From 07bdf7ce0e2806fd386939941a931bd087f6fa17 Mon Sep 17 00:00:00 2001
From: hitonanode <32937551+hitonanode@users.noreply.github.com>
Date: Sun, 28 Apr 2024 17:28:53 +0900
Subject: [PATCH] acl_beats: add comments

---
 segmenttree/acl_beats.hpp | 34 ++++++++++++++++++++++++----------
 1 file changed, 24 insertions(+), 10 deletions(-)

diff --git a/segmenttree/acl_beats.hpp b/segmenttree/acl_beats.hpp
index 77c5e926..4396b131 100644
--- a/segmenttree/acl_beats.hpp
+++ b/segmenttree/acl_beats.hpp
@@ -1,7 +1,6 @@
 #pragma once
 #include "acl_lazysegtree.hpp"
 
-// CUT begin
 template <class S, S (*op)(S, S), S (*e)(), class F, S (*mapping)(F, S), F (*composition)(F, F),
           F (*id)()>
 class segtree_beats : public atcoder::lazy_segtree<S, op, e, F, mapping, composition, id> {
@@ -19,12 +18,14 @@ class segtree_beats : public atcoder::lazy_segtree<S, op, e, F, mapping, composi
 namespace RangeChMinMaxAddSum {
 #include <algorithm>
 
-template <typename Num>
-inline Num second_lowest(Num a, Num a2, Num c, Num c2) noexcept { // a < a2, c < c2
+template <typename Num> inline Num second_lowest(Num a, Num a2, Num c, Num c2) noexcept {
+    assert(a <= a2); // a < a2 or a == a2 == INF
+    assert(c <= c2); // c < c2 or c == c2 == -INF
     return a == c ? std::min(a2, c2) : a2 <= c ? a2 : c2 <= a ? c2 : std::max(a, c);
 }
-template <typename Num>
-inline Num second_highest(Num a, Num a2, Num b, Num b2) noexcept { // a > a2, b > b2
+template <typename Num> inline Num second_highest(Num a, Num a2, Num b, Num b2) noexcept {
+    assert(a >= a2); // a > a2 or a == a2 == -INF
+    assert(b >= b2); // b > b2 or b == b2 == INF
     return a == b ? std::max(a2, b2) : a2 >= b ? a2 : b2 >= a ? b2 : std::min(a, b);
 }
 
@@ -36,7 +37,7 @@ struct S {
     unsigned sz, nlo, nhi;
     bool fail;
     S() : lo(BINF), hi(-BINF), lo2(BINF), hi2(-BINF), sum(0), sz(0), nlo(0), nhi(0), fail(0) {}
-    S(BNum x, unsigned sz_ = 1)
+    S(BNum x, unsigned sz_)
         : lo(x), hi(x), lo2(BINF), hi2(-BINF), sum(x * sz_), sz(sz_), nlo(sz_), nhi(sz_), fail(0) {}
     friend std::ostream &operator<<(std::ostream &os, const S s) {
         return os << "[(" << s.lo << "x" << s.nlo << ", " << s.lo2 << ", " << s.hi2 << ", " << s.hi
@@ -49,10 +50,12 @@ S op(S l, S r) {
     if (l.lo > l.hi) return r;
     if (r.lo > r.hi) return l;
     S ret;
-    ret.lo = std::min(l.lo, r.lo), ret.hi = std::max(l.hi, r.hi);
+    ret.lo = std::min(l.lo, r.lo);
+    ret.hi = std::max(l.hi, r.hi);
     ret.lo2 = second_lowest(l.lo, l.lo2, r.lo, r.lo2),
     ret.hi2 = second_highest(l.hi, l.hi2, r.hi, r.hi2);
-    ret.sum = l.sum + r.sum, ret.sz = l.sz + r.sz;
+    ret.sum = l.sum + r.sum;
+    ret.sz = l.sz + r.sz;
     ret.nlo = l.nlo * (l.lo <= r.lo) + r.nlo * (r.lo <= l.lo);
     ret.nhi = l.nhi * (l.hi >= r.hi) + r.nhi * (r.hi >= l.hi);
     return ret;
@@ -76,21 +79,32 @@ F composition(F fnew, F fold) {
 F id() { return F(); }
 S mapping(F f, S x) {
     if (x.sz == 0) return e();
-    if (x.lo == x.hi or f.lb == f.ub or f.lb >= x.hi or f.ub <= x.lo) 
+
+    // f の作用後 x の要素が 1 種類だけになるケース
+    if (x.lo == x.hi or f.lb == f.ub or f.lb >= x.hi or f.ub <= x.lo) {
         return S(std::min(std::max(x.lo, f.lb), f.ub) + f.bias, x.sz);
+    }
+
+    // 2 種類 -> 1 種類
     if (x.lo2 == x.hi) {
-        x.lo = x.hi2 = std::max(x.lo, f.lb) + f.bias, x.hi = x.lo2 = std::min(x.hi, f.ub) + f.bias;
+        x.lo = x.hi2 = std::max(x.lo, f.lb) + f.bias;
+        x.hi = x.lo2 = std::min(x.hi, f.ub) + f.bias;
         x.sum = x.lo * x.nlo + x.hi * x.nhi;
         return x;
     }
+
+    // lo と lo2, hi と hi2 が潰れないケース
     if (f.lb < x.lo2 and f.ub > x.hi2) {
         BNum nxt_lo = std::max(x.lo, f.lb), nxt_hi = std::min(x.hi, f.ub);
         x.sum += (nxt_lo - x.lo) * x.nlo - (x.hi - nxt_hi) * x.nhi + f.bias * x.sz;
         x.lo = nxt_lo + f.bias, x.hi = nxt_hi + f.bias, x.lo2 += f.bias, x.hi2 += f.bias;
         return x;
     }
+
     x.fail = 1;
     return x;
 }
+
 using segtree = segtree_beats<S, op, e, F, mapping, composition, id>;
+
 } // namespace RangeChMinMaxAddSum