Problem D
Zipf's Song
At first, you figure that the most listened to songs must be the best songs. However, you quickly realize that this approach is flawed. Even if all songs of the album are equally good, the early songs are more likely to be listened to more often than the later ones, because monkeys will tend to start listening to the first song, listen for a few songs and then, when their fickle ears start craving something else, stop listening. Instead, if all songs are equal, you expect that their play frequencies should follow Zipf’s Law.
Zipf’s Law is an empirical law originally formulated about word frequencies in natural languages, but it has been observed that many natural phenomena, such as population sizes and incomes, approximately follow the same law. It predicts that the relative frequency of the $i$’th most common object (in this case, a song) should be proportional to $1/i$.
To illustrate this in our setting, suppose we have an album where all songs are equally good. Then by Zipf’s Law, you expect that the first song is listened to twice as often as the second song, and more generally that the first song is listened to $i$ times as often as the $i$’th song. When some songs are better than others, those will be listened to more often than predicted by Zipf’s Law, and those are the songs your program should select as the good songs. Specifically, suppose that song $i$ has been played $f_ i$ times but that Zipf’s Law predicts that it would have been played $z_ i$ times. Then you define the quality of song $i$ to be $q_ i = f_ i/z_ i$. Your software should select the songs with the highest values of $q_ i$.
Input
The first line of input contains two integers $n$ and $m$ ($1 \le n \le 50\, 000$, $1 \le m \le n$), the number of songs on the album, and the number of songs to select. Then follow $n$ lines. The $i$’th of these lines contains an integer $f_ i$ and string $s_ i$, where $0 \le f_ i \le 10^{12}$ is the number of times the $i$’th song was listened to, and $s_ i$ is the name of the song. Each song name is at most $30$ characters long and consists only of the characters ‘a’-‘z’, ‘0’-‘9’, and underscore (‘_’).
Output
Output a list of the $m$ songs with the highest quality $q_ i$, in decreasing order of quality. If two songs have the same quality, give precedence to the one appearing first on the album (presumably there was a reason for the producers to put that song before the other).
| Sample Input 1 | Sample Output 1 |
|---|---|
4 2 30 one 30 two 15 three 25 four |
four two |
| Sample Input 2 | Sample Output 2 |
|---|---|
15 3 197812 re_hash 78906 5_4 189518 tomorrow_comes_today 39453 new_genious 210492 clint_eastwood 26302 man_research 22544 punk 19727 sound_check 17535 double_bass 18782 rock_the_house 198189 19_2000 13151 latin_simone 12139 starshine 11272 slow_country 10521 m1_a1 |
19_2000 clint_eastwood tomorrow_comes_today |
