Random Tree Distance

AtCoder

IDarc195_e

Time2000ms

Memory256MB

Difficulty—

There is an integer sequence $A = (A_2,A_3,\ldots,A_N)$. Also, for an integer sequence $P=(P_2, P_3, \ldots ,P_N)$ where $1 \leq P_i \leq i-1$ for each $i$ $(2 \leq i \leq N)$, define the weighted tree $T(P)$ with $N$ vertices, rooted at vertex $1$, as follows: * A rooted tree where, for each $i$ $(2 \leq i \leq N)$, the parent of $i$ is $P_i$, and the weight of the edge between $i$ and $P_i$ is $A_i$. You are given $Q$ queries. Process them in order. The $i$\-th query is as follows: * You are given integers $u_i$ and $v_i$, each between $1$ and $N$. For each of the possible $(N-1)!$ sequences $P$, take the tree $T(P)$ and consider the distance between vertices $u_i$ and $v_i$ in this tree. Output the sum, modulo $998244353$, of these distances over all $T(P)$. Here, the distance between two vertices $u_i$ and $v_i$ is the sum of the weights of the edges on the unique path (not visiting the same vertex more than once) that connects them. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq Q \leq 2 \times 10^5$ * $1 \leq A_i \leq 10^9$ * $1 \leq u_i < v_i \leq N$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $Q$ $A_2$ $A_3$ $\ldots$ $A_N$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_Q$ $v_Q$ [samples]

Samples

Input #1

Output #1

2
3

*   If $P = (1,1)$, then in the tree $T(P)$, the distance between vertices $1$ and $2$ is $1$, and the distance between vertices $1$ and $3$ is $1$.
*   If $P = (1,2)$, then in the tree $T(P)$, the distance between vertices $1$ and $2$ is $1$, and the distance between vertices $1$ and $3$ is $2$.

Therefore, the total distance between vertices $1$ and $2$ over all $T(P)$ is $2$, and the total distance between vertices $1$ and $3$ over all $T(P)$ is $3$.

Input #2

2 1
100
1 2

Output #2

Input #3

9 6
765689282 93267307 563699854 951829154 801512848 389123318 924504746 596035433
3 8
2 5
5 8
2 9
8 9
5 7

Output #3

55973424
496202632
903509579
343265517
550981449
68482696

Remember to take the sum modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Random Tree Distance",
    "description": {
      "content": "There is an integer sequence $A = (A_2,A_3,\\ldots,A_N)$. Also, for an integer sequence $P=(P_2, P_3, \\ldots ,P_N)$ where $1 \\leq P_i \\leq i-1$ for each $i$ $(2 \\leq i \\leq N)$, define the weighted tre",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc195_e"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is an integer sequence $A = (A_2,A_3,\\ldots,A_N)$. Also, for an integer sequence $P=(P_2, P_3, \\ldots ,P_N)$ where $1 \\leq P_i \\leq i-1$ for each $i$ $(2 \\leq i \\leq N)$, define the weighted tre...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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