CIAC 2006: Algorithms and Complexity pp 272-283

# How to Pack Directed Acyclic Graphs into Small Blocks

• Yuichi Asahiro
• Tetsuya Furukawa
• Keiichi Ikegami
• Eiji Miyano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3998)

## Abstract

The paper studies the following variant of clustering or laying out problems of graphs: Given a directed acyclic graph (DAG for short), the objective is to find a mapping of its nodes into blocks of size at most B that minimizes the maximum number of external arcs during traversals of the acyclic structure by following paths from the roots to the leaves. An external arc is defined as an arc connecting two distinct blocks. The problem can be shown to be NP-hard generally, and to remain intractable even if B = 2 and the height of DAGs is three. In this paper we provide a $$\frac{3}{2}$$ factor linear time approximation algorithm for B = 2, and prove that the $$\frac{3}{2}$$ ratio is optimal in terms of approximation guarantee. In the case of B ≥ 3, we also show that there is no $$\frac{3}{2} - \varepsilon$$ factor approximation algorithm assuming P ≠ NP, where ε is arbitrarily small positive. Furthermore, we give a 2 factor approximation algorithm for B = 3 if the input is restricted to a set of layered graphs.

## Keywords

Approximation Algorithm Layered Graph Small Block External Memory Truth Assignment
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.

## References

1. 1.
Abello, J., Pardalos, P.M., Resende, M.G.C. (eds.): Handbook of massive data sets. Kluwer Academic Pub., Dordrecht (2002)
2. 2.
Aggarwal, A., Vitter, J.S.: The input/output complexity of sorting and related problems. Commun. ACM 31(9), 1116–1127 (1988)
3. 3.
Alstrup, S., Bender, M.A., Demaine, E.D., Farach-Colton, M., Rauhe, T., Thorup, M.: Efficient tree layout in a multilevel memory hierarchy. CoRR cs.DS/0211010 (2002)Google Scholar
4. 4.
Amer-Yahia, S., Koudas, N., Marian, A., Srivastava, D., Toman, D.: Structure and content scoring for XML. In: Proc. 31st VLDB, pp. 361–372 (2005)Google Scholar
5. 5.
Arge, L., Danner, A., Teh, S.-M.: I/O-efficient point location using persistent B-trees. In: Proc. 5th ALENEX, pp. 82–92 (2003)Google Scholar
6. 6.
Clark, D., Munro, J.: Efficient suffix trees on secondary storage. In: Proc. 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 383–391 (1996)Google Scholar
7. 7.
Diwan, A.A., Rane, S., Seshadri, S., Sudarshan, S.: Clustering techniques for minimizing external path length. In: Proc. 22nd VLDB, pp. 432–353 (1996)Google Scholar
8. 8.
Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5, 691–703 (1976)
9. 9.
Gil, J., Itai, A.: Packing trees. In: Proc. 3rd Annual European Symposium on Algorithms, pp. 113–127 (1995); Full version: J. Algorithms 32(2), pp.108–132 (1999)Google Scholar
10. 10.
Kirkpatrick, D.G.: Optimal search in planar subdivisions. SIAM J. Comput. 12(28), 28–35 (1983)
11. 11.
Varman, P.J., Verma, R.M.: An efficient multiversion access structure. IEEE Trans. on Knowledge and Data Engineering 9(3) (1997)Google Scholar
12. 12.
Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
13. 13.
Vitter, J.S.: External memory algorithms and data structures: Dealing with massive data. ACM Comput. Surveys 33(2), 209–271 (2001)

## Authors and Affiliations

• Yuichi Asahiro
• 1
• Tetsuya Furukawa
• 2
• Keiichi Ikegami
• 3
• Eiji Miyano
• 3
1. 1.Department of Social Information SystemsKyushu Sangyo UniversityFukuokaJapan
2. 2.Department of Economic EngineeringKyushu UniversityFukuokaJapan
3. 3.Department of Systems Innovation and InformaticsKyushu Institute of TechnologyFukuokaJapan