So, we are keeping a track of the Adjacency List of each Vertex. The O(|V | 2) memory space required is the main limitation of the adjacency matrices. The weights can also be stored in the Linked List Node. Finding an edge is fast. Each Node in this Linked list represents the reference to the other vertices which share an edge with the current vertex. Note that in the below implementation, we use dynamic arrays (vector in C++/ArrayList in Java) to represent adjacency lists instead of the linked list. If a graph G = (V,E) has |V| vertices and |E| edges, then what is the amount of space needed to store the graph using the adjacency list representation? You analysis is correct for a completely connected graph. • Depending on problems, both representations are useful. 5. So the amount of space that's required is going to be n plus m for the edge list and the implementation list. First is the variables dependence on which you are studying; second are those variables that are considered constant; and third are kind of "free" variables, which you usually assume to take the worst-case values. Time needed to find all neighbors in O(n). This representation takes O(V+2E) for undirected graph, and O(V+E) for directed graph. Now, if we consider 'm' to be the length of the Linked List. Every possible node -> node relationship is represented. The space complexity is also . Even on recent GPUs, they allow handling of fairly small graphs. In the above code, we initialize a vector and push elements into it using the … 3. ∑deg(v)=2|E| . Given a graph, to build the adjacency matrix, we need to create a square matrix and fill its values with 0 and 1. Size of array is |V| (|V| is the number of nodes). The space required by the adjacency matrix representation is O(V 2), so adjacency matrices can waste a lot of space if the number of edges |E| is O(V).Such graphs are said to be sparse.For example, graphs in which in-degree or out-degree are bounded by a constant are sparse. For example, for sorting obviously the bigger, If its not idiotic can you please explain, https://stackoverflow.com/questions/33499276/space-complexity-of-adjacency-list-representation-of-graph/61200377#61200377, Space complexity of Adjacency List representation of Graph. In contrast, using any index will have complexity O(n log n). Adjacency matrices are a good choice when the graph is dense since we need O(V2) space anyway. Figure 1 and 2 show the adjace… If the number of edges are increased, then the required space will also be increased. But I think I need some more reading to wrap my head around your explanation :), @CodeYogi, yes, but before jumping to the worst case, you need to assume which variables you study the dependence on and which you completely fix. The complexity of Adjacency List representation This representation takes O (V+2E) for undirected graph, and O (V+E) for directed graph. But if the graph is undirected, then the total number of items in these adjacency lists will be 2|E| because for any edge (i, j), i will appear in adjacency list j and vice-versa. This can be done in O(1)time. What is the space exact space (in Bytes) needed for each of these representations: Adjacency List, Adjacency Matrix. Input: Output: Algorithm add_edge(adj_list, u, v) Input − The u and v of an edge {u,v}, and the adjacency list Adjacency matrices require significantly more space (O (v 2)) than an adjacency list would. 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