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F - Make Bipartite

Score : $500$ points

Problem Statement

Given is an undirected graph with $N+1$ vertices.
The vertices are called Vertex $0$, Vertex $1$, $\ldots$, Vertex $N$.

For each $i=1,2,\ldots,N$, the graph has an undirected edge with a weight of $A_i$ connecting Vertex $0$ and Vertex $i$.

Additionally, for each $i=1,2,\ldots,N$, the graph has an undirected edge with a weight of $B_i$ connecting Vertex $i$ and Vertex $i+1$. (Here, Vertex $N+1$ stands for Vertex $1$.)

The graph has no edge other than these $2N$ edges above.

Let us delete some of the edges from this graph so that the graph will be bipartite.
What is the minimum total weight of the edges that have to be deleted?

Constraints

• $3 \leq N \leq 2 \times 10^5$
• $1 \leq A_i \leq 10^9$
• $1 \leq B_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\dots$ $A_N$

$B_1$ $B_2$ $\dots$ $B_N$

Output

Print the answer.

5
31 4 159 2 65
5 5 5 5 10

16

Deleting the edge connecting Vertices $0,2$ (weight: $4$), the edge connecting Vertices $0,4$ (weight: $2$), and the edge connecting Vertices $1,5$ (weight: $10$) makes the graph bipartite.

4
100 100 100 1000000000
1 2 3 4

10