cp-library
This documentation is automatically generated by online-judge-tools/verification-helper
test/library-checker/convolution/min_plus_convolution_convex_convex.test.py
- View this file on GitHub
- Last update: 2025-08-24 20:27:05+09:00
- Problem: https://judge.yosupo.jp/problem/min_plus_convolution_convex_convex
Depends on
- cp_library/alg/dp/max2_fn.py
- cp_library/alg/dp/monotone_minima_fn.py
- cp_library/ds/list/elist_fn.py
- cp_library/io/io_base_cls.py
- cp_library/io/io_cls.py
- cp_library/io/parsable_cls.py
- cp_library/io/parser_cls.py
- cp_library/io/read_fn.py
- cp_library/io/write_fn.py
- cp_library/math/conv/minplus_conv_fn.py
Code
# verification-helper: PROBLEM https://judge.yosupo.jp/problem/min_plus_convolution_convex_convex
def main():
N, M = read(tuple[int, ...])
A = read(list[int])
B = read(list[int])
C = minplus_conv_cnvx(A,B)
write(*C)
from cp_library.io.read_fn import read
from cp_library.io.write_fn import write
from cp_library.math.conv.minplus_conv_fn import minplus_conv_cnvx
if __name__ == "__main__":
main()# verification-helper: PROBLEM https://judge.yosupo.jp/problem/min_plus_convolution_convex_convex
def main():
N, M = read(tuple[int, ...])
A = read(list[int])
B = read(list[int])
C = minplus_conv_cnvx(A,B)
write(*C)
'''
╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸
https://kobejean.github.io/cp-library
'''
from typing import Type, Union, overload
from typing import TypeVar
_S = TypeVar('S'); _T = TypeVar('T'); _U = TypeVar('U'); _T1 = TypeVar('T1'); _T2 = TypeVar('T2'); _T3 = TypeVar('T3'); _T4 = TypeVar('T4'); _T5 = TypeVar('T5'); _T6 = TypeVar('T6')
@overload
def read() -> list[int]: ...
@overload
def read(spec: Type[_T], char=False) -> _T: ...
@overload
def read(spec: _U, char=False) -> _U: ...
@overload
def read(*specs: Type[_T], char=False) -> tuple[_T, ...]: ...
@overload
def read(*specs: _U, char=False) -> tuple[_U, ...]: ...
def read(*specs: Union[Type[_T],_T], char=False):
IO.stdin.char = char
if not specs: return IO.stdin.readnumsinto([])
parser: _T = Parser.compile(specs[0] if len(specs) == 1 else specs)
return parser(IO.stdin)
from os import read as os_read, write as os_write, fstat as os_fstat
import sys
from __pypy__.builders import StringBuilder
def max2(a, b): return a if a > b else b
class IOBase:
@property
def char(io) -> bool: ...
@property
def writable(io) -> bool: ...
def __next__(io) -> str: ...
def write(io, s: str) -> None: ...
def readline(io) -> str: ...
def readtoken(io) -> str: ...
def readtokens(io) -> list[str]: ...
def readints(io) -> list[int]: ...
def readdigits(io) -> list[int]: ...
def readnums(io) -> list[int]: ...
def readchar(io) -> str: ...
def readchars(io) -> str: ...
def readinto(io, lst: list[str]) -> list[str]: ...
def readcharsinto(io, lst: list[str]) -> list[str]: ...
def readtokensinto(io, lst: list[str]) -> list[str]: ...
def readintsinto(io, lst: list[int]) -> list[int]: ...
def readdigitsinto(io, lst: list[int]) -> list[int]: ...
def readnumsinto(io, lst: list[int]) -> list[int]: ...
def wait(io): ...
def flush(io) -> None: ...
def line(io) -> list[str]: ...
class IO(IOBase):
BUFSIZE = 1 << 16; stdin: 'IO'; stdout: 'IO'
__slots__ = 'f', 'file', 'B', 'O', 'V', 'S', 'l', 'p', 'char', 'sz', 'st', 'ist', 'writable', 'encoding', 'errors'
def __init__(io, file):
io.file = file
try: io.f = file.fileno(); io.sz, io.writable = max2(io.BUFSIZE, os_fstat(io.f).st_size), ('x' in file.mode or 'r' not in file.mode)
except: io.f, io.sz, io.writable = -1, io.BUFSIZE, False
io.B, io.O, io.S = bytearray(), [], StringBuilder(); io.V = memoryview(io.B); io.l = io.p = 0
io.char, io.st, io.ist, io.encoding, io.errors = False, [], [], 'ascii', 'ignore'
def _dec(io, l, r): return io.V[l:r].tobytes().decode(io.encoding, io.errors)
def readbytes(io, sz): return os_read(io.f, sz)
def load(io):
while io.l >= len(io.O):
if not (b := io.readbytes(io.sz)):
if io.O[-1] < len(io.B): io.O.append(len(io.B))
break
pos = len(io.B); io.B.extend(b)
while ~(pos := io.B.find(b'\n', pos)): io.O.append(pos := pos+1)
def __next__(io):
if io.char: return io.readchar()
else: return io.readtoken()
def readchar(io):
io.load(); r = io.O[io.l]
c = chr(io.B[io.p])
if io.p >= r-1: io.p = r; io.l += 1
else: io.p += 1
return c
def write(io, s: str): io.S.append(s)
def readline(io): io.load(); l, io.p = io.p, io.O[io.l]; io.l += 1; return io._dec(l, io.p)
def readtoken(io):
io.load(); r = io.O[io.l]
if ~(p := io.B.find(b' ', io.p, r)): s = io._dec(io.p, p); io.p = p+1
else: s = io._dec(io.p, r-1); io.p = r; io.l += 1
return s
def readtokens(io): io.st.clear(); return io.readtokensinto(io.st)
def readints(io): io.ist.clear(); return io.readintsinto(io.ist)
def readdigits(io): io.ist.clear(); return io.readdigitsinto(io.ist)
def readnums(io): io.ist.clear(); return io.readnumsinto(io.ist)
def readchars(io): io.load(); l, io.p = io.p, io.O[io.l]; io.l += 1; return io._dec(l, io.p-1)
def readinto(io, lst):
if io.char: return io.readcharsinto(lst)
else: return io.readtokensinto(lst)
def readcharsinto(io, lst): lst.extend(io.readchars()); return lst
def readtokensinto(io, lst):
io.load(); r = io.O[io.l]
while ~(p := io.B.find(b' ', io.p, r)): lst.append(io._dec(io.p, p)); io.p = p+1
lst.append(io._dec(io.p, r-1)); io.p = r; io.l += 1; return lst
def _readint(io, r):
while io.p < r and io.B[io.p] <= 32: io.p += 1
if io.p >= r: return None
minus = x = 0
if io.B[io.p] == 45: minus = 1; io.p += 1
while io.p < r and io.B[io.p] >= 48: x = x * 10 + (io.B[io.p] & 15); io.p += 1
io.p += 1
return -x if minus else x
def readintsinto(io, lst):
io.load(); r = io.O[io.l]
while io.p < r and (x := io._readint(r)) is not None: lst.append(x)
io.l += 1; return lst
def _readdigit(io): d = io.B[io.p] & 15; io.p += 1; return d
def readdigitsinto(io, lst):
io.load(); r = io.O[io.l]
while io.p < r and io.B[io.p] > 32: lst.append(io._readdigit())
if io.B[io.p] == 10: io.l += 1
io.p += 1
return lst
def readnumsinto(io, lst):
if io.char: return io.readdigitsinto(lst)
else: return io.readintsinto(lst)
def line(io): io.st.clear(); return io.readinto(io.st)
def wait(io):
io.load(); r = io.O[io.l]
while io.p < r: yield
def flush(io):
if io.writable: os_write(io.f, io.S.build().encode(io.encoding, io.errors)); io.S = StringBuilder()
sys.stdin = IO.stdin = IO(sys.stdin); sys.stdout = IO.stdout = IO(sys.stdout)
import typing
from numbers import Number
from types import GenericAlias
from typing import Callable, Collection
class Parsable:
@classmethod
def compile(cls):
def parser(io: 'IOBase'): return cls(next(io))
return parser
@classmethod
def __class_getitem__(cls, item): return GenericAlias(cls, item)
class Parser:
def __init__(self, spec): self.parse = Parser.compile(spec)
def __call__(self, io: IOBase): return self.parse(io)
@staticmethod
def compile_type(cls, args = ()):
if issubclass(cls, Parsable): return cls.compile(*args)
elif issubclass(cls, (Number, str)):
def parse(io: IOBase): return cls(next(io))
return parse
elif issubclass(cls, tuple): return Parser.compile_tuple(cls, args)
elif issubclass(cls, Collection): return Parser.compile_collection(cls, args)
elif callable(cls):
def parse(io: IOBase): return cls(next(io))
return parse
else: raise NotImplementedError()
@staticmethod
def compile(spec=int):
if isinstance(spec, (type, GenericAlias)):
cls, args = typing.get_origin(spec) or spec, typing.get_args(spec) or tuple()
return Parser.compile_type(cls, args)
elif isinstance(offset := spec, Number):
cls = type(spec)
def parse(io: IOBase): return cls(next(io)) + offset
return parse
elif isinstance(args := spec, tuple): return Parser.compile_tuple(type(spec), args)
elif isinstance(args := spec, Collection): return Parser.compile_collection(type(spec), args)
elif isinstance(fn := spec, Callable):
def parse(io: IOBase): return fn(next(io))
return parse
else: raise NotImplementedError()
@staticmethod
def compile_line(cls, spec=int):
if spec is int:
def parse(io: IOBase): return cls(io.readnums())
elif spec is str:
def parse(io: IOBase): return cls(io.line())
else:
fn = Parser.compile(spec)
def parse(io: IOBase): return cls((fn(io) for _ in io.wait()))
return parse
@staticmethod
def compile_repeat(cls, spec, N):
fn = Parser.compile(spec)
def parse(io: IOBase): return cls([fn(io) for _ in range(N)])
return parse
@staticmethod
def compile_children(cls, specs):
fns = tuple((Parser.compile(spec) for spec in specs))
def parse(io: IOBase): return cls([fn(io) for fn in fns])
return parse
@staticmethod
def compile_tuple(cls, specs):
if isinstance(specs, (tuple,list)) and len(specs) == 2 and specs[1] is ...: return Parser.compile_line(cls, specs[0])
else: return Parser.compile_children(cls, specs)
@staticmethod
def compile_collection(cls, specs):
if not specs or len(specs) == 1 or isinstance(specs, set):
return Parser.compile_line(cls, *specs)
elif (isinstance(specs, (tuple,list)) and len(specs) == 2 and isinstance(specs[1], int)):
return Parser.compile_repeat(cls, specs[0], specs[1])
else:
raise NotImplementedError()
def write(*args, **kwargs):
'''Prints the values to a stream, or to stdout_fast by default.'''
sep, file = kwargs.pop("sep", " "), kwargs.pop("file", IO.stdout)
at_start = True
for x in args:
if not at_start: file.write(sep)
file.write(str(x))
at_start = False
file.write(kwargs.pop("end", "\n"))
if kwargs.pop("flush", False): file.flush()
def monotone_minima(N: int, M: int, func: Callable[[int,int,int],bool]):
'''
Finds row minima in a totally monotone N×M matrix using the SMAWK algorithm.
The matrix is defined implicitly through the comparison function.
A matrix is totally monotone if the minimum in row i occurs at column j,
then the minimum in row i+1 must occur at column j' where j ≤ j'.
Time: O(N log M), Space: O(N)
Args:
N: Number of rows
M: Number of columns
func(i,j,k): Returns True if element (i,j) < element (i,k)
Returns:
List of column indices containing the minimum value for each row
Example:
# Find minima where each element is (i-j)²
min_indices = monotone_minima(5, 5, lambda i,j,k: (i-j)**2 < (i-k)**2)
'''
min_j, st = [0] * N, elist(N)
st.append((0, N, 0, M))
while st:
li, ri, lj, rj = st.pop()
if li == ri: continue
mi, mj = li + ri >> 1, lj
for j in range(lj + 1, rj):
if func(mi, mj, j): mj = j
min_j[mi] = mj
st.append((li, mi, lj, mj+1))
st.append((mi+1, ri, mj, rj))
return min_j
def elist(hint: int) -> list: ...
try:
from __pypy__ import newlist_hint
except:
def newlist_hint(hint): return []
elist = newlist_hint
'''
╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸
x₀ ────────●─●────────●───●────────●───────●────────► X₀
╳ ╲ ╱ ╲ ╱
x₄ ────────●─●────────●─╳─●────────●─╲───╱─●────────► X₁
╳ ╳ ╲ ╲ ╱ ╱
x₂ ────────●─●────────●─╳─●────────●─╲─╳─╱─●────────► X₂
╳ ╱ ╲ ╲ ╳ ╳ ╱
x₆ ────────●─●────────●───●────────●─╳─╳─╳─●────────► X₃
╳ ╳ ╳ ╳
x₁ ────────●─●────────●───●────────●─╳─╳─╳─●────────► X₄
╳ ╲ ╱ ╱ ╳ ╳ ╲
x₅ ────────●─●────────●─╳─●────────●─╱─╳─╲─●────────► X₅
╳ ╳ ╱ ╱ ╲ ╲
x₃ ────────●─●────────●─╳─●────────●─╱───╲─●────────► X₆
╳ ╱ ╲ ╱ ╲
x₇ ────────●─●────────●───●────────●───────●────────► X₇
╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╸
Math - Convolution
'''
def minplus_conv_arb_cnvx(arb: list[int], cnvx: list[int]) -> list[int]:
N, M = len(cnvx), len(arb)
def cmp(i, j, k):
return i >= k and (i-j >= N or (cnvx[i-j] + arb[j] >= cnvx[i-k] + arb[k]))
cols = monotone_minima(N+M-1, M, cmp)
return [arb[j] + cnvx[i-j] for i, j in enumerate(cols)]
def minplus_conv_cnvx(A: list[int], B: list[int]) -> list[int]:
if not (N := len(A)) | (M := len(B)): return []
C = [0] * (K:=N+M-1)
C[0], I, J = A[i := 0] + B[j := 0], N-1, M-1
for k in range(1, K):
if j == J or (i != I and A[i+1] + B[j] < A[i] + B[j+1]): i += 1
else: j += 1
C[k] = A[i] + B[j]
return C
def minplus_iconv(A: list[int], B: list[int]):
N, M = len(A), len(B)
for i in range(N-1,-1,-1):
A[i] = min(B[j] + A[i-j] for j in range(min(M,i+1)))
if __name__ == "__main__":
main()