まぁ、キーを変えられる特殊なデータ構造を実装してもよいのだが、
いずれか片方をキーとしてライブラリを使って、
もう片方を求める際には、自前の二分探索で求めるのが手っ取り早いだろう。(多少、遅くなるが)
# https://github.com/tatyam-prime/SortedSet/blob/main/SortedSet.py
import math
from bisect import bisect_left, bisect_right
from typing import Generic, Iterable, Iterator, TypeVar, Union, List
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
def ccw(ax, ay, bx, by, cx, cy):
"""
C :1
A------>B C :0
C :-1
"""
dx1 = bx - ax
dy1 = by - ay
dx2 = cx - ax
dy2 = cy - ay
if dy1 * dx2 > dy2 * dx1:
return -1
if dx1 * dy2 > dy1 * dx2:
return 1
if dx1 * dx2 < 0 or dy1 * dy2 < 0:
return 1
if dx1 * dx1 + dy1 * dy1 < dx2 * dx2 + dy2 * dy2:
return -1
return 0
sst = SortedSet()
ssb = SortedSet()
INF30 = 1 << 30
INF60 = 1 << 61
sst.add(-INF30)
sst.add(INF30)
ssb.add(-INF30)
ssb.add(INF30)
x2y_t = {-INF30: -INF60, INF30: -INF60}
x2y_b = {-INF30: INF60, INF30: INF60}
buf = []
q = int(input())
for i in range(q):
x, y, a, b = map(int, input().split())
# print(x, y, a, b, buf)
# print(sst, x2y_t)
# print(ssb, x2y_b)
# Top
if x in x2y_t and x2y_t[x] > y:
pass
else:
lx = sst.lt(x)
rx = sst.gt(x)
ly = x2y_t[lx]
ry = x2y_t[rx]
if ccw(lx, ly, rx, ry, x, y) > 0:
sst.add(x)
x2y_t[x] = y
left_count = sst.index(x)
right_count = sst.size - sst.index_right(x)
while left_count >= 2:
llx = sst.lt(lx)
lly = x2y_t[llx]
if ccw(llx, lly, x, y, lx, ly) > 0:
break
sst.discard(lx)
del x2y_t[lx]
left_count -= 1
lx, ly = llx, lly
while right_count >= 2:
rrx = sst.gt(rx)
rry = x2y_t[rrx]
if ccw(x, y, rrx, rry, rx, ry) > 0:
break
sst.discard(rx)
del x2y_t[rx]
right_count -= 1
rx, ry = rrx, rry
# Bottom
if x in x2y_b and x2y_b[x] < y:
pass
else:
lx = ssb.lt(x)
rx = ssb.gt(x)
ly = x2y_b[lx]
ry = x2y_b[rx]
if ccw(lx, ly, rx, ry, x, y) < 0:
ssb.add(x)
x2y_b[x] = y
left_count = ssb.index(x)
right_count = ssb.size - ssb.index_right(x)
while left_count >= 2:
llx = ssb.lt(lx)
lly = x2y_b[llx]
if ccw(llx, lly, x, y, lx, ly) < 0:
break
ssb.discard(lx)
del x2y_b[lx]
left_count -= 1
lx, ly = llx, lly
while right_count >= 2:
rrx = ssb.gt(rx)
rry = x2y_b[rrx]
if ccw(x, y, rrx, rry, rx, ry) < 0:
break
ssb.discard(rx)
del x2y_b[rx]
right_count -= 1
rx, ry = rrx, rry
if b == 0:
if a == 0:
buf.append(0)
elif a > 0:
x = sst.lt(INF30)
buf.append(a * x)
else:
x = sst.gt(-INF30)
buf.append(a * x)
elif b > 0:
l = 0
r = sst.size
while l + 1 < r:
m = (l + r) // 2
x1 = sst[m]
y1 = x2y_t[x1]
x2 = sst[m + 1]
y2 = x2y_t[x2]
dx = x2 - x1
dy = y2 - y1
if b * dy >= -a * dx:
l = m
else:
r = m
x1 = sst[l]
x2 = sst[r]
ans1 = -INF60 if x1 == -INF30 else a * x1 + b * x2y_t[x1]
ans2 = -INF60 if x2 == INF30 else a * x2 + b * x2y_t[x2]
buf.append(max(ans1, ans2))
else:
l = 0
r = ssb.size
while l + 1 < r:
m = (l + r) // 2
x1 = ssb[m]
y1 = x2y_b[x1]
x2 = ssb[m + 1]
y2 = x2y_b[x2]
dx = x2 - x1
dy = y2 - y1
if b * dy >= -a * dx:
l = m
else:
r = m
x1 = ssb[l]
x2 = ssb[r]
ans1 = -INF60 if x1 == -INF30 else a * x1 + b * x2y_b[x1]
ans2 = -INF60 if x2 == INF30 else a * x2 + b * x2y_b[x2]
buf.append(max(ans1, ans2))
print('\n'.join(map(str, buf)))