Abstract
Continuing the recent trend, in this article we design several space-efficient algorithms for two well-known graph search methods. Both these search methods share the same name breadth-depth search (henceforth BDS), although they work entirely in different fashion. The classical implementation for these graph search methods takes \(O(m+n)\) time and \(O(n \lg n)\) bits of space in the standard word RAM model (with word size being \(\varTheta (\lg n)\) bits), where m and n denotes the number of edges and vertices of the input graph respectively. Our goal here is to beat the space bound of the classical implementations, and design \(o(n \lg n)\) space algorithms for these search methods by paying little to no penalty in the running time. Note that our space bounds (i.e., with \(o(n \lg n)\) bits of space) do not even allow us to explicitly store the required information to implement the classical algorithms, yet our algorithms visits and reports all the vertices of the input graph in correct order.
This work was partially supported by JST CREST Grant Number JPMJCR1402.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We use \(\lg \) to denote logarithm to the base 2.
- 2.
Our algorithm performs atmost \(O(m+n)\) insertion/deletion/retrieval during its entire execution using the dictionary of Theorem 2 which takes O(1) time with a probability of \((1-1/n^c)\) (where \(c \ge 3\)) for each insertion/deletion/retrieval. Thus, the probability that our algorithm takes more than \(O(m+n)\) time is \((1/n^{c-2})\) by union bound rule.
References
Asano, T., et al.: Depth-first search using \(O(n)\) bits. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 553–564. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13075-0_44
Banerjee, N., Chakraborty, S., Raman, V., Satti, S.R.: Space efficient linear time algorithms for BFS, DFS and applications. Theory Comput. Syst. (2018)
Banyassady, B., et al.: Improved time-space trade-offs for computing voronoi diagrams. In: 34th STACS, pp. 9:1–9:14 (2017)
Chakraborty, S.: Space efficient graph algorithms. Ph.D. thesis. The Institute of Mathematical Sciences, HBNI, India (2018)
Chakraborty, S., Jo, S., Satti, S.R.: Improved space-efficient linear time algorithms for some classical graph problems. CoRR, abs/1712.03349 (2017)
Chakraborty, S., Mukherjee, A., Raman, V., Satti, S.R.: A framework for in-place graph algorithms. In: 26th ESA, pp. 13:1–13:16 (2018)
Chakraborty, S., Raman, V., Satti, S.R.: Biconnectivity, st-numbering and other applications of DFS using \({\rm O}(n)\) bits. J. Comput. Syst. Sci. 90, 63–79 (2017)
Chakraborty, S., Satti, S.R.: Space-efficient algorithms for maximum cardinality search, its applications, and variants of BFS. J. Comb. Optim. 37(2), 465–481 (2018)
Corneil, D.G., Krueger, R.: A unified view of graph searching. SIAM J. Discrete Math. 22(4), 1259–1276 (2008)
Demaine, E.D., der Heide, F.M., Pagh, R., Pǎtraşcu, M.: De dictionariis dynamicis pauco spatio utentibus. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 349–361. Springer, Heidelberg (2006). https://doi.org/10.1007/11682462_34
Elmasry, A., Hagerup, T., Kammer, F.: Space-efficient basic graph algorithms. In: 32nd STACS, pp. 288–301 (2015)
Greenlaw, R.: Breadth-depth search is P-complete. Parallel Process. Lett. 3, 209–222 (1993)
Horowitz, E., Sahni, S.: Fundamentals of Computer Algorithms. Computer Science Press (1978)
Jiang, B.: I/O-and CPU-optimal recognition of strongly connected components. Inf. Process. Lett. 45(3), 111–115 (1993)
Kiyomi, M., Ono, H., Otachi, Y., Schweitzer, P., Tarui, J.: Space-efficient algorithms for longest increasing subsequence. In: 35th STACS, pp. 44:1–44:15 (2018)
Lincoln, A., Williams, V.V., Wang, J.R., Williams, R.R.: Deterministic time-space trade-offs for k-sum. In: 43rd ICALP, pp. 58:1–58:14 (2016)
Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Chakraborty, S., Mukherjee, A., Satti, S.R. (2019). Space Efficient Algorithms for Breadth-Depth Search. In: Gąsieniec, L., Jansson, J., Levcopoulos, C. (eds) Fundamentals of Computation Theory. FCT 2019. Lecture Notes in Computer Science(), vol 11651. Springer, Cham. https://doi.org/10.1007/978-3-030-25027-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-25027-0_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-25026-3
Online ISBN: 978-3-030-25027-0
eBook Packages: Computer ScienceComputer Science (R0)