Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On Complexity of 1-Center in Various Metrics (2112.03222v3)

Published 6 Dec 2021 in cs.CC, cs.CG, cs.DS, and cs.LG

Abstract: We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=\Omega(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+\epsilon)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (36)
  1. Scheduling lower bounds via AND subset sum. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 4:1–4:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
  2. Distributed PCP theorems for hardness of approximation in P. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 25–36, 2017.
  3. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377–391. SIAM, 2016.
  4. Polyline simplification has cubic complexity. In Gill Barequet and Yusu Wang, editors, 35th International Symposium on Computational Geometry, SoCG 2019, June 18-21, 2019, Portland, Oregon, USA, volume 129 of LIPIcs, pages 18:1–18:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
  5. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 51–58. ACM, 2015.
  6. Quadratic conditional lower bounds for string problems and dynamic time warping. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 79–97. IEEE Computer Society, 2015.
  7. Banishing bias from consensus sequences. In Alberto Apostolico and Jotun Hein, editors, Combinatorial Pattern Matching, 8th Annual Symposium, CPM 97, Aarhus, Denmark, June 30 - July 2, 1997, Proceedings, volume 1264 of Lecture Notes in Computer Science, pages 247–261. Springer, 1997.
  8. Approximating the median under the ulam metric. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 761–775. SIAM, 2021.
  9. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In Madhu Sudan, editor, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 261–270. ACM, 2016.
  10. Lijie Chen. On the hardness of approximate and exact (bichromatic) maximum inner product. Theory Comput., 16:1–50, 2020.
  11. Sublinear optimization for machine learning. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 449–457. IEEE Computer Society, 2010.
  12. On approximability of clustering problems without candidate centers. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2635–2648. SIAM, 2021.
  13. Geometric median in nearly linear time. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 9–21. ACM, 2016.
  14. On the complexity of closest pair via polar–pair of point–sets. SIAM J. Discrete Math., 33(1):509–527, 2019.
  15. Colin de la Higuera and Francisco Casacuberta. Topology of strings: Median string is np-complete. Theoretical Computer Science, 230(1):39–48, 2000.
  16. On covering problems of codes. Theory Comput. Syst., 30(2):113–119, 1997.
  17. Some np-complete geometric problems. In Ashok K. Chandra, Detlef Wotschke, Emily P. Friedman, and Michael A. Harrison, editors, Proceedings of the 8th Annual ACM Symposium on Theory of Computing, May 3-5, 1976, Hershey, Pennsylvania, USA, pages 10–22. ACM, 1976.
  18. Efficient approximation algorithms for the hamming center problem. In Robert Endre Tarjan and Tandy J. Warnow, editors, Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 17-19 January 1999, Baltimore, Maryland, USA, pages 905–906. ACM/SIAM, 1999.
  19. On the asymptotic behaviour of some towers of function fields over finite fields. Journal of Number Theory, 61(2):248 – 273, 1996.
  20. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1–33:38, 2019.
  21. Super-logarithmic depth lower bounds via direct sum in communication coplexity. In Proceedings of the Sixth Annual Structure in Complexity Theory Conference, Chicago, Illinois, USA, June 30 - July 3, 1991, pages 299–304. IEEE Computer Society, 1991.
  22. On a speculated relation between chvátal–sankoff constants of several sequences. Combinatorics, Probability and Computing, 18(4):517–532, 2009.
  23. Distinguishing string selection problems. Inf. Comput., 185(1):41–55, 2003.
  24. Finding similar regions in many sequences. J. Comput. Syst. Sci., 65(1):73–96, 2002.
  25. On the closest string and substring problems. J. ACM, 49(2):157–171, 2002.
  26. Ashley Montanaro. Metric Embeddings. http://people.maths.bris.ac.uk/~csxam/presentations/embeddings.pdf, 2008. [Online; accessed 12-December-2008].
  27. Dynamic algorithms for LIS and distance to monotonicity. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 671–684. ACM, 2020.
  28. Complexities of the centre and median string problems. In Proceedings of the 14th Annual Conference on Combinatorial Pattern Matching, CPM’03, page 315–327, Berlin, Heidelberg, 2003. Springer-Verlag.
  29. Hardness results for the center and median string problems under the weighted and unweighted edit distances. Journal of discrete algorithms, 3(2-4):390–415, 2005.
  30. Accurate and nearly optimal sublinear approximations to ulam distance. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2012–2031. SIAM, 2017.
  31. Aviad Rubinstein. Hardness of approximate nearest neighbor search. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 1260–1268. ACM, 2018.
  32. A low-complexity algorithm for the construction of algebraic-geometric codes better than the Gilbert-Varshamov bound. IEEE Trans. Information Theory, 47(6):2225–2241, 2001.
  33. A linear-time algorithm for the 1-mismatch problem. In Frank K. H. A. Dehne, Andrew Rau-Chaplin, Jörg-Rüdiger Sack, and Roberto Tamassia, editors, Algorithms and Data Structures, 5th International Workshop, WADS ’97, Halifax, Nova Scotia, Canada, August 6-8, 1997, Proceedings, volume 1272 of Lecture Notes in Computer Science, pages 126–135. Springer, 1997.
  34. The consensus string problem for a metric is np-complete. J. Discrete Algorithms, 1(1):111–117, 2003.
  35. Ryan Williams. On the difference between closest, furthest, and orthogonal pairs: Nearly-linear vs barely-subquadratic complexity. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1207–1215. SIAM, 2018.
  36. Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the International Congress of Mathematicians: Rio de Janeiro 2018, pages 3447–3487. World Scientific, 2018.
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now