import math

def mrr(runs):
    total = 0.0
    for ranked in runs:
        for i, rel in enumerate(ranked, start=1):
            if rel:
                total += 1 / i
                break
    return total / len(runs)

def ndcg_at_k(ranked_relevances, k):
    dcg = sum(r / math.log2(i + 1) for i, r in enumerate(ranked_relevances[:k], start=1))
    ideal = sorted(ranked_relevances, reverse=True)[:k]
    idcg = sum(r / math.log2(i + 1) for i, r in enumerate(ideal, start=1))
    return dcg / idcg if idcg else 0.0