Succinct Representation of Data Structures:Graph Representations

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Graph Representations

In this section, we briefly describe some space efficient representations of graphs. In particular, we consider representations that take close to the information theoretic minimum space and support degree and adjacency queries efficiently. A degree query asks for the degree of a given node in the graph, and an adjacency query asks whether two given vertices are adjacent or not in the graph. In addition, we would also like to support listing all the vertices adjacent to a given vertex.

Tura´n [57] gave a linear time constructible representation of an arbitrary planar graph that takes at most 12 bits per node. Though this gives a space efficient representation of planar graphs, it does not support the queries efficiently. Kannan et al. [34] have given an implicit (linear space) graph representation that supports adjacency queries using O(lg n) bit probes.

Jacobson [33] gave a representation that takes O(n) bits of space to represent a planar graph on n nodes and supports degree and adjacency queries in constant time. It uses a simple mapping of one-page graphs to sequences of balanced parentheses, and the fact that a planar graph always has a 4-page embedding. By storing auxiliary structures to support some natural operations on these sequences (see Section 37.4.2), one can also support the navigational operations in optimal time.

Munro and Raman [41] improved the space to 8n + 2m + o(n) bits, for a planar graph on n vertices with m edges, still supporting the queries in constant time. In general, their representation takes 2kn + 2m + o(nk + m) bits to store a k page graph on n vertices and m edges and supports degree and adjacency queries in O(k) time.

There have been several improvements [7, 8, 29, 37, 38], improving the space close to the information theoretic-lower bound, simultaneously expanding the class of graphs for which the scheme works. In particular, Lu [38] gave an optimal space (within lower-order terms) representation that can be constructed in linear time. This supports degree and adjacency queries in O(1) time for constant-genus graphs.

The main idea is to partition the given graph G on n vertices into o(n/ lg n) disjoint subgraphs of size O(lg6 n) by removing a subgraph H of size o(n/ lg n). This is done using a ‘planarization algorithm’ for bounded genus graphs, and an algorithm to construct a ‘separator decomposition tree’ of a planar graph. The representation of G is obtained by storing a rerepresentation of H, and recursing on each of the smaller subgraphs upto a constant number of levels, after which one can use a precomputed table to operate on the small subgraphs. See [38] for details.

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