Khái niệm cốt lõi
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.
Tóm tắt
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.
Thống kê
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).
Trích dẫn
"The delay of these algorithms is O(k²Δ)."
"The delay of the proposed algorithm is O(kΔ)."