Consider a graph G with n nodes and m edges, where n m (i.e. G is a sparse graph).

Assume we want to be able to quickly find all the neighbors of a given node in G. Which of the following representations for this graph would support this operation in the most computationally efficient way?

Question

Consider a graph G with n nodes and m edges, where n >> m (i.e. G is a sparse graph).

Assume we want to be able to quickly find all the neighbors of a given node in G. Which of the following representations for this graph would support this operation in the most computationally efficient way?

BlackTom AI · Accepted Answer

The problem asks which graph representation allows quickly finding all neighbors of a given node in a sparse graph where n >> m.
First, I note that the provided data indicates the correct answer is adjacency list, but the actual list of answer options is missing from the input, so I cannot comment on other choices or map them to option labels.
In general, adjacency list representations store for each node the direct neighbors as a list or set. This means that retrieving all neighbors of a node v requires time proportional to its degree deg(v), plus the overhead of accessing the list for v. In sparse graphs, where the average degree is low, this is highly efficient because you only traverse the small list of neighbors rather than inspecting many non-neighbor candidates.
By contrast, an adjacency matrix would store a 0/1 entry for every possible edge, taking O(n^2) space and requiring O(n) time to scan a row to obtain all neighbors, which is inefficient for large n and sparse graphs, since you would inspect many non-edges.
Other representations not shown here might offer various trade-offs, but for the operation of listing all neighbors quickly in a sparse graph, the adjacency list is the standard optimal choice due to its linear-in-degree access pattern and space efficiency.
If there were multiple options, we would evaluate each by the time complexity of listing neighbors, the space requirements, and how well they scale with sparse connectivity; however, without the other options provided, we can still justify adjacency lists as the preferred representation for fast neighbor enumeration in sparse graphs.

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