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A Local Approach to Studying the Time and Space Complexity of Deterministic and Nondeterministic Decision Trees (2311.17306v1)

Published 29 Nov 2023 in cs.CC

Abstract: In this paper, we study arbitrary infinite binary information systems each of which consists of an infinite set called universe and an infinite set of two-valued functions (attributes) defined on the universe. We consider the notion of a problem over information system, which is described by a finite number of attributes and a mapping associating a decision to each tuple of attribute values. As algorithms for problem solving, we investigate deterministic and nondeterministic decision trees that use only attributes from the problem description. Nondeterministic decision trees are representations of decision rule systems that sometimes have less space complexity than the original rule systems. As time and space complexity, we study the depth and the number of nodes in the decision trees. In the worst case, with the growth of the number of attributes in the problem description, (i) the minimum depth of deterministic decision trees grows either as a logarithm or linearly, (ii) the minimum depth of nondeterministic decision trees either is bounded from above by a constant or grows linearly, (iii) the minimum number of nodes in deterministic decision trees has either polynomial or exponential growth, and (iv) the minimum number of nodes in nondeterministic decision trees has either polynomial or exponential growth. Based on these results, we divide the set of all infinite binary information systems into three complexity classes. This allows us to identify nontrivial relationships between deterministic decision trees and decision rules systems represented by nondeterministic decision trees. For each class, we study issues related to time-space trade-off for deterministic and nondeterministic decision trees.

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References (38)
  1. Springer, Cham (2019)
  2. Springer, Cham (2020)
  3. Ben-Or, M.: Lower bounds for algebraic computation trees (preliminary report). In: 15th Annual ACM Symposium on Theory of Computing, STOC 1983, pp. 80–86 (1983)
  4. Math. Program. 79, 163–190 (1997)
  5. IEEE Trans. Knowl. Data Eng. 12(2), 292–306 (2000)
  6. Wadsworth, Belmont, CA (1984)
  7. Springer, Berlin Heidelberg (2013)
  8. J. Comput. Syst. Sci. 16(3), 413–417 (1978)
  9. J. Comput. Syst. Sci. 18(1), 86–91 (1979)
  10. In: 2023 IEEE International Conference on Big Data. 9th Special Session on Information Granulation in Data Science and Scalable Computing (2023). (to appear)
  11. Cognitive Technologies. Springer, Berlin Heidelberg (2012)
  12. Found. Comput. Math. 17(1), 61–72 (2017)
  13. In: 36th Annual Symposium on Foundations of Computer Science, FOCS 1995, pp. 258–265 (1995)
  14. Computational Complexity 7(3), 193–203 (1998)
  15. URL christophm.github.io/interpretable-ml-book/
  16. Morávek, J.: A localization problem in geometry and complexity of discrete programming. Kybernetika 8(6), 498–516 (1972)
  17. Moshkov, M.: Decision Trees. Theory and Applications (in Russian). Nizhny Novgorod University Publishers, Nizhny Novgorod (1994)
  18. Moshkov, M.: Optimization problems for decision trees. Fundam. Inform. 21(4), 391–401 (1994)
  19. Moshkov, M.: Two approaches to investigation of deterministic and nondeterministic decision trees complexity. In: 2nd World Conference on the Fundamentals of Artificial Intelligence, WOCFAI 1995, pp. 275–280 (1995)
  20. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Global approach. Fundam. Inform. 25(2), 201–214 (1996)
  21. Moshkov, M.: Comparative analysis of deterministic and nondeterministic decision tree complexity. Local approach. In: Trans. Rough Sets IV, Lecture Notes in Computer Science, vol. 3700, pp. 125–143. Springer, Berlin Heidelberg (2005)
  22. Moshkov, M.: Time complexity of decision trees. In: Trans. Rough Sets III, Lecture Notes in Computer Science, vol. 3400, pp. 244–459. Springer, Berlin Heidelberg (2005)
  23. Moshkov, M.: On the class of restricted linear information systems. Discret. Math. 307(22), 2837–2844 (2007)
  24. Moshkov, M.: Time and space complexity of deterministic and nondeterministic decision trees. Ann. Math. Artif. Intell. 91(1), 45–74 (2023)
  25. Springer, Berlin Heidelberg (2008)
  26. Springer, Berlin Heidelberg (2011)
  27. Discr. Math. 154, 203–216 (1996)
  28. Pawlak, Z.: Information systems theoretical foundations. Inf. Syst. 6(3), 205–218 (1981)
  29. Kluwer (1991)
  30. In: B.W. Wah (ed.) Wiley Encyclopedia of Computer Science and Engineering. John Wiley & Sons, Inc. (2008). URL https://doi.org/10.1002/9780470050118.ecse466
  31. Inf. Sci. 177(1), 3–27 (2007)
  32. WorldScientific, Singapore (2007)
  33. Sauer, N.: On the density of families of sets. J. of Combinatorial Theory (A) 13, 145–147 (1972)
  34. Shelah, S.: A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. of Mathematics 41, 241–261 (1972)
  35. In: R. Slowinski (ed.) Intelligent Decision Support - Handbook of Applications and Advances of the Rough Sets Theory, Theory and Decision Library, vol. 11, pp. 331–362. Springer (1992)
  36. J. Algorithms 3(1), 1–8 (1982)
  37. Yao, A.C.: Algebraic decision trees and Euler characteristics. In: 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992, pp. 268–277 (1992)
  38. Yao, A.C.: Decision tree complexity and Betti numbers. In: 26th Annual ACM Symposium on Theory of Computing, STOC 1994, pp. 615–624 (1994)
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