Kth Largest Element in a Stream

An arcade machine shows a "score to beat" banner: the k-th highest score recorded so far. Scores keep arriving one at a time, and after each one the banner must be refreshed.

Build a class ScoreBoard:

• ScoreBoard(k, scores) starts with the fixed number k and a list of scores already recorded (possibly empty).
• add(score) records a new score and returns the k-th highest of all scores recorded so far, counting repeats separately (so in [7, 7, 3] the 2nd highest is 7).

board = ScoreBoard(3, [40, 10, 25, 55])
board.add(30)    # 30   (55, 40, 30, 25, 10)
board.add(5)     # 30
board.add(60)    # 40   (60, 55, 40, ...)
board.add(40)    # 40   (60, 55, 40, 40, ...)
board.add(70)    # 55

solo = ScoreBoard(1, [])
solo.add(-4)     # -4
solo.add(-9)     # -4

Constraints:

• 1 <= k <= 10^5 and 0 <= len(scores) <= 10^5; scores are in [-10^9, 10^9].
• Up to 10^5 calls to add. Whenever add is called, at least k scores exist once the new one is counted.
• Sorting all scores on every call is far too slow at this size. Aim for O(log k) per add after an O(n log k) (or better) setup.