A graph is a set of vertices connected by edges. The two standard representations are the adjacency list (each vertex stores its neighbors) and the adjacency matrix (a V×V grid of booleans). The choice depends on graph density.
A graph is a set of vertices connected by edges. The two standard representations are the adjacency list (each vertex stores its neighbors) and the adjacency matrix (a V×V grid of booleans). The choice depends on graph density.
Graph: 0 - 1
| |
2 - 3
Adjacency list: Adjacency matrix:
0: [1, 2] 0 1 2 3
1: [0, 3] 0 [0 1 1 0]
2: [0, 3] 1 [1 0 0 1]
3: [1, 2] 2 [1 0 0 1]
3 [0 1 1 0]
# Adjacency list (dict of lists) — preferred for sparse graphs
adj = {0: [1, 2], 1: [0, 3], 2: [0, 3], 3: [1, 2]}
neighbors = adj[1] # O(1) to get a vertex's neighbors
# Adjacency matrix
matrix = [[0]*4 for _ in range(4)]
matrix[0][1] = matrix[1][0] = 1
has_edge = matrix[0][1] == 1 # O(1) edge lookup
| Adjacency list | Adjacency matrix | |
|---|---|---|
| Space | O(V + E) | O(V²) |
| Edge exists? | O(degree) | O(1) |
| Iterate neighbors | O(degree) | O(V) |
| Best for | sparse graphs | dense graphs |
Most real-world graphs (social networks, road maps, dependency graphs) are sparse, so adjacency lists save enormous space and speed up traversals like BFS/DFS.
Knowing the trade-off lets you pick the representation that keeps your graph algorithms efficient instead of accidentally using O(V²) memory.
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