2007 Canadian Computing Competition, Stage 2

Day 2, Problem 1: Gerrymandering

Politicians like to get elected. They will do just about anything to get elected. Including changing the rules of the voting: who can vote, where you can vote, when you can vote, etc. One very common practice is called gerrymandering, where the boundaries of "ridings" are redrawn to favour a particular candidate (the one doing the redrawing, of course).

Your task is to help determine how to do some simple, linear gerrymandering.

You will be given the information about N ridings (2 ≤ N ≤ 1,000) where there are candidates from p (2 ≤ p ≤ 10) different parties. These N ridings are linear, in the sense that they are side-by-side; there are two ridings (on the ends) that have only one adjacent riding, with the rest of the ridings having two adjacent ridings. A picture is shown below for N = 4 and p = 2 (which is also the sample data):

  Riding 1 Riding 2 Riding 3 Riding 4
Votes for Party 1 1 4 1 6
Votes for Party 2 5 3 2 1

Note that Riding 1 and Riding 2 are adjacent, Riding 2 and 3 are adjacent, Riding 3 and 4 are adjacent. No other ridings are adjacent.

You have some financial backing that will let you bribe the people in charge of setting the boundaries of ridings: in particular, there is a fixed rate to merge two adjacent boundaries. When you merge two ridings, the votes of the ridings merge together, in the sense that the number of votes of party 1 is the sum of the votes of party 1 in each riding, and likewise for all other parties.

Your task is to merge the minimum number of regions such that the first party (your party!) has a majority of the ridings. Note that to win a riding, the party must have more votes than any other party in that riding. Also note that to have a majority of ridings, if there are Q ridings (where QN), then your party has won at least floor(Q/2)+1 of the ridings.

Input

The first line of input will consist of the integer N. The second line of input will consist of the integer p. The next N lines will each contain p non-negative integers (where each integer is at most 10,000), separated by one space character. Specifically, the p integers on each line are v1 v2 ... vp where v1 is the number of votes that party 1 will receive in this riding, v2 is the number of votes that party 2 will receive in this riding, etc. You may also assume that the total number of voters is less than 2 billion.

Output

One line, consisting of an integer, which gives the minimum number of ridings that need to be merged in order for the first party to win a majority of ridings. If the first party cannot win, even with any number of mergers, output -1.

Sample Input 1

4
2
1 5
4 3
1 2
6 1

Sample Output 1

1

Sample Input 2

3
3
2 0 1
1 3 0
0 0 1

Sample Output 2

-1

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Point Value: 20 (partial)
Time Limit: 10.00s
Memory Limit: 256M
Added: Dec 09, 2008

Languages Allowed:
C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3

Comments (Search)

On the official PDF, http://cemc.uwaterloo.ca/contests/computing/2007/stage2/day2.pdf, N is less than equal to 100 000. Why is this smaller on PEG?