TY - JOUR

T1 - An Optimal Algorithm for Finding Champions in Tournament Graphs

AU - Beretta, Lorenzo

AU - Nardini, Franco Maria

AU - Trani, Roberto

AU - Venturini, Rossano

N1 - Publisher Copyright: © 1989-2012 IEEE.

PY - 2023

Y1 - 2023

N2 - A tournament graph is a complete directed graph, which can be used to model a round-robin tournament between n players. In this paper, we address the problem of finding a champion of the tournament, also known as Copeland winner, which is a player that wins the highest number of matches. In detail, we aim to investigate algorithms that find the champion by playing a low number of matches. Solving this problem allows us to speed up several Information Retrieval and Recommender System applications, including question answering, conversational search, etc. Indeed, these applications often search for the champion inducing a round-robin tournament among the players by employing a machine learning model to estimate who wins each pairwise comparison. Our contribution, thus, allows finding the champion by performing a low number of model inferences. We prove that any deterministic or randomized algorithm finding a champion with constant success probability requires Ω(ℓn) comparisons, where ℓ is the number of matches lost by the champion. We then present an asymptotically-optimal deterministic algorithm matching this lower bound without knowing ℓ, and we extend our analysis to three variants of the problem. Lastly, we conduct a comprehensive experimental assessment of the proposed algorithms on a question answering task on public data. Results show that our proposed algorithms speed up the retrieval of the champion up to 13× with respect to the state-of-the-art algorithm that perform the full tournament.

AB - A tournament graph is a complete directed graph, which can be used to model a round-robin tournament between n players. In this paper, we address the problem of finding a champion of the tournament, also known as Copeland winner, which is a player that wins the highest number of matches. In detail, we aim to investigate algorithms that find the champion by playing a low number of matches. Solving this problem allows us to speed up several Information Retrieval and Recommender System applications, including question answering, conversational search, etc. Indeed, these applications often search for the champion inducing a round-robin tournament among the players by employing a machine learning model to estimate who wins each pairwise comparison. Our contribution, thus, allows finding the champion by performing a low number of model inferences. We prove that any deterministic or randomized algorithm finding a champion with constant success probability requires Ω(ℓn) comparisons, where ℓ is the number of matches lost by the champion. We then present an asymptotically-optimal deterministic algorithm matching this lower bound without knowing ℓ, and we extend our analysis to three variants of the problem. Lastly, we conduct a comprehensive experimental assessment of the proposed algorithms on a question answering task on public data. Results show that our proposed algorithms speed up the retrieval of the champion up to 13× with respect to the state-of-the-art algorithm that perform the full tournament.

KW - copeland winner

KW - minimum selection

KW - pairwise ranking

KW - round-robin tournament

KW - Tournament graph

U2 - 10.1109/TKDE.2023.3267345

DO - 10.1109/TKDE.2023.3267345

M3 - Journal article

AN - SCOPUS:85153493799

VL - 35

SP - 10197

EP - 10209

JO - IEEE Transactions on Knowledge and Data Engineering

JF - IEEE Transactions on Knowledge and Data Engineering

SN - 1041-4347

IS - 10

ER -