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Problem solution · Python

ABC281 E — Least Elements

ABC281 E — Least Elements : a Python solution using sliding window or two pointers. Learn the idea, check the complexity, and read the full code, with credit to KATO-Hiro AtCoder Solutions.

Technique
Sliding window or two pointers
Source
KATO-Hiro AtCoder Solutions
Length
231 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Sliding window or two pointers

For ABC281 E — Least Elements , the implementation maintains a moving interval and updates only the information that enters or leaves the window.

  1. Choose the invariant that makes a window valid or useful.
  2. Advance the right boundary and add the new element.
  3. Move the left boundary only as needed while maintaining the invariant and updating the answer.

Code notes

  • 231 lines of Python from the credited upstream file abc281_e.py.
  • The implementation visibly relies on sequence storage, ordered lookup.
  • No explicit loop blocks detected.

Complexity

Confirm that neither pointer moves backwards; if so, the scan is usually linear apart from the window’s data-structure operations.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from KATO-Hiro AtCoder Solutions by KATO-Hiro and is used under the CC0-1.0 licence.

Open problem ↗Original source ↗
Full codeABC281 E — Least Elements · PythonPython
Use this to learn the idea, then write your own version.
# -*- coding: utf-8 -*-  import mathfrom bisect import bisect_left, bisect_right, insortfrom typing import Generic, Iterable, Iterator, TypeVar, Union, ListT = TypeVar('T')  class SortedMultiset(Generic[T]):    """Sorted multi set (set) in C++.     See:    https://qiita.com/tatyam/items/492c70ac4c955c055602    https://github.com/tatyam-prime/SortedSet/blob/main/SortedMultiset.py    """     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 SortedMultiset from iterable. / O(N) if sorted / O(N log N)"        a = list(a)         if not all(a[i] <= a[i + 1] for i in range(len(a) - 1)):  # type: ignore            a = sorted(a)  # type: ignore         self._build(a)     def __iter__(self) -> Iterator[T]:        for i in self.a:            for j in i:                yield j  # type: ignore     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 "SortedMultiset" + 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]:  # type: ignore                return a        return a  # type: ignore     def __contains__(self, x: T) -> bool:        if self.size == 0:            return False         a = self._find_bucket(x)        i = bisect_left(a, x)  # type: ignore        return i != len(a) and a[i] == x     def count(self, x: T) -> int:        "Count the number of x."        return self.index_right(x) - self.index(x)     def add(self, x: T) -> None:        "Add an element. / O(√N)"        if self.size == 0:            self.a = [[x]]            self.size = 1            return         a = self._find_bucket(x)        insort(a, x)  # type: ignore        self.size += 1         if len(a) > len(self.a) * self.REBUILD_RATIO:            self._build()     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)  # type: ignore         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:  # type: ignore                return a[bisect_left(a, x) - 1]  # type: ignore        return None     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:  # type: ignore                return a[bisect_right(a, x) - 1]  # type: ignore        return None     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:  # type: ignore                return a[bisect_right(a, x)]  # type: ignore        return None     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:  # type: ignore                return a[bisect_left(a, x)]  # type: ignore        return None     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]  # type: ignore             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:  # type: ignore                return ans + bisect_left(a, x)  # type: ignore            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:  # type: ignore                return ans + bisect_right(a, x)  # type: ignore            ans += len(a)        return ans  def main():    import sys     input = sys.stdin.readline     n, m, k = map(int, input().split())    a = list(map(int, input().split()))    b = sorted(a[:m])    left = SortedMultiset(b[:k])    right = SortedMultiset(b[k:])    summed = sum(b[:k])    ans = [summed]     # 差分を更新    # multisetを2つ使ってm個の要素のうち先頭k個、それ以外を管理    for i in range(n - m):        # m個の右端を追加        aj = a[i + m]        left_max = left[~0]         if aj <= left_max:            left.add(aj)            summed += aj        else:            right.add(aj)         # m個の左端を削除        ai = a[i]         if ai in left:            left.discard(ai)            summed -= ai        else:            right.discard(ai)                # 左側がk個、右側がm - k個となるように調整        while len(left) > k:            left_max = left[~0]            left.discard(left_max)            summed -= left_max             right.add(left_max)         while len(right) > m - k:            right_min = right[0]            right.discard(right_min)             left.add(right_min)            summed += right_min         ans.append(summed)     print(*ans)  if __name__ == "__main__":    main()

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