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ZOJ Problem Set - 2452
Cartesian Tree

Time Limit: 10 Seconds      Memory Limit: 32768 KB      Special Judge

Let us consider a special type of a binary search tree, called a cartesian tree. Recall that a binary search tree is a rooted ordered binary tree, such that for its every node x the following condition is satisfied: each node in its left subtree has the key less then the key of x, and each node in its right subtree has the key greater then the key of x.

That is, if we denote left subtree of the node x by L(x), its right subtree by R(x) and its key by kx, then for each node x we have

  • if y L(x) then ky < kx
  • if z R(x) then kz > kx

The binary search tree is called cartesian if its every node x in addition to the main key kx also has an auxiliary key that we will denote by ax, and for these keys the heap condition is satisfied, that is

  • if y is the parent of x then ay < ax

Thus a cartesian tree is a binary rooted ordered tree, such that each of its nodes has a pair of two keys (k, a) and three conditions described are satisfied.

Given a set of pairs, construct a cartesian tree out of them, or detect that it is not possible.


Input consists of multiple test cases.
For each case, the first line contains an integer N -- the number of pairs you should build cartesian tree out of (1 N 50 000). The following N lines contain two numbers each -- given pairs (ki, ai). For each pair |ki|, |ai| 30 000. All main keys and all auxiliary keys are different, i.e. ki kj and ai aj for each i j.


For each test case, print in the first line YES if it is possible to build a cartesian tree out of given pairs or NO if it is not. If the answer is positive, on the following N lines output the tree. Let nodes be numbered from 1 to N corresponding to pairs they contain as they are given in the input file. For each node output three numbers -- its parent, its left child and its right child. If the node has no parent or no corresponding child, output 0 instead.
If there are several possible trees, output any one.

Sample Input

5 4
2 2
3 9
0 5
1 3
6 6
4 11

Possible Output for Sample Input

2 3 6
0 5 1
1 0 7
5 0 0
2 4 0
1 0 0
3 0 0

Source: Northeastern Europe 2002, Northern Subregion
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