4047: Brackets
Description
A two dimensional array is called a bracket array if each grid contains one of the two possible brackets − "(" or ")". A path through the two dimensional array cells is called monotonous if any two consecutive cells in the path are side-adjacent and each cell of the path is located below or to the right from the previous one.
A two dimensional array whose size equals n×m is called a correct bracket array, if any string formed by writing out the brackets on some monotonous way from cell (1,1) to cell (n,m) forms a correct bracket sequence.
Let's define the operation of comparing two correct bracket arrays of equal size (a and b) like that. Let's consider a given two dimensional array of priorities (c) − a two dimensional array of same size, containing different integers from 1 to nm. Let's find such position (i,j) in the two dimensional array, that ai,j≠bi,j. If there are several such positions, let's choose the one where number ci,j is minimum. If ai,j="(", then a<b, otherwise a>b. If the position (i,j) is not found, then the arrays are considered equal.
Your task is to find a k-th two dimensional correct bracket array. It is guaranteed that for the given sizes of n and m there will be no less than k two dimensional correct bracket arrays.
The first line contains integers n, m and k − the sizes of the array and the number of the sought correct bracket array (1≤n,m≤100, 1≤k≤1018). Then an array of priorities is given, n lines each containing m numbers, number pi,j shows the priority of character j in line i (1≤pi,j≤nm, all pi,j are different).
Please do not use the %lld specificator to read or write 64-bit integers in C++. It is preferred to use the cin, cout streams or the %I64d specificator.
Print the k-th two dimensional correct bracket array.
1 2 1
1 2
()
2 3 1
1 2 3
4 5 6
(()
())
3 2 2
3 6
1 4
2 5
()
)(
()
In the first sample exists only one correct two-dimensional bracket array.
In the second and in the third samples two arrays exist.
A bracket sequence is called regular if it is possible to obtain correct arithmetic expression by inserting characters «+» and «1» into this sequence. For example, sequences «(())()», «()» and «(()(()))» are regular, while «)(», «(()» and «(()))(» are not.