codeforces 990D、Graph And Its Complement(思维)

                                                             D. Graph And Its Complement

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

Given three numbers n,a,bn,a,b. You need to find an adjacency matrix of such an undirected graph that the number of components in it is equal to aa, and the number of components in its complement is bb. The matrix must be symmetric, and all digits on the main diagonal must be zeroes.

In an undirected graph loops (edges from a vertex to itself) are not allowed. It can be at most one edge between a pair of vertices.

The adjacency matrix of an undirected graph is a square matrix of size nn consisting only of "0" and "1", where nn is the number of vertices of the graph and the ii-th row and the ii-th column correspond to the ii-th vertex of the graph. The cell (i,j)(i,j) of the adjacency matrix contains 11 if and only if the ii-th and jj-th vertices in the graph are connected by an edge.

A connected component is a set of vertices XX such that for every two vertices from this set there exists at least one path in the graph connecting this pair of vertices, but adding any other vertex to XX violates this rule.

The complement or inverse of a graph GG is a graph HH on the same vertices such that two distinct vertices of HH are adjacent if and only if they are not adjacent in GG.

Input

In a single line, three numbers are given n,a,b(1≤n≤1000,1≤a,b≤n)n,a,b(1≤n≤1000,1≤a,b≤n): is the number of vertexes of the graph, the required number of connectivity components in it, and the required amount of the connectivity component in it's complement.

Output

If there is no graph that satisfies these constraints on a single line, print "NO" (without quotes).

Otherwise, on the first line, print "YES"(without quotes). In each of the next nn lines, output nn digits such that jj-th digit of ii-th line must be 11 if and only if there is an edge between vertices ii and jj in GG (and 00 otherwise). Note that the matrix must be symmetric, and all digits on the main diagonal must be zeroes.

If there are several matrices that satisfy the conditions — output any of them.

Examples

input

Copy

3 1 2

output

Copy

YES
001
001
110

input

Copy

3 3 3

output

Copy

NO

 

一、原题地址

点我传送

 

二、大致题意

给出n,a,b。表示有n个城镇现在要构造一副图，使得原图连通块的数量为a。把原图上所有边去掉再补上原来图上没有的边（即补图）的连通块数量为b。

 

三、大致思路

  首先在a和b中一定会有一个数的值为1。证明很简单，对于一副图中的一个连通块 x ，对于这个连通块中的点，当这个点取补图的时候，它会与其他连通块的所有点连一条边，这个连通块 x 当中的其他点也是如此，那么其实就相当于把所有的点都汇在了一起变成一个连通块。

那么其实情况就不会很多了，要么a=1,要么b=1，要么a和b都等于1。

1、当b=1时，只需要考虑原本的图是怎么样的就行，那么只需要把前a-1个点独立出去，然后再把后面的点连通成一块就行。

2、当a=1时，情况其实和1是一样的，可以选择去构造补图，相当于所有操作全部取反就行。

3、a和b都等于1时，n=1肯定是可以的，n=2或者n=3都不行，但是n>=4时只需要把图构造成一条链，又可以了。

4、a和b都不等于1已经讨论过了不行。

 

四、代码

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