Consider a graph G with 4 vertices and 5 edges.For each of the three graph representations in C from the lecture (array of edges, adjacency matrix, adjacency list), calculate the storage costs for G under the same assumptions as in Exercise 3b. For the array-of-edges representation, assume that memory for the maximum possible number of edges, i.e. V·(V-1)/2, is allocated.How do the representations compare?

Question

BlackTom AI · Accepted Answer

We need to compare storage costs for three representations of a graph with V = 4 vertices and E = 5 edges, under the same assumptions as in the referenced exercise. The prompt also specifies that for the array-of-edges representation, memory is allocated for the maximum possible number of edges, which is V*(V-1)/2 = 4*3/2 = 6. We will reason about typical storage cost models for each representation and then evaluate the given answer choices.

First, recall the rough storage cost models commonly used in graph representations:
- Array of edges (edge list) with allocated capacity for all possible edges: storage scales with the allocated capacity, which in this case is 6 edge slots. So |array-of-edges| ≈ 6 units (for the purpose of comparison, counting each potential edge slot as one unit).
- Adjacency matrix: an adjacency matrix for an undirected graph with V = 4 uses a 4x4 matrix, which has 16 entries. Even if many entries are zero (no edge), the storage cost is proportional to V^2, so |adjacency-matrix| ≈ 16 units.
- Adjacency list: storage includes a list for each vertex, containing the adjacent vertices. In a straightforward accounting, you store each edge twice (once for each endpoint) in the lists, so there are 2E adjacency entries. Here 2E = 10. There may also be per-vertex head pointers or similar overhead, but the dominant term is the 2E entries. Thus, |adjacency-list| ≈ 10 units (plus small constant overhead).

With these rough counts in mind, compare the three: 6 (array of edges) vs. 10 (adjacency list) vs. 16 (adjacency matrix).
- If we rank strictly by these totals, we get: array-of-edges < adjacency-list < adjacency-matrix.
- Translating this into the provided answer templates, we would expect the correct ordering to reflect that the array-of-edges is the smallest, the adjacency list is intermediate, and the adjacency matrix is the largest.

Now, evaluate each option in light of the above reasoning:
- Option A: "|adjacency-matrix| < |adjacency list| < |array-of-edges|" claims the matrix is the smallest, then the list, then the array. This contradicts the counts above (matrix is largest, not smallest), so this option is incorrect.
- Option B: "|adjacency-matrix| = |adjacency-list| < |array-of-edges|" asserts matrix and list are equal and both smaller than the array. Our counts do not support equality; matrix is larger than list, and both are larger than the array. This option is incorrect.
- Option C: "|adjacency-matrix| < |array-of-edges| < |adjacency-list|" asserts matrix is smallest, array is middle, and list is largest. This contradicts the typical ordering (matrix should be largest, not smallest). So this is incorrect.
- Option D: "|adjacency-matrix| < |array-of-edges| = |adjacency-list|" asserts matrix is smallest and array equals list and both are larger than matrix. Again, this does not align with the expected sizes (matrix is not smallest, and array and list are not necessarily equal here). This is incorrect.
- Option E: "|array-of-edges| = |adjacency-matrix| < |adjacency-list|" asserts the array and matrix are equal in size and both smaller than the list. Our counts show the matrix is much larger than the array, so this is incorrect.

In summary, under the standard interpretation of storage costs for these representations and the given allocation for the array-of-edges, none of the provided options match the expected ordering (array-of-edges smallest, adjacency-list intermediate, adjacency-matrix largest). The closest qualitative description is that the matrix requires the most space, the array-of-edges is the least, and the adjacency list sits in between, but no option exactly captures that exact ordering.

If you’re allowed to, you might raise this as a potential mismatch between the computed ordering and the answer choices, or double-check the exact cost model used in Exercise 3b (for example, whether the adjacency list cost includes only actual edges or also per-vertex overhead, and whether the edge list’s allocation is counted per-slot rather than per-element).

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