Các Cây Con Biên Phân Biệt Trong Các Cây Ngẫu Nhiên

Louisa Seelbach Benkner1, Stephan Wagner2
1Department of Electrical Engineering and Computer Science, University of Siegen, Hölderlinstrasse 3, 57076, Siegen, Germany
2Department of Mathematics, Uppsala Universitet, Box 480, 751 06 Uppsala, Sweden

Tóm tắt

Abstract Một cây con biên là một cây con được sinh ra từ một đỉnh và tất cả các đỉnh con của nó trong một cây có gốc. Chúng tôi xem xét vấn đề ước lượng số lượng các cây con biên phân biệt trong các cây ngẫu nhiên theo một khái niệm tổng quát về sự phân biệt, cho phép nhiều cách diễn giải khác nhau về cái gọi là cây “phân biệt”. Các mô hình cây ngẫu nhiên được xem xét là cây sinh ngẫu nhiên và các họ cây tăng dần (cây đệ quy, cây tăng dần bậc d và cây đệ quy định hướng phẳng tổng quát). Chúng tôi chứng minh rằng bậc độ lớn của số lượng cây con biên phân biệt (dưới một số điều kiện nhẹ nhàng về nghĩa của 'phân biệt') trong các cây ngẫu nhiên với n đỉnh là $$n/\sqrt{\log n}$$ n / log n đối với các cây sinh ngẫu nhiên và $$n/\log n$$ n / log n đối với các cây tăng dần.

Từ khóa


Tài liệu tham khảo

Abiteboul, S., Bourhis, P., Vianu, V.: Highly expressive query languages for unordered data trees. Theory of Comput. Syst. 57(4), 927–966 (2015). https://doi.org/10.1007/s00224-015-9617-5

Aho, A.V., Sethi, R., Ullman, J.D.: Compilers: Principles, Techniques, and Tools. Addison-Wesley series in computer science / World student series edition. Addison-Wesley, Reading, MA, USA (1986)

Aldous, D.: Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1(2), 228–266 (1991). https://doi.org/10.1214/aoap/1177005936

Artin, M.: Algebra. Pearson Prentice Hall, Upper Saddle River, NJ, USA (2011)

Bergeron, F., Flajolet, P., Salvy, B.: Varieties of increasing trees. In: CAAP ’92 (Rennes, 1992), Lecture Notes in Comput. Sci., vol. 581, pp. 24–48. Springer, Berlin (1992). https://doi.org/10.1007/3-540-55251-0_2

Bodini, O., Genitrini, A., Gittenberger, B., Larcher, I., Naima, M.: Compaction for two models of logarithmic-depth trees: Analysis and experiments. Random Struct. Algorithms 61(1), 31–61 (2022). https://doi.org/10.1002/rsa.21056

Bóna, M., Flajolet, P.: Isomorphism and symmetries in random phylogenetic trees. J. Appl. Probab. 46(4), 1005–1019 (2009). https://doi.org/10.1239/jap/1261670685

Boneva, I., Ciucanu, R., Staworko, S.: Schemas for unordered XML on a DIME. Theory of Compuing Systems 57(2), 337–376 (2015). https://doi.org/10.1007/s00224-014-9593-1

Bousquet-Mélou, M., Lohrey, M., Maneth, S., Noeth, E.: XML compression via DAGs. Theory of Comput. Syst. 57(4), 1322–1371 (2015)

Bryant, R.E.: Symbolic boolean manipulation with ordered binary-decision diagrams. ACM Comput. Surv. 24(3), 293–318 (1992)

Buneman, P., Grohe, M., Koch, C.: Path queries on compressed XML. In: J.C. Freytag, et al. (eds.) Proceedings of the 29th Conference on Very Large Data Bases, VLDB 2003, pp. 141–152. Morgan Kaufmann (2003)

Cai, X.S.: A study of large fringe and non-fringe subtrees in conditional galton-watson trees. Ph.D. thesis, McGill University (2016)

Dennert, F., Grübel, R.: On the subtree size profile of binary search trees. Comb. Probab. Comput. 19(4), 561–578 (2010). https://doi.org/10.1017/S0963548309990630

Devroye, L.: On the richness of the collection of subtrees in random binary search trees. Inf. Process. Lett. 65(4), 195–199 (1998)

Devroye, L., Janson, S.: Protected nodes and fringe subtrees in some random trees. Electron. Commun. Probab. 19, 1–10 (2014). https://doi.org/10.1214/ECP.v19-3048

Drmota, M.: Random Trees: An Interplay Between Combinatorics and Probability, 1st edn. Springer, Vienna, Austria (2009)

Feng, Q., Mahmoud, H.M.: On the variety of shapes on the fringe of a random recursive tree. J. Appl. Probab. 47(1), 191–200 (2010). https://doi.org/10.1239/jap/1269610825

Fill, J.A., Kapur, N.: Limiting distributions for additive functionals on Catalan trees. Theoret. Comput. Sci. 326(1–3), 69–102 (2004). https://doi.org/10.1016/j.tcs.2004.05.010

Finch, S.R.: Mathematical Constants. Encyclopedia of Mathematics and its Applications 94. Cambridge University Press (2003). https://www.cambridge.org/academic/subjects/mathematics/recreationalmathematics/mathematical-constants

Flajolet, P., Gourdon, X., Martínez, C.: Patterns in random binary search trees. Random Structures & Algorithms 11(3), 223–244 (1997)

Flajolet, P., Sedgewick, R.: Mellin transforms and asymptotics: finite differences and Rice’s integrals. Theoret. Comput. Sci. 144(1–2), 101–124 (1995). https://doi.org/10.1016/0304-3975(94)00281-M. Special volume on mathematical analysis of algorithms

Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press (2009). https://www.cambridge.org/core/books/analytic-combinatorics/7E37474C43E9B95C90BEDE082CF28708

Flajolet, P., Sipala, P., Steyaert, J.M.: Analytic variations on the common subexpression problem. In: Proceedings of the 17th International Colloquium on Automata, Languages and Programming, ICALP 1990, Lecture Notes in Computer Science, vol. 443, pp. 220–234. Springer (1990)

Frick, M., Grohe, M., Koch, C.: Query evaluation on compressed trees (extended abstract). In: Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science, LICS 2003, pp. 188–197. IEEE Computer Society Press (2003)

Fuchs, M.: Limit theorems for subtree size profiles of increasing trees. Comb. Probab. Comput. 21(3), 412–441 (2012). https://doi.org/10.1017/S096354831100071X

Ganardi, M., Hucke, D., Lohrey, M., Benkner, L.S.: Universal tree source coding using grammar-based compression. IEEE Trans. Inf. Theory 65(10), 6399–6413 (2019). https://doi.org/10.1109/TIT.2019.2919829

Gut, A.: Probability: A Graduate Course. Springer, New York (2005)

Holmgren, C., Janson, S., Šileikis, M.: Multivariate normal limit laws for the numbers of fringe subtrees in $$m$$-ary search trees and preferential attachment trees. Electron. J. Combin. 24(2), Paper No. 2.51, 49 pp. (2017). https://doi.org/10.37236/6374

Janson, S.: Random cutting and records in deterministic and random trees. Random Structures & Algorithms 29(2), 139–179 (2006). https://doi.org/10.1002/rsa.20086. https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.20086

Janson, S.: Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Probab. Surv. 9, 103–252 (2012). https://doi.org/10.1214/11-PS188

Janson, S.: Asymptotic normality of fringe subtrees and additive functionals in conditioned galton-watson trees. Random Struct. Algorithms 48(1), 57–101 (2016). https://doi.org/10.1002/rsa.20568

Jordan, C.: Sur les assemblages de lignes. Journal für die reine und angewandte Mathematik 70, 185–190 (1869)

Kieffer, J.C., Yang, E.H., Szpankowski, W.: Structural complexity of random binary trees. In: Proceedings of the 2009 IEEE International Symposium on Information Theory, ISIT 2009, pp. 635–639. IEEE (2009)

Knuth, D.E.: The Art of Computer Programming, vol. 3. Addison-Wesley, Reading, MA (1998)

Kolchin, V.F.: Random mappings / Valentin F. Kolchin. Translations series in mathematics and engineering. Optimization Software, New York (1986)

Lohrey, M., Maneth, S., Reh, C.P.: Compression of unordered XML trees. In: 20th International Conference on Database Theory, ICDT 2017, March 21-24, 2017, Venice, Italy, pp. 18:1–18:17 (2017). https://doi.org/10.4230/LIPIcs.ICDT.2017.18

Meir, A., Moon, J.W.: On the altitude of nodes in random trees. Can. J. Math. 30(5), 997–1015 (1978). https://doi.org/10.4153/CJM-1978-085-0

Olsson, C., Wagner, S.: Automorphisms of random trees. In: 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, LIPIcs. Leibniz Int. Proc. Inform., vol. 225, pp. 16:1–16:16. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2022)

Otter, R.: The number of trees. Ann. of Math. 2(49), 583–599 (1948). https://doi.org/10.2307/1969046

Panholzer, A., Prodinger, H.: Level of nodes in increasing trees revisited. Random Structures Algorithms 31(2), 203–226 (2007). https://doi.org/10.1002/rsa.20161

Paterson, M., Wegman, M.N.: Linear unification. J. Comput. Syst. Sci. 16(2), 158–167 (1978)

Ralaivaosaona, D., Wagner, S.: A central limit theorem for additive functionals of increasing trees. Combin. Probab. Comput. 28(4), 618–637 (2019). https://doi.org/10.1017/s0963548318000585

Ralaivaosaona, D., Wagner, S.G.: Repeated fringe subtrees in random rooted trees. In: Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2015, pp. 78–88. SIAM (2015)

Ruskey, F.: Generating linear extensions of posets by transpositions. J. Combin. Theory Ser. B 54(1), 77–101 (1992)

Seelbach Benkner, L., Lohrey, M.: Average case analysis of leaf-centric binary tree sources. In: 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, pp. 16:1–16:15 (2018). https://doi.org/10.4230/LIPIcs.MFCS.2018.16

Seelbach Benkner, L., Wagner, S.G.: On the collection of fringe subtrees in random binary trees. In: Y. Kohayakawa, F.K. Miyazawa (eds.) LATIN 2020: Theoretical Informatics - 14th Latin American Symposium, São Paulo, Brazil, January 5-8, 2021, Proceedings, Lecture Notes in Computer Science, vol. 12118, pp. 546–558. Springer (2020). https://doi.org/10.1007/978-3-030-61792-9_43

Zhang, J., Yang, E.H., Kieffer, J.C.: A universal grammar-based code for lossless compression of binary trees. IEEE Trans. Inf. Theory 60(3), 1373–1386 (2014)

Zhang, S., Du, Z., Wang, J.T.: New techniques for mining frequent patterns in unordered trees. IEEE Transactions on Cybernetics 45(6), 1113–1125 (2015). https://doi.org/10.1109/TCYB.2014.2345579