Question

I’ m trying to implement a weighted graph. I know that there are two ways to implement a weighted graph. Either with a two dimensional array(adjacency matrix) or with an array of linked lists(adjacency list). Which one of the two is more efficient and faster?

Answer 1

Which one of the two is more efficient and faster?

That depends on your usage and the kinds of graphs you want to store.

Let n be the number of nodes and m be the number of edges. If you want to know whether two nodes u and v are connected (and the weight of the edge), an adjacency matrix allows you to determine this in constant time (in O-notation, O(1)), simply by retrieving the entry A[u,v]. With an adjacency list, you will have to look at every entry in u’s list, or v’s list – in the worst case, there could be n entries. So edge lookup for an adjacency list is in O(n).

The main downside of an adjacency matrix is the memory required. Alltogether, you need to store n^2 entries. With an adjacency list, you need to store only the edges that actually exist (m entries, asuming a directed graph). So if your graph is sparse, adjacency lists clearly occupy much less memory.

My conclusion would be: Use an adjacency matrix if your main operation is retrieving the edge weight for two specific nodes; under the condition that your graphs are small enough so that n^2 entries fit in memory. Otherwise, use the adjacency list.