Rank songs by pairwise user preferences

You are given preference rankings for n users over m songs. Each song is labeled from 0 to m - 1.

For each user i (0-indexed), you are given an array pref[i] of length m that is a permutation of [0, 1, ..., m-1]. This array represents user i's ranking of all songs:

• If song a appears before song b in pref[i] , then user i prefers song a over song b .

Using these preference lists, define the following:

1. Song x beats y :
   • Let count_pref_x be the number of users who prefer x over y .
   • Song x beats song y if either:
     • count_pref_x is strictly greater than half of all users (i.e., count_pref_x > n / 2 ), or
     • count_pref_x is exactly half of all users (i.e., count_pref_x * 2 == n ) and x < y (by song ID).
2. Song x is more popular than y :
   • For each song, define its beat count as the number of distinct other songs it beats according to rule (1).
   • Song x is more popular than song y if:
     • The beat count of x is greater than the beat count of y , or
     • The beat counts are equal and x < y (by song ID).

Task: Given n, m, and all preference lists pref[0..n-1], compute the popularity ranking of all songs. Return a list of all song IDs from most popular to least popular, sorted by the "more popular than" relation defined above.

Assume:

• 1 <= m <= M_max , 1 <= n <= N_max (choose reasonable constraints if implementing).
• All pref[i] are valid permutations of [0, ..., m-1] .

Your function should output an array of length m containing the song IDs in descending order of popularity. Break ties using the song ID (smaller ID is considered more popular when beat counts are equal).