Adjacency Matrix - Data Structures

Data Structures

For use as a data structure, the main alternative to the adjacency matrix is the adjacency list. Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only bytes of contiguous space, where is the number of vertices. Besides avoiding wasted space, this compactness encourages locality of reference.

However, if the graph is sparse, adjacency lists require less storage space, because they do not waste any space to represent edges that are not present. Using a naïve array implementation on a 32-bit computer, an adjacency list for an undirected graph requires about bytes of storage, where is the number of edges.

Noting that a simple graph can have at most edges, allowing loops, we can let denote the density of the graph. Then, or the adjacency list representation occupies more space precisely when . Thus a graph must be sparse indeed to justify an adjacency list representation.

Besides the space tradeoff, the different data structures also facilitate different operations. Finding all vertices adjacent to a given vertex in an adjacency list is as simple as reading the list. With an adjacency matrix, an entire row must instead be scanned, which takes O(n) time. Whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list.