Constant Sum Subsequence

AtCoder

IDarc150_f

Time5000ms

Memory256MB

Difficulty—

We have a sequence of positive integers of length $N^2$, $A=(A_1,\ A_2,\ \dots,\ A_{N^2})$, and a positive integer $S$. For this sequence of positive integers, $A_i=A_{i+N}$ holds for positive integers $i\ (1\leq i \leq N^2-N)$, and only $A_1,\ A_2,\ \dots,\ A_N$ are given as the input. Find the minimum integer $L$ such that every sequence $B$ of positive integers totaling $S$ is a (not necessarily contiguous) subsequence of the sequence $(A_1,\ A_2,\ \dots,\ A_L)$ of positive integers. It can be shown that at least one such $L$ exists under the Constraints of this problem. ## Constraints * $1 \leq N \leq 1.5 \times 10^6$ * $1 \leq S \leq \min(N,2 \times 10^5)$ * $1 \leq A_i \leq S$ * For every positive integer $x\ (1\leq x \leq S)$, $(A_1,\ A_2,\ \dots,\ A_N)$ contains at least one occurrence of $x$. * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $S$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

6 4
1 1 2 1 4 3

Output #1

9

There are eight sequences $B$ to consider: $B=(1,\ 1,\ 1,\ 1),\ (1,\ 1,\ 2),\ (1,\ 2,\ 1),\ (1,\ 3),\ (2,\ 1,\ 1),\ (2,\ 2),\ (3,\ 1),\ (4)$.
For $L=8$, for instance, $B=(2,\ 2)$ is not a subsequence of $(A_1,A_2,\ \dots,\ A_8)=(1,\ 1,\ 2,\ 1,\ 4,\ 3,\ 1,\ 1)$.
For $L=9$, on the other hand, every $B$ is a subsequence of $(A_1,A_2,\ \dots,\ A_9)=(1,\ 1,\ 2,\ 1,\ 4,\ 3,\ 1,\ 1,\ 2)$.

Input #2

14 5
1 1 1 2 3 1 2 4 5 1 1 2 3 1

Output #2

Input #3

19 10
1 6 2 7 4 8 5 9 1 10 4 1 3 1 3 2 2 2 1

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Constant Sum Subsequence",
    "description": {
      "content": "We have a sequence of positive integers of length $N^2$, $A=(A_1,\\ A_2,\\ \\dots,\\ A_{N^2})$, and a positive integer $S$. For this sequence of positive integers, $A_i=A_{i+N}$ holds for positive integer",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc150_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a sequence of positive integers of length $N^2$, $A=(A_1,\\ A_2,\\ \\dots,\\ A_{N^2})$, and a positive integer $S$. For this sequence of positive integers, $A_i=A_{i+N}$ holds for positive integer...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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