Comparing and implementing adjacency lists and adjacency matrices.
Choosing the right graph representation is a critical first step in solving any graph problem, as it significantly impacts the performance and space requirements of the algorithm. The two most common representations are the adjacency matrix and the adjacency list. An adjacency matrix is an `V x V` matrix (where V is the number of vertices), where `matrix[i][j] = 1` indicates an edge between vertex `i` and `j`. This representation offers fast O(1) time complexity to check for an edge between two given vertices. However, it requires O(V^2) space, which can be prohibitive for sparse graphs (graphs with few edges). The adjacency list is often more efficient for sparse graphs. It consists of an array of linked lists, where the list at index `i` stores all vertices adjacent to vertex `i`. The space complexity is O(V + E), where E is the number of edges, which is much better for sparse graphs. Checking for an edge between two vertices takes O(degree(i)) time, which can be slower than the matrix representation. The choice between them depends on the graph's density and the operations you need to perform. For dense graphs, a matrix might be better, while for the sparse graphs common in real-world problems, an adjacency list is typically preferred.