63 lines
2.8 KiB
Text
63 lines
2.8 KiB
Text
Farmer John is worried for the health of his cows (conveniently numbered
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$1 ... N$ as always) after an outbreak of the highly contagious bovine
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disease COWVID-19.
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Recently, Farmer John tested all of his cows and found some of them to be
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positive for the disease. Using video footage from inside his barn, he is able
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to review recent interactions between pairs of cows --- it turns out that when
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cows greet each-other, they shake hooves, a gesture that can unfortunately
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spread the infection from one cow to another. Farmer John assembles a
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time-stamped list of interacting pairs of cows, with entries of the form
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$(t, x, y)$, meaning that at time $t$, cow $x$ shook hooves with cow $y$.
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Farmer John also knows the following:
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(i) Exactly one cow on his farm could have started out carrying the disease
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(we'll call this cow "patient zero").
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(ii) Once a cow is infected, she passes the infection along with her next $K$
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hoof shakes (possibly including the same partner cow several times). After
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shaking hooves $K$ times, she no longer passes the infection along with
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subsequent hoof shakes (since at this point she realizes she is spreading the
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infection and washes her hooves carefully).
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(iii) Once a cow is infected, she stays infected.
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Unfortunately, Farmer John doesn't know which of his $N$ cows is patient zero,
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nor does he know the value of $K$! Please help him narrow down the
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possibilities for these unknowns based on his data. It is guaranteed that at
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least one possibility is valid.
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INPUT FORMAT
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The first line of the input file contains $N$ ($2 <= N <= 100$) and $T$
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($1 <= T <= 250$). The next line contains a string of length $N$ whose
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entries are 0s and 1s, describing the current state of Farmer John's $N$ cows
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--- 0 represents a healthy cow and 1 represents a cow presently with the
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disease. Each of the next $T$ lines describes a record in Farmer John's list of
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interactions and consists of three integers $t$, $x$, and $y$, where $t$ is a
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positive integer time of the interaction ($t <= 250$) and $x$ and $y$ are
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distinct integers in the range $1 \ldots N$, indicating which cows shook hands
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at time $t$. At most one interaction happens at each point in time.
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OUTPUT FORMAT
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Print a single line with three integers $x$, $y$, and $z$, where $x$ is the
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number of possible cows who could have been patient zero, $y$ is the smallest
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possible value of $K$ consistent with the data, and $z$ is the largest possible
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value of $K$ consistent with the data (if there is no upper bound on $K$ that
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can be deduced from the data, print "Infinity" for $z$). Note that it might be
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possible to have $K=0$.
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SAMPLE INPUT
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4 3
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1100
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7 1 2
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5 2 3
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6 2 4
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SAMPLE OUTPUT
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1 1 Infinity
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The only candidate for patient zero is cow 1. For all $K>0$, cow 1 infects cow 2
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at time 7, while cows 3 and 4 remain uninfected.
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