Three ways to store a graph in memory, advantages and disadvantages

Question

There are three ways to store a graph in memory:

1. Nodes as objects and edges as pointers
2. A matrix containing all edge weights between numbered node x a

Answer 1

There is another option: nodes as objects, edges as objects too, each edge being at the same time in two doubly-linked lists: the list of all edges coming out from the same node and the list of all edges going into the same node.

struct Node {
    ... node payload ...
    Edge *first_in;    // All incoming edges
    Edge *first_out;   // All outgoing edges
};

struct Edge {
    ... edge payload ...
    Node *from, *to;
    Edge *prev_in_from, *next_in_from; // dlist of same "from"
    Edge *prev_in_to, *next_in_to;     // dlist of same "to"
};

The memory overhead is big (2 pointers per node and 6 pointers per edge) but you get

• O(1) node insertion
• O(1) edge insertion (given pointers to "from" and "to" nodes)
• O(1) edge deletion (given the pointer)
• O(deg(n)) node deletion (given the pointer)
• O(deg(n)) finding neighbors of a node

The structure also can represent a rather general graph: oriented multigraph with loops (i.e. you can have multiple distinct edges between the same two nodes including multiple distinct loops - edges going from x to x).

A more detailed explanation of this approach is available here.