Flip Cards

AtCoder

IDabc291_d

Time2000ms

Memory256MB

Difficulty—

$N$ cards, numbered $1$ through $N$, are arranged in a line. For each $i\ (1\leq i < N)$, card $i$ and card $(i+1)$ are adjacent to each other. Card $i$ has $A_i$ written on its front, and $B_i$ written on its back. Initially, all cards are face up. Consider flipping zero or more cards chosen from the $N$ cards. Among the $2^N$ ways to choose the cards to flip, find the number, modulo $998244353$, of such ways that: * when the chosen cards are flipped, for every pair of adjacent cards, the integers written on their face-up sides are different. ## Constraints * $1\leq N \leq 2\times 10^5$ * $1\leq A_i,B_i \leq 10^9$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $B_1$ $A_2$ $B_2$ $\vdots$ $A_N$ $B_N$ [samples]

Samples

Input #1

Output #1

4

Let $S$ be the set of card numbers to flip.
For example, when $S={2,3}$ is chosen, the integers written on their visible sides are $1,2$, and $4$, from card $1$ to card $3$, so it satisfies the condition.
On the other hand, when $S={3}$ is chosen, the integers written on their visible sides are $1,4$, and $4$, from card $1$ to card $3$, where the integers on card $2$ and card $3$ are the same, violating the condition.
Four $S$ satisfy the conditions: ${},{1},{2}$, and ${2,3}$.

Input #2

Output #2

Input #3

8
877914575 602436426
861648772 623690081
476190629 262703497
971407775 628894325
822804784 450968417
161735902 822804784
161735902 822804784
822804784 161735902

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Flip Cards",
    "description": {
      "content": "$N$ cards, numbered $1$ through $N$, are arranged in a line. For each $i\\ (1\\leq i < N)$, card $i$ and card $(i+1)$ are adjacent to each other. Card $i$ has $A_i$ written on its front, and $B_i$ writt",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc291_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "$N$ cards, numbered $1$ through $N$, are arranged in a line. For each $i\\ (1\\leq i < N)$, card $i$ and card $(i+1)$ are adjacent to each other. Card $i$ has $A_i$ written on its front, and $B_i$ writt...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments