Segtree

It is the data structure for monoids $(S, \cdot: S \times S \to S, e \in S)$, i.e., the algebraic structure that satisfies the following properties.

• associativity: $(a \cdot b) \cdot c$ = $a \cdot (b \cdot c)$ for all $a, b, c \in S$
• existence of the identity element: $a \cdot e$ = $e \cdot a$ = $a$ for all $a \in S$

Given an array $S$ of length $n$, it processes the following queries in $O(\log n)$ time (see Appendix for further details).

• Updating an element
• Calculating the product of the elements of an interval

For simplicity, in this document, we assume that the oracles op and e work in constant time. If these oracles work in $O(T)$ time, each time complexity appear in this document is multipled by $O(T)$.

Constructor

(1) segtree<S, op, e> seg(int n)
(2) segtree<S, op, e> seg(vector<S> v)

The following should be defined.

• The type S
• The binary operation S op(S a, S b)
• The identity element S e()

For example, for Range Minimum Query, the definitions are as follows.

int op(int a, int b) {
    return min(a, b);
}

int e() {
    return (int)(1e9);
}

segtree<int, op, e> seg(10);

• (1): It creates an array a of length n. All the elements are initialized to e().
• (2): It creates an array a of length n = v.size(), initialized to v.

See examples and here for further details.

Constraints

• $0 \leq n \leq 10^8$

Complexity

• $O(n)$

set

void seg.set(int p, S x)

It assigns $x$ to a[p].

Constraints

• $0 \leq p < n$

Complexity

• $O(\log n)$

get

S seg.get(int p)

It returns a[p].

Constraints

• $0 \leq p < n$

Complexity

• $O(1)$

prod

S seg.prod(int l, int r)

It returns op(a[l], ..., a[r - 1]), assuming the properties of the monoid. It returns e() if $l = r$.

Constraints

• $0 \leq l \leq r \leq n$

Complexity

• $O(\log n)$

all_prod

S seg.all_prod()

It returns op(a[0], ..., a[n - 1]), assuming the properties of the monoid. It returns e() if $n = 0$.

Complexity

• $O(1)$

max_right

(1) int seg.max_right<f>(int l)
(2) int seg.max_right<F>(int l, F f)

• (1): It applies binary search on the segment tree. The function bool f(S x) should be defined.
• (2): The function object that takes S as the argument and returns bool should be defined.

It returns an index r that satisfies both of the following.

• r = l or f(op(a[l], a[l + 1], ..., a[r - 1])) = true
• r = n or f(op(a[l], a[l + 1], ..., a[r])) = false

If f is monotone, this is the maximum r that satisfies f(op(a[l], a[l + 1], ..., a[r - 1])) = true.

Constraints

• if f is called with the same argument, it returns the same value, i.e., f has no side effect.
• f(e()) = true
• $0 \leq l \leq n$

Complexity

• $O(\log n)$

min_left

(1) int seg.min_left<f>(int r)
(2) int seg.min_left<F>(int r, F f)

• (1): It applies binary search on the segment tree. The function bool f(S x) should be defined.
• (2): The function object that takes S as the argument and returns bool should be defined.

It returns an index l that satisfies both of the following.

• l = r or f(op(a[l], a[l + 1], ..., a[r - 1])) = true
• l = 0 or f(op(a[l - 1], a[l], ..., a[r - 1])) = false

If f is monotone, this is the minimum l that satisfies f(op(a[l], a[l + 1], ..., a[r - 1])) = true.

Constraints

• if f is called with the same argument, it returns the same value, i.e., f has no side effect.
• f(e()) = true
• $0 \leq r \leq n$

Complexity

• $O(\log n)$

Examples

AC code of https://atcoder.jp/contests/practice2/tasks/practice2_j