What is better, adjacency lists or adjacency matrices for graph problems in C++?

Question

What is better, adjacency lists or adjacency matrix, for graph problems in C++? What are the advantages and disadvantages of each?

Answer 1

To add to keyser5053's answer about memory usage.

For any directed graph, an adjacency matrix (at 1 bit per edge) consumes n^2 * (1) bits of memory.

For a complete graph, an adjacency list (with 64 bit pointers) consumes n * (n * 64) bits of memory, excluding list overhead.

For an incomplete graph, an adjacency list consumes 0 bits of memory, excluding list overhead.

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For an adjacency list, you can use the follow formula to determine the maximum number of edges (e) before an adjacency matrix is optimal for memory.

edges = n^2 / s to determine the maximum number of edges, where s is the pointer size of the platform.

If you're graph is dynamically updating, you can maintain this efficiency with an average edge count (per node) of n / s.

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Some examples with 64 bit pointers and dynamic graph (A dynamic graph updates the solution of a problem efficiently after changes, rather than recomputing it from scratch each time after a change has been made.)

For a directed graph, where n is 300, the optimal number of edges per node using an adjacency list is:

= 300 / 64
= 4

If we plug this into keyser5053's formula, d = e / n^2 (where e is the total edge count), we can see we are below the break point (1 / s):

d = (4 * 300) / (300 * 300)
d < 1/64
aka 0.0133 < 0.0156

However, 64 bits for a pointer can be overkill. If you instead use 16bit integers as pointer offsets, we can fit up to 18 edges before breaking point.

= 300 / 16
= 18

d = ((18 * 300) / (300^2))
d < 1/16
aka 0.06 < 0.0625

Each of these examples ignore the overhead of the adjacency lists themselves (64*2 for a vector and 64 bit pointers).