Min Cost Cycle

AtCoder

IDagc028_c

Time2000ms

Memory256MB

Difficulty—

We have a directed weighted graph with $N$ vertices. Each vertex has two integers written on it, and the integers written on Vertex $i$ are $A_i$ and $B_i$. In this graph, there is an edge from Vertex $x$ to Vertex $y$ for all pairs $1 \leq x,y \leq N$, and its weight is ${\rm min}(A_x,B_y)$. We will consider a directed cycle in this graph that visits every vertex exactly once. Find the minimum total weight of the edges in such a cycle. ## Constraints * $2 \leq N \leq 10^5$ * $1 \leq A_i \leq 10^9$ * $1 \leq B_i \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $B_1$ $A_2$ $B_2$ $:$ $A_N$ $B_N$ [samples]

Samples

Input #1

Output #1

7

Consider the cycle $1→3→2→1$. The weights of those edges are ${\rm min}(A_1,B_3)=1$, ${\rm min}(A_3,B_2)=2$ and ${\rm min}(A_2,B_1)=4$, for a total of $7$. As there is no cycle with a total weight of less than $7$, the answer is $7$.

Input #2

Output #2

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Min Cost Cycle",
    "description": {
      "content": "We have a directed weighted graph with $N$ vertices. Each vertex has two integers written on it, and the integers written on Vertex $i$ are $A_i$ and $B_i$. In this graph, there is an edge from Vertex",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc028_c"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a directed weighted graph with $N$ vertices. Each vertex has two integers written on it, and the integers written on Vertex $i$ are $A_i$ and $B_i$.\nIn this graph, there is an edge from Vertex...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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