning:
            spell.add(lightning[0])
            if lightning[0] > spell[k]:
                s += lightning[0] - spell[k]
        if power > 0:
            lightning.add(power)
            spell.add(power)
            if power > spell[k]:
                s += power - spell[k]
        else:
            lightning.discard(-power)
            if -power > spell[k]:
                s -= -power - spell[k]
            spell.discard(-power)
        if lightning:
            if lightning[0] > spell[k]:
                s -= lightning[0] - spell[k]
            spell.discard(lightning[0])
    elif power > 0:
        spell.add(power)
        if power > spell[k]:
            s += power - spell[k]
    else:
        if -power > spell[k]:
            s -= -power - spell[k]
        spell.discard(-power)
    k = max(k, -len(spell))
    if -k > len(lightning) + 1:
        k += 1
        s -= spell[k]
    elif -k < len(lightning) + 1 and spell[k] > 0:
        s += spell[k]
        k -= 1
    print(s)