Kernkonzepte
The proposed algorithm enumerates all connected induced subgraphs of size k in a graph with a delay of O(kΔ), where k is the subgraph size and Δ is the maximum degree of the graph.
Zusammenfassung
The paper presents a new algorithm for enumerating all connected induced subgraphs of size k in an undirected graph. The key highlights are:
- The proposed algorithm has a delay of O(kΔ), which improves upon the current best delay bound of O(k²Δ) in the literature.
- The algorithm works by visiting the vertices in reverse order of a depth-first search and expanding the subgraphs by adding neighboring vertices in a depth-first manner. Visited vertices are marked as closed to avoid redundant enumeration.
- The algorithm uses global data structures like arrays and boolean flags to efficiently track the visited vertices and enable fast lookups during the enumeration process.
- The correctness of the algorithm is proven, showing that each connected induced subgraph of size k is enumerated exactly once.
- The space complexity of the algorithm is shown to be O(|V| + |E|), where |V| is the number of vertices and |E| is the number of edges in the input graph.
Statistiken
The number of connected induced subgraphs of size k in a graph G with n vertices and maximum degree Δ is upper bounded by n * (eΔ)^k / (Δ-1)^k.
The delay of the proposed algorithm is O(kΔ).
Zitate
"The delay of these algorithms is O(k²Δ)."
"The delay of the proposed algorithm is O(kΔ)."