Codeforces Round 663 (Div. 2) |
---|

Finished |

bitmasks

brute force

constructive algorithms

dp

greedy

implementation

*2000

No tag edit access

The problem statement has recently been changed. View the changes.

×
D. 505

time limit per test

1 secondmemory limit per test

256 megabytesinput

standard inputoutput

standard outputA binary matrix is called good if every even length square sub-matrix has an odd number of ones.

Given a binary matrix $$$a$$$ consisting of $$$n$$$ rows and $$$m$$$ columns, determine the minimum number of cells you need to change to make it good, or report that there is no way to make it good at all.

All the terms above have their usual meanings — refer to the Notes section for their formal definitions.

Input

The first line of input contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n \leq m \leq 10^6$$$ and $$$n\cdot m \leq 10^6$$$) — the number of rows and columns in $$$a$$$, respectively.

The following $$$n$$$ lines each contain $$$m$$$ characters, each of which is one of 0 and 1. If the $$$j$$$-th character on the $$$i$$$-th line is 1, then $$$a_{i,j} = 1$$$. Similarly, if the $$$j$$$-th character on the $$$i$$$-th line is 0, then $$$a_{i,j} = 0$$$.

Output

Output the minimum number of cells you need to change to make $$$a$$$ good, or output $$$-1$$$ if it's not possible at all.

Examples

Input

3 3 101 001 110

Output

2

Input

7 15 000100001010010 100111010110001 101101111100100 010000111111010 111010010100001 000011001111101 111111011010011

Output

-1

Note

In the first case, changing $$$a_{1,1}$$$ to $$$0$$$ and $$$a_{2,2}$$$ to $$$1$$$ is enough.

You can verify that there is no way to make the matrix in the second case good.

Some definitions —

- A binary matrix is one in which every element is either $$$1$$$ or $$$0$$$.
- A sub-matrix is described by $$$4$$$ parameters — $$$r_1$$$, $$$r_2$$$, $$$c_1$$$, and $$$c_2$$$; here, $$$1 \leq r_1 \leq r_2 \leq n$$$ and $$$1 \leq c_1 \leq c_2 \leq m$$$.
- This sub-matrix contains all elements $$$a_{i,j}$$$ that satisfy both $$$r_1 \leq i \leq r_2$$$ and $$$c_1 \leq j \leq c_2$$$.
- A sub-matrix is, further, called an even length square if $$$r_2-r_1 = c_2-c_1$$$ and $$$r_2-r_1+1$$$ is divisible by $$$2$$$.

Codeforces (c) Copyright 2010-2023 Mike Mirzayanov

The only programming contests Web 2.0 platform

Server time: Dec/08/2023 18:37:51 (j2).

Desktop version, switch to mobile version.

Supported by

User lists

Name |
---|