Subsegments with Small Sums

AtCoder

IDarc169_b

Time2000ms

Memory256MB

Difficulty—

You are given a positive integer $S$. For a sequence of positive integers $x$, we define $f(x)$ as follows: * Consider decomposing $x$ into several contiguous subsequences. For each contiguous subsequence, the sum of its elements must be **at most** $S$. The minimum number of contiguous subsequences in such a decomposition is the value of $f(x)$. It can be proved that the value of $f$ is always definable under the constraints of this problem. You are given a sequence of positive integers $A=(A_1,A_2,\cdots,A_N)$. Find the value of $\sum_{1 \leq l \leq r \leq N} f((A_l,A_{l+1},\cdots,A_r))$. ## Constraints * $1 \leq N \leq 250000$ * $1 \leq S \leq 10^{15}$ * $1 \leq A_i \leq \min(S,10^9)$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $S$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples]

Samples

Input #1

3 3
1 2 3

Output #1

8

For example, consider $x=(1,2,3)$. The decomposition $(1,2),(3)$ satisfies the condition, and no decomposition into fewer than two contiguous subsequences satisfies the condition, so $f((1,2,3))=2$.
Shown below are the possible $l,r$ and the corresponding values of $f$:

*   $(l,r)=(1,1)$: $f((1))=1$
*   $(l,r)=(1,2)$: $f((1,2))=1$
*   $(l,r)=(1,3)$: $f((1,2,3))=2$
*   $(l,r)=(2,2)$: $f((2))=1$
*   $(l,r)=(2,3)$: $f((2,3))=2$
*   $(l,r)=(3,3)$: $f((3))=1$

Thus, the answer is $1+1+2+1+2+1=8$.

Input #2

5 1
1 1 1 1 1

Output #2

Input #3

5 15
5 4 3 2 1

Output #3

Input #4

20 1625597454
786820955 250480341 710671229 946667801 19271059 404902145 251317818 22712439 520643153 344670307 274195604 561032101 140039457 543856068 521915711 857077284 499774361 419370025 744280520 249168130

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Subsegments with Small Sums",
    "description": {
      "content": "You are given a positive integer $S$. For a sequence of positive integers $x$, we define $f(x)$ as follows: *   Consider decomposing $x$ into several contiguous subsequences. For each contiguous subs",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc169_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a positive integer $S$. For a sequence of positive integers $x$, we define $f(x)$ as follows:\n\n*   Consider decomposing $x$ into several contiguous subsequences. For each contiguous subs...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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