Abs Abs Function

AtCoder

IDarc155_b

Time2000ms

Memory256MB

Difficulty—

For a set $S$ of pairs of non-negative integers, and a non-negative integer $x$, let $f_S(x)$ defined as $\displaystyle f_S(x)=\min_{(a, b) \in S} \left| \left| x-a \right| - b \right|$. We have a set $T$ of pairs of non-negative integers. Initially, $T=\lbrace (A, B)\rbrace$. Process $Q$ queries. The $i$\-th query gives you three non-negative integers $t_i$, $a_i$, and $b_i$, and asks you to do the following. * If $t_i=1$, add to $T$ the pair $(a_i, b_i)$ of non-negative integers. * If $t_i=2$, print the minimum value of $f_{T}(x)$ for a non-negative integer $x$ such that $a_i \leq x \leq b_i$. ## Constraints * $1 \leq Q \leq 2 \times 10^5$ * $0 \leq A,B \leq 10^{9}$ * $t_i$ is $1$ or $2$. * $0 \leq a_i,b_i \leq 10^{9}$ * If $t_i=2$, then $a_i \leq b_i$. * There is at least one query with $t_i=2$. * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $Q$ $A$ $B$ $t_1$ $a_1$ $b_1$ $t_2$ $a_2$ $b_2$ $\vdots$ $t_Q$ $a_Q$ $b_Q$ [samples]

Samples

Input #1

Output #1

2
1

In the second query, $T=\lbrace(0, 5), (3, 11) \rbrace$. For $x=7$, we have $f_T(7)=\min \lbrace \left| \left|7-0\right|-5\right|, \left| \left|7-3\right|-11\right| \rbrace=\min \lbrace 2, 7 \rbrace=2$. Similarly, $f_T(8)=3$. Thus, the answer is $\min \lbrace 2, 3 \rbrace =2$.
In the fourth query, $T=\lbrace(0, 5), (3, 11), (8, 2) \rbrace$. In $8 \leq x \leq 9$, $f_T(x)$ takes the minimum value $f_T(9)=1$ at $x=9$.

Input #2

2 1 2
1 2 3
2 2 6

Output #2

Input #3

20 795629912 123625148
2 860243184 892786970
2 645778367 668513124
1 531411849 174630323
1 635062977 195695960
2 382061637 411843651
1 585964296 589553566
1 310118888 68936560
1 525351160 858166280
2 395304415 429823333
2 583145399 703645715
2 97768492 218377432
1 707220749 459967102
1 210842017 363390878
2 489541834 553583525
2 731279777 811513313
1 549864943 493384741
1 815378318 826084592
2 369622093 374205455
1 78240781 821999998
2 241667193 243982581

Output #3

API Response (JSON)

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    "name": "Abs Abs Function",
    "description": {
      "content": "For a set $S$ of pairs of non-negative integers, and a non-negative integer $x$, let $f_S(x)$ defined as $\\displaystyle f_S(x)=\\min_{(a, b) \\in S} \\left| \\left| x-a \\right| - b \\right|$. We have a set",
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      "statement_type": "Markdown",
      "content": "For a set $S$ of pairs of non-negative integers, and a non-negative integer $x$, let $f_S(x)$ defined as $\\displaystyle f_S(x)=\\min_{(a, b) \\in S} \\left| \\left| x-a \\right| - b \\right|$.\nWe have a set...",
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      "language": "English"
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