Compact Implicit Representation of Graphs
How to represent a graph in memory is a fundamental data structuring problem. In the usual representations, a graph is stored by representing explicitly all vertices and all edges. The names (labels) assigned to vertices are used only to encode the edges and betray nothing about the structure of the graph itself and hence are a “waste” of space. In this context, we present a general framework for labeling any graph so that adjacency between any two given vertices can be tested in constant time. The labeling schema assigns to each vertex x of a general graph a O(δ(x)log3n) bit label, where n is the number of vertices and δ(x) is x’s degree. The adjacency test can be performed in 5 steps and the schema can be computed in polynomial time. This representation strictly contrasts with usual representations, i.e. adjacency matrix and adjacency list representations, which require O(nlog n) bit label per vertex and constant time adjacency test, and O(δ(x)log n) bit label per vertex and O(logδ (x)) steps to test adjacency, respectively. Additionally, the labeling schema is implicit, that is: no pointers are used.
KeywordsBipartite Graph General Graph Label Schema Implicit Representation Node Basis
Unable to display preview. Download preview PDF.
- 5.Frederickson, G.N., Janardan, R.: Optimal message routing without complete routing tables. In: Proc. 5th Annual ACM Symposium on Principles of Distributed Computing, Calgary, August 1986, pp. 88–97 (1986)Google Scholar
- 17.Talamo, M., Vocca, P.: A time optimal digraph browsing on a sparse representation. Technical Report 8, Math Department, University of Rome "Tor Vergata", Submitted to JGAA (1997)Google Scholar
- 18.Turan, G.: Succint representation of graphs. Discrete Appl. Math. 8, 289–284 (1984)Google Scholar
- 20.van Leeuwen, J., Tan, R.B.: Computer networks with compact routing tables. In: Rozenberg, G., Salomaa, A. (eds.) The Book of L, vol. 790. Springer, Heidelberg (1986)Google Scholar
- 22.van Leeuwen, J., Tan, R.B.: Compact routing methods: A survey. In: Proc. Colloquium on Structural Information and Communication Complexity (SICC 1994). Carleton University Press, Ottawa (1994)Google Scholar