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

I am just going to touch on overcoming the trade-off of regular adjacency list representation, since other answers have covered other aspects.

It is possible to represent a graph in adjacency list with EdgeExists query in amortized constant time, by taking advantage of Dictionary and HashSet data structures. The idea is to keep vertices in a dictionary, and for each vertex, we keep a hash set referencing to other vertices it has edges with.

One minor trade-off in this implementation is that it will have space complexity O(V + 2E) instead of O(V + E) as in regular adjacency list, since edges are represented twice here (because each vertex have its own hash set of edges). But operations such as AddVertex, AddEdge, RemoveEdge can be done in amortized time O(1) with this implementation, except for RemoveVertex which takes O(V) like adjacency matrix. This would mean that other than implementation simplicity, adjacency matrix don't have any specific advantage. We can save space on sparse graph with almost the same performance in this adjacency list implementation.

Take a look at implementations below in Github C# repository for details. Note that for weighted graph it uses a nested dictionary instead of dictionary-hash set combination so as to accommodate weight value. Similarly for directed graph there is separate hash sets for in & out edges.

Advanced-Algorithms

Note: I believe using lazy deletion we can further optimize RemoveVertex operation to O(1) amortized, even though I haven't tested that idea. For example, upon deletion just mark the vertex as deleted in dictionary, and then lazily clear orphaned edges during other operations.