数列 - 遅延伝播セグメントツリー (遅延セグ木)

概要

モノイド \(X\) とその上のモノイド（右）作用 \(M\) があるとする. このとき次のことが高速に出来る. \(X\) 上の数列 \(v\) について,

  • 区間取得
    • 添字区間 \(I\) について \(\prod_{i \in I} x_i\)
  • 区間更新
    • 添字区間 \(I\) と作用 \(m \in M\) について
      • 各 \(i \in I\) に対して \(x_i \leftarrow x_i \ast m\) (作用)

すなわち数列に対する更新操作をモノイド作用だということにしている. 計算量はこの2つの操作がともに \(O(\log N)\).

例題

/// Sequence - Lazy Segment Tree
use crate::algebra::act::*;
use crate::algebra::monoid::*;

#[derive(Debug, Clone)]
pub struct LazySegmentTree<X, M> {
    length: usize,       // of leaves
    length_upper: usize, // power of 2
    size: usize,         // of nodes
    data: Vec<X>,
    act: Vec<M>,
}
impl<X: Copy + Monoid, M: Copy + Monoid + Act<X>> LazySegmentTree<X, M> {
    pub fn new(length: usize) -> Self {
        let mut length_upper = 1;
        while length_upper < length {
            length_upper *= 2;
        }
        let size = length_upper * 2 - 1;
        let data = vec![X::unit(); size];
        let act = vec![M::unit(); size];
        LazySegmentTree {
            length,
            length_upper,
            size,
            data,
            act,
        }
    }
    pub fn from(xs: Vec<X>) -> Self {
        let mut tree = Self::new(xs.len());
        for i in 0..xs.len() {
            tree.data[tree.size / 2 + i] = xs[i];
        }
        for i in (0..tree.size / 2).rev() {
            tree.data[i] = tree.data[2 * i + 1] * tree.data[2 * i + 2];
        }
        tree
    }
    fn propagation(&mut self, idx: usize) {
        if idx < self.size / 2 {
            self.act[idx * 2 + 1] = self.act[idx * 2 + 1] * self.act[idx];
            self.act[idx * 2 + 2] = self.act[idx * 2 + 2] * self.act[idx];
        }
        self.data[idx] = self.act[idx].act(self.data[idx]);
        self.act[idx] = M::unit();
    }
    fn update_sub(
        &mut self,
        range: std::ops::Range<usize>,
        m: M,
        idx: usize,
        focus: std::ops::Range<usize>,
    ) {
        self.propagation(idx);
        if focus.end <= range.start || range.end <= focus.start {
            return;
        }
        if range.start <= focus.start && focus.end <= range.end {
            self.act[idx] = self.act[idx] * m;
            self.propagation(idx);
        } else if idx < self.data.len() / 2 {
            let mid = (focus.start + focus.end) / 2;
            self.update_sub(range.clone(), m, idx * 2 + 1, focus.start..mid);
            self.update_sub(range.clone(), m, idx * 2 + 2, mid..focus.end);
            self.data[idx] = self.data[idx * 2 + 1] * self.data[idx * 2 + 2];
        }
    }
    pub fn update(&mut self, range: std::ops::Range<usize>, m: M) {
        self.update_sub(range, m, 0, 0..self.length_upper);
    }
    fn product_sub(
        &mut self,
        range: std::ops::Range<usize>,
        idx: usize,
        focus: std::ops::Range<usize>,
    ) -> X {
        self.propagation(idx);
        if focus.end <= range.start || range.end <= focus.start {
            X::unit()
        } else if range.start <= focus.start && focus.end <= range.end {
            self.data[idx]
        } else {
            let mid = (focus.start + focus.end) / 2;
            let a = self.product_sub(range.clone(), idx * 2 + 1, focus.start..mid);
            let b = self.product_sub(range.clone(), idx * 2 + 2, mid..focus.end);
            a * b
        }
    }
    pub fn product(&mut self, range: std::ops::Range<usize>) -> X {
        self.product_sub(range, 0, 0..self.length_upper)
    }
    pub fn index(&mut self, i: usize) -> X {
        self.product(i..i + 1)
    }
    pub fn to_vec(&mut self) -> Vec<X> {
        (0..self.length).map(|i| self.index(i)).collect()
    }
}
impl<X: std::fmt::Debug, M: std::fmt::Debug> LazySegmentTree<X, M> {
    pub fn debug(&self) {
        #[cfg(debug_assertions)]
        for i in 0..self.size {
            if i > 0 && (i + 1).count_ones() == 1 {
                eprintln!();
            }
            eprint!("{:?} / {:?}; ", &self.data[i], &self.act[i]);
        }
        eprintln!();
    }
}