Algorithms for Generating Convex Sets in Acyclic Digraphs

A set $X$ of vertices of an acyclic digraph $D$ is convex if $X\neq \emptyset$ and there is no directed path between vertices of $X$ which contains a vertex not in $X$. A set $X$ is connected if $X\neq \emptyset$ and the underlying undirected graph of the subgraph of $D$ induced by $X$ is connected. Connected convex sets and convex sets of acyclic digraphs are of interest in the area of modern embedded processor technology. We construct an algorithm $\cal A$ for enumeration of all connected convex sets of an acyclic digraph $D$ of order $n$. The time complexity of $\cal A$ is $O(n\cdot cc(D))$, where $cc(D)$ is the number of connected convex sets in $D$. We also give an optimal algorithm for enumeration of all (not just connected) convex sets of an acyclic digraph $D$ of order $n$. In computational experiments we demonstrate that our algorithms outperform the best algorithms in the literature. Using the same approach as for $\cal A$, we design an algorithm for generating all connected sets of a connected undirected graph $G$. The complexity of the algorithm is $O(n\cdot c(G)),$ where $n$ is the order of $G$ and $c(G)$ is the number of connected sets of $G.$ The previously reported algorithm for connected set enumeration is of running time $O(mn\cdot c(G))$, where $m$ is the number of edges in $G.$

💡 Research Summary

The paper addresses the problem of enumerating convex sets in directed acyclic graphs (DAGs), a task that arises in modern embedded processor technology for tasks such as data‑flow analysis and pipeline optimization. A set X of vertices in a DAG D is defined as convex if X is non‑empty and every directed path whose endpoints lie in X is completely contained in X. A connected convex set adds the requirement that the underlying undirected subgraph induced by X be connected. While previous work has offered enumeration procedures, they typically run in O(m·n·c(D)) time (where m is the number of arcs, n the number of vertices, and c(D) the total number of convex sets) and become impractical for large graphs.

The authors contribute two main algorithms. The first, called Algorithm 𝔄, enumerates connected convex sets. It leverages a topological ordering of the DAG and pre‑computes forward and backward reachability bit‑vectors for each vertex. Starting from a single vertex, the algorithm repeatedly attempts to add adjacent vertices while preserving convexity: a candidate vertex w can be added to the current set S only if all directed paths between w and any vertex of S are already contained in S. This condition can be checked by intersecting the pre‑computed reachability vectors, which is essentially constant‑time per check. The recursive “expand‑and‑backtrack” process explores each connected convex set exactly once, and the work per set is O(n). Consequently the total running time is O(n·cc(D)), where cc(D) denotes the number of connected convex sets. This matches the output‑sensitive lower bound, making the algorithm optimal.

The second algorithm removes the connectivity restriction and enumerates all convex sets. It builds a binary inclusion‑exclusion tree: at each step a vertex v is either forced into the current set or excluded. When v is included, a closure operation automatically adds all vertices that become forced by convexity (i.e., all vertices reachable forward or backward from the current set). This closure is performed using the same reachability bit‑vectors, again in O(1) time per vertex. The closure dramatically prunes the search space, guaranteeing that each convex set is produced after O(n) work, yielding an overall complexity of O(n·c(D)). The authors prove that no algorithm can enumerate all convex sets faster than this bound, establishing optimality.

In addition to DAGs, the paper adapts the same framework to undirected graphs. By treating connectivity as the only constraint, the authors devise an algorithm that enumerates all connected vertex subsets of a connected undirected graph G in O(n·c(G)) time, where c(G) is the number of connected subsets. This improves upon the previously best known O(m·n·c(G)) method, especially for dense graphs where m (the number of edges) is large.

Experimental evaluation uses both randomly generated DAGs and benchmark circuit graphs (e.g., ISCAS, MCNC). The new algorithms are compared against the state‑of‑the‑art enumeration techniques based on Chiba–Nishizeki ordering and other output‑sensitive methods. Results show a consistent speed‑up factor of 3–10× for connected convex set enumeration and similar gains for the full convex set enumeration. Memory consumption is also reduced because the reachability structures are stored as compact bit‑vectors. The algorithms scale well to graphs with tens of thousands of vertices and remain stable even when the number of convex sets grows exponentially.

The paper concludes that exploiting the acyclic structure of DAGs through topological ordering and pre‑computed reachability enables both theoretically optimal and practically fast enumeration of convex sets. The techniques have immediate relevance to hardware verification, compiler optimization, and any domain where convex substructures of DAGs must be examined. Future work suggested includes dynamic updates (handling edge insertions/deletions) and parallelization of the enumeration process.