A12493 | Number of Components
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题目描述
The Kingdom of Kremland is a tree (a connected undirected graph without cycles) consisting of $n$ vertices. Each vertex $i$ has its own value $a_i$ . All vertices are connected in series by edges. Formally, for every $1 \leq i < n$ there is an edge between the vertices of $i$ and $i+1$ .
Denote the function $f(l, r)$ , which takes two integers $l$ and $r$ ( $l \leq r$ ):
- We leave in the tree only vertices whose values range from $l$ to $r$ .
- The value of the function will be the number of connected components in the new graph.
Your task is to calculate the following sum: $$$$\sum_{l=1}^{n} \sum_{r=l}^{n} f(l, r) $$$$
Denote the function $f(l, r)$ , which takes two integers $l$ and $r$ ( $l \leq r$ ):
- We leave in the tree only vertices whose values range from $l$ to $r$ .
- The value of the function will be the number of connected components in the new graph.
Your task is to calculate the following sum: $$$$\sum_{l=1}^{n} \sum_{r=l}^{n} f(l, r) $$$$
输入格式
The first line contains a single integer $n$ ( $1 \leq n \leq 10^5$ ) — the number of vertices in the tree.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq n$ ) — the values of the vertices.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq n$ ) — the values of the vertices.
输出格式
Print one number — the answer to the problem.
输入输出样例
输入 #1
3 2 1 3
输出 #1
7
输入 #2
4 2 1 1 3
输出 #2
11
输入 #3
10 1 5 2 5 5 3 10 6 5 1
输出 #3
104
In the first example, the function values will be as follows:
- $f(1, 1)=1$ (there is only a vertex with the number $2$ , which forms one component)
- $f(1, 2)=1$ (there are vertices $1$ and $2$ that form one component)
- $f(1, 3)=1$ (all vertices remain, one component is obtained)
- $f(2, 2)=1$ (only vertex number $1$ )
- $f(2, 3)=2$ (there are vertices $1$ and $3$ that form two components)
- $f(3, 3)=1$ (only vertex $3$ )
Totally out $7$ .In the second example, the function values will be as follows:
- $f(1, 1)=1$
- $f(1, 2)=1$
- $f(1, 3)=1$
- $f(1, 4)=1$
- $f(2, 2)=1$
- $f(2, 3)=2$
- $f(2, 4)=2$
- $f(3, 3)=1$
- $f(3, 4)=1$
- $f(4, 4)=0$ (there is no vertex left, so the number of components is $0$ )
Totally out $11$ .
- $f(1, 1)=1$ (there is only a vertex with the number $2$ , which forms one component)
- $f(1, 2)=1$ (there are vertices $1$ and $2$ that form one component)
- $f(1, 3)=1$ (all vertices remain, one component is obtained)
- $f(2, 2)=1$ (only vertex number $1$ )
- $f(2, 3)=2$ (there are vertices $1$ and $3$ that form two components)
- $f(3, 3)=1$ (only vertex $3$ )
Totally out $7$ .In the second example, the function values will be as follows:
- $f(1, 1)=1$
- $f(1, 2)=1$
- $f(1, 3)=1$
- $f(1, 4)=1$
- $f(2, 2)=1$
- $f(2, 3)=2$
- $f(2, 4)=2$
- $f(3, 3)=1$
- $f(3, 4)=1$
- $f(4, 4)=0$ (there is no vertex left, so the number of components is $0$ )
Totally out $11$ .
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评测结果：Accepted