Abstract
There are many representations of planar graphs but few are as elegant as Turán’s (1984): it is simple and practical, uses only four bits per edge, can handle multi-edges and can store any specified embedding. Its main disadvantage has been that “it does not allow efficient searching” (Jacobson, 1989). In this paper we show how to add a sublinear number of bits to Turán’s representation such that it supports fast navigation, thus overcoming this disadvantage. Other data structures for planar embeddings may be asymptotically faster or smaller but ours is simpler, and that can be a theoretical as well as a practical advantage: e.g., we show how our structure can be built efficiently in parallel.
The second and fifth authors received travel funding from EU grant H2020-MSCA-RISE-2015 BIRDS GA No. 690941. The second, third and fifth authors received funding from Basal Funds FB0001, Conicyt, Chile. The third author received funding from Academy of Finland grant 268324. Early parts of this work were done while the third author was at the University of Helsinki and while the third and fifth authors were visiting the University of A Coruña. Many thanks to Jérémy Barbay, Luca Castelli Aleardi, Arash Farzan, Ian Munro, Pat Nicholson and Julian Shun. The third author is grateful to the late David Gregory for his course on graph theory.
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References
Awerbuch, B., Shiloach, Y.: New connectivity and MSF algorithms for shuffle-exchange network and PRAM. IEEE Trans. Computers 36(10), 1258–1263 (1987)
Bader, D.A., Cong, G.: A fast, parallel spanning tree algorithm for symmetric multiprocessors (SMPs). J. Parallel and Distributed Computing 65, 994–1006 (2005)
Barbay, J., Aleardi, L.C., He, M., Munro, J.I.: Succinct representation of labeled graphs. Algorithmica 62, 224–257 (2012)
Biggs, N.: Spanning trees of dual graphs. J. Comb. Theory, Series B, 11: 127–131 (1971)
Blandford, D.K., Blelloch, G.E., Kash, I.A.: Compact representations of separable graphs. In: SODA, pp. 679–688 (2003)
Blelloch, G.E., Farzan, A.: Succinct representations of separable graphs. In: Amir, A., Parida, L. (eds.) CPM 2010. LNCS, vol. 6129, pp. 138–150. Springer, Heidelberg (2010). doi:10.1007/978-3-642-13509-5_13
Aleardi, L.C., Devillers, O., Schaeffer, G.: Succinct representations of planar maps. TCS 408, 174–187 (2008)
Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications. SIAM J. Comp. 34, 924–945 (2005)
Cong, G., Bader, D.A.: The Euler tour technique and parallel rooted spanning tree. In: ICPP, pp. 448–457 (2004)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Multithreaded algorithms. In: Introduction to Algorithms, pp. 772–812. MIT Press (2009)
Eppstein, D.: Dynamic generators of topologically embedded graphs. In: SODA, pp. 599–608 (2003)
Ferres, L., Fuentes-Sepúlveda, J., He, M., Zeh, N.: Parallel construction of succinct trees. In: Bampis, E. (ed.) SEA 2015. LNCS, vol. 9125, pp. 3–14. Springer, Cham (2015). doi:10.1007/978-3-319-20086-6_1
Fusy, É., Schaeffer, G., Poulalhon, D.: Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling. TALG 4, 19 (2008)
He, X., Kao, M., Lu, H.: Linear-time succinct encodings of planar graphs via canonical orderings. SIAM J. Discrete Math. 12, 317–325 (1999)
Jacobson, G.: Space-efficient static trees and graphs. In: FOCS, pp. 549–554 (1989)
Kao, M., Teng, S., Toyama, K.: An optimal parallel algorithm for planar cycle separators. Algorithmica 14, 398–408 (1995)
Keeler, K., Westbrook, J.: Short encodings of planar graphs and maps. DAM 58, 239–252 (1995)
Labeit, J., Shun, J., Blelloch, G.E.: Parallel lightweight wavelet tree, suffix array and FM-index construction. In: DCC, pp. 33–42 (2016)
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Applied Math. 36, 177–189 (1979)
Munro, J.I., Nicholson, P.K.: Compressed representations of graphs. In: Encyclopedia of Algorithms, pp. 382–386. Springer (2016)
Navarro, G.: Compact Data Structures: A Practical Approach. Cambridge University Press (2016)
Riley, T.R., Thurston, W.P.: The absence of efficient dual pairs of spanning trees in planar graphs. Electronic J. Comb. 13 (2006)
Shiloach, Y., Vishkin, U.: An o(log n) parallel connectivity algorithm. J. Algorithms 3(1), 57–67 (1982)
Shun, J., Dhulipala, L., Blelloch, G.E.: A simple and practical linear-work parallel algorithm for connectivity. In: SPAA, pp. 143–153 (2014)
Turán, A.: On the succinct representation of graphs. DAM 8, 289–294 (1984)
Tutte, W.T.: A census of planar maps. Canadian J. Math. 15, 249–271 (1963)
Yannakakis, M.: Embedding planar graphs in four pages. JCSS 38, 36–67 (1989)
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Ferres, L., Fuentes, J., Gagie, T., He, M., Navarro, G. (2017). Fast and Compact Planar Embeddings. In: Ellen, F., Kolokolova, A., Sack, JR. (eds) Algorithms and Data Structures. WADS 2017. Lecture Notes in Computer Science(), vol 10389. Springer, Cham. https://doi.org/10.1007/978-3-319-62127-2_33
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DOI: https://doi.org/10.1007/978-3-319-62127-2_33
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