On the Heaviest Increasing or Decreasing Subsequence of a Permutation, and Paths and Matchings on Weighted Point Sets

  • Toshinori Sakai
  • Jorge Urrutia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Let S = {s(1), …, s(n) } be a permutation of the integers {1,…, n}. A subsequence of S with elements {s(i1), …, s(i k )} is called an increasing subsequence if s(i1) < ⋯ < s(i k ); It is called a decreasing subsequence if s(i1) > ⋯ > s(i k ). The weight of a subsequence of S, is the sum of its elements. In this paper, we prove that any permutation of {1, …, n} contains an increasing or a decreasing subsequence of weight greater than \(n\sqrt{2n/3}\).

Our motivation to study the previous problem arises from the following problem: Let P be a set of n points on the plane in general position, labeled with the integers {1, …,n} in such a way that the labels of different points are different. A non-crossing path Π with vertices in P is an increasing path if when we travel along it, starting at one of its end-points, the labels of its vertices always increase. The weight of an increasing path, is the sum of the labels of its vertices. Determining lower bounds on the weight of the heaviest increasing path a point set always has.

We also study the problem of finding a non-crossing matching of the elements of P of maximum weight, where the weight of an edge with endpoints i, j ∈ P is min{i,j}.


General Position Perfect Match Maximum Weight Geometric Graph Disjoint Interior 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Rajagopalan, S., Suri, S.: Long non-crossing configurations in the plane. Fundamenta Informaticae 22, 385–394 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chung, F.R.K.: On unimodal subsequences. J. Combin. Theory Ser. A 29, 267–279 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dumitrescu, A., Tóth, C.: Long non-crossing configurations in the plane. Discrete Comput. Geom. 44, 727–752 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2, 463–470 (1935)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Czyzowicz, J., Kranakis, E., Krizanc, D., Urrutia, J.: Maximal length common non-intersecting paths. In: Proc. Eighth Canadian Conference on Computational Geometry, Ottawa, pp. 180–189 (August 1996)Google Scholar
  6. 6.
    Károlyi, G., Pach, J., Tóth, G.: Ramsey-type results for geometric graphs I. Discrete and Computational Geometry 18, 247–255 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sakai, T., Urrutia, J.: Monotonic Polygons and Paths in Weighted Point Sets. In: Akiyama, J., Bo, J., Kano, M., Tan, X. (eds.) CGGA 2010. LNCS, vol. 7033, pp. 164–175. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Tutte, W.T.: The quest of the perfect square. Amer. Math. Monthly 72, 29–35 (1965)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Toshinori Sakai
    • 1
  • Jorge Urrutia
    • 2
  1. 1.Tokai UniversityShibuya-kuJapan
  2. 2.Universidad Nacional Autónoma de MéxicoMéxico D.F.México

Personalised recommendations