A Tight Bound for Shortest Augmenting Paths on Trees

  • Bartłomiej Bosek
  • Dariusz Leniowski
  • Piotr Sankowski
  • Anna Zych-Pawlewicz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree \(T=(W\uplus B, E)\) is being revealed online, i.e., in each round one vertex from \(B\) with its incident edges arrives. It was conjectured by Chaudhuri et al. [7] that the total length of all shortest augmenting paths found is \(O(n \log n)\). In this paper we prove a tight \(O(n \log n)\) upper bound for the total length of shortest augmenting paths for trees improving over \(O(n \log ^2 n)\) bound [5].


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Bartłomiej Bosek
    • 1
  • Dariusz Leniowski
    • 2
  • Piotr Sankowski
    • 2
  • Anna Zych-Pawlewicz
    • 2
  1. 1.Theoretical Computer Science Department, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Institute of Computer ScienceUniversity of WarsawWarsawPoland

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