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Academic Commons Search Resultsen-usOnline Maintenance of Minimal Directed Hypergraphs
https://academiccommons.columbia.edu/catalog/ac:142917
Italiano, Giuseppe F.; Nanni, Umberto10.7916/D8F196T2Mon, 19 Jun 2017 14:40:27 +0000In this paper we deal with directed hypergraphs, a generalization of directed graphs previously introduced in the literature. In particular, we investigate the problem of maintaining efficiently information about minimal hyperpaths while new hyperarcs are inserted. We consider several definitions of minimal hyperpath and we prove that accordingly to some of such definitions the problem of finding the minimal hyperpath is NP-complete, while hyperpaths with minimal GAP and minimal RANK can be found in polynomial time. We deal with this problem in an online fashion, by allowing insertions of hyperarcs in the hypergraphs. We present data structures and algorithms which allow to return a hyperpath with minimal GAP or RANK between an arbitrarily given pair of nodes in a time which is linear in its size. The total time required to maintain the data structure during the insertion of new hyperarcs is (9(m n2) for min-GAP and (9(m n2 log n) for min-RANK (where m is the total size of the description of the hyperarcs and n is the number of nodes). These results are useful in applications where directed hypergraphs are known to be a suitable model (e.g. transition networks, rewriting systems, database schemes, logic programming and problem solving).Computer scienceComputer ScienceReportsDynamic Data Structures for Series Parallel Digraphs
https://academiccommons.columbia.edu/catalog/ac:142873
Italiano, Giuseppe F.; Spaccamela, Alberto Marchetti; Nanni, Umberto10.7916/D8C82JCHMon, 19 Jun 2017 14:40:14 +0000We consider the problem of dynamically maintaining general series parallel directed acyclic graphs (GSP dags), two-terminal series parallel directed acyclic graphs (TTSP dags) and looped series parallel directed graphs (looped SP digraphs). We present data structures for updating (by both inserting and deleting either a group of edges or vertices) GSP dags, TTSP clags and looped SP digraphs of m edges and n vertices in O( log n) worst-case time. The time required to check whether there is a path between two given vertices is O(log n), while a path of length k can be traced out in O(k + log n) time. For GSP and TTSP dags, our data structures are able to report a regular expression describing all the paths between two vertices x and y in O(h + log n), where h ≤ n is the total number of vertices which are contained in paths from x to y. Although GSP dags can have as many as O(n2) edges, we use an implicit representation which requires only O(n) space. Motivations for studying dynamic graphs arise in several areas, such as communication networks, Incremental compilation environments and the design of very high level languages, while the dynamic maintenance of series parallel graphs is also relevant in reducible flow diagrams.Computer scienceComputer ScienceReports