Graph Representation in Memory

Internal

• Graphs

Overview

CLRS page 589, DDIA location 1392.

Graph Input Size

The input size of a graph is given by two parameters: the number of vertices (n) and the number of edges (m). For a connected, undirected graph with n vertices, the number of edges ranges from n - 1 and n⋅(n - 1)/2.

Adjacency Lists

An adjacency list graph is a data structure that keeps track of vertices and edges as independent entities. The structure can be used to store both directed and undirected graphs.

In its most generic form, it has an array or a list of vertices and an array or a list of edges. These two arrays cross-reference each other. Each edge has two pointers, one for each of its vertex endpoints. For directed graphs we keep track of which one is the head and which one is the tail. Each vertex points to all of the edges of which it is a member. For directed graphs, a vertex keeps track of each of the edges it is the tail.

A practical implementation (see CLRS page 590) consists of an array adj of n lists, where the n is the number of vertices. Each array element corresponds to one of the graph's vertices, numbered from 0 to n-1, and contains an adjacency list: the list that contains all the vertices in the graph for which there's an edge to the current vertex. That is, adj[i] contains all the vertices adjacent to i in the graph.

Note that for an undirected graph, each edge is represented twice. For a (u, v) pair of vertices, the edge is represented as "v" in the u's adjacency list and as as "u" in v's adjacency list. For a directed graph, that is not the case, as u → v ≠ v → u.

The space requirements for this data structure is Θ(n + m). An adjacency list representation provides a memory-efficient way of representing sparse graphs, so it is usually the method of choice. Adjacency lists are very well suited for graph search.

Adjacency Matrices

The edges of a graph are represented using a matrix. Applies to directed and undirected graphs. The matrix is denoted by A, and it is a nxn square matrix, where n is the number of vertices in the graph. The semantics is A_ij=1 if and only if there's an edge between vertices i and j in the graph. It can be extended for parallel arcs and weighted graphs. For parallel arcs, A_ij denotes the number of edges. For weighted graphs, A_ij denotes the weight of the edge. For directed graphs, if the arc is directed from i to j, A_ij=+1 and if it is directed from j to i, A_ij=-1.

Space requirements: an adjacency matrix requires space quadratic in the number of vertices Θ(n²). For dense graphs this is fine but for sparse graphs, it is wasteful.

Cases when adjacency matrix representation is preferred:

• dense graphs.
• When we need to tell quickly if there is an edge connecting two given vertices.

More details in CLRS page 590.