Negative Cycle

AtCoder

IDagc036_d

Time2000ms

Memory256MB

Difficulty—

We have a weighted directed graph with $N$ vertices numbered $0$ to $N-1$. The graph initially has $N-1$ edges. The $i$\-th edge ($0 \leq i \leq N-2$) is directed from Vertex $i$ to Vertex $i+1$ and has a weight of $0$. Snuke will now add a new edge $(i → j)$ for every pair $i, j$ ($0 \leq i,j \leq N-1,\ i \neq j$). The weight of the edge will be $-1$ if $i < j$, and $1$ otherwise. Ringo is a boy. A negative cycle (a cycle whose total weight is negative) in a graph makes him sad. He will delete some of the edges added by Snuke so that the graph will no longer contain a negative cycle. The cost of deleting the edge $(i → j)$ is $A_{i,j}$. He cannot delete edges that have been present from the beginning. Find the minimum total cost required to achieve Ringo's objective. ## Constraints * $3 \leq N \leq 500$ * $1 \leq A_{i,j} \leq 10^9$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_{0,1}$ $A_{0,2}$ $A_{0,3}$ $\cdots$ $A_{0,N-1}$ $A_{1,0}$ $A_{1,2}$ $A_{1,3}$ $\cdots$ $A_{1,N-1}$ $A_{2,0}$ $A_{2,1}$ $A_{2,3}$ $\cdots$ $A_{2,N-1}$ $\vdots$ $A_{N-1,0}$ $A_{N-1,1}$ $A_{N-1,2}$ $\cdots$ $A_{N-1,N-2}$ [samples]

Samples

Input #1

Output #1

2

If we delete the edge $(0 → 1)$ added by Snuke, the graph will no longer contain a negative cycle. The cost will be $2$ in this case, which is the minimum possible.

Input #2

Output #2

2

If we delete the edges $(1 → 2)$ and $(3 → 0)$ added by Snuke, the graph will no longer contain a negative cycle. The cost will be $1+1=2$ in this case, which is the minimum possible.

Input #3

10
190587 2038070 142162180 88207341 215145790 38 2 5 20
32047998 21426 4177178 52 734621629 2596 102224223 5 1864
41 481241221 1518272 51 772 146 8805349 3243297 449
918151 126080576 5186563 46354 6646 491776 5750138 2897 161
3656 7551068 2919714 43035419 495 3408 26 3317 2698
455357 3 12 1857 5459 7870 4123856 2402 258
3 25700 16191 102120 971821039 52375 40449 20548149 16186673
2 16 130300357 18 6574485 29175 179 1693 2681
99 833 131 2 414045824 57357 56 302669472 95
8408 7 1266941 60620177 129747 41382505 38966 187 5151064

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Negative Cycle",
    "description": {
      "content": "We have a weighted directed graph with $N$ vertices numbered $0$ to $N-1$. The graph initially has $N-1$ edges. The $i$\\-th edge ($0 \\leq i \\leq N-2$) is directed from Vertex $i$ to Vertex $i+1$ and h",
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc036_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a weighted directed graph with $N$ vertices numbered $0$ to $N-1$.\nThe graph initially has $N-1$ edges. The $i$\\-th edge ($0 \\leq i \\leq N-2$) is directed from Vertex $i$ to Vertex $i+1$ and h...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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