As an employee of the world’s most respected political polling corporation, you must take complex, real-world issues and simplify them down to a few numbers. It isn’t always easy. A big election is coming up and, at the request of Candidate X, you have just finished polling $n$ people. You have gathered three pieces of information from each person, with the values for the $i^\text {th}$ person recorded as:
$a_ i$ – the number of digits of $\pi $ they have memorized
$b_ i$ – the number of hairs on their head
$c_ i$ – whether they will vote for Candidate X
Unfortunately, you are beginning to wonder if these are really the most relevant questions to ask. In fact, you cannot see any correlation between $a$, $b$, and $c$ in the data. Of course, you cannot just contradict your customer – that is a good way to lose your job!
Perhaps the answer is to find some weighting formula to make the results look meaningful. You will pick two real values $S$ and $T$, and sort the poll results $(a_ i, b_ i, c_ i)$ by the measure $a_ i \cdot S + b_ i \cdot T$. The sort will look best if the results having $c_ i$ true are clustered as close to each other as possible. More precisely, if $j$ and $k$ are the indices of the first and last results with $c_ i$ true, you want to minimize the cluster size which is $k-j+1$. Note that some choices of $S$ and $T$ will result in ties among the $(a_ i,b_ i,c_ i)$ triples. When this happens, you should assume the worst possible ordering occurs (that which maximizes the cluster size for this $(S, T)$ pair).
The input starts with a line containing $n$ ($1 \leq n \leq 250\, 000$), which is the number of people polled. This is followed by one line for each person polled. Each of those lines contains integers $a_ i$ ($0 \leq a_ i \leq 2\, 000\, 000$), $b_ i$ ($0 \leq b_ i \leq 2\, 000\, 000$), and $c_ i$, where $c_ i$ is $1$ if the person will vote for Candidate X and $0$ otherwise. The input is guaranteed to contain at least one person who will vote for Candidate X.
Display the smallest possible cluster size over all possible $(S, T)$ pairs.
Sample Input 1 | Sample Output 1 |
---|---|
6 0 10 0 10 0 1 12 8 1 5 5 0 11 2 1 11 3 0 |
4 |
Sample Input 2 | Sample Output 2 |
---|---|
10 6 1 1 0 2 0 2 1 1 6 1 1 8 2 0 4 4 0 4 0 0 2 3 1 6 1 0 6 3 1 |
8 |
Sample Input 3 | Sample Output 3 |
---|---|
5 5 7 0 3 4 0 5 7 0 5 7 1 9 4 0 |
1 |