of Results 1. We define a graph problem which we refer to as the c o m p o n e n t merging problem. Versions of the problem appear as bottlenecks in various graph algorithms. We show how to solve an important special cilye of the problem in time t ( m , n ) = O(m logloglog, n), where the graph hasn vertices, and m edges, and d = m a x ( m / n , 2 ) . The solut,ion makes use of a special data structure. The performance of the data structure is sped up by introducing a new algorithmic tool called packets.2. An immediate application of the first result is an O(t(rn, n ) ) algorithm for finding minimum spanning trees in undirected graphs without using F-heaps. This time bound is inferior to Fredman and Tarjan's O(mp(rn, n)) algorithm [7] which uses F-heaps, where p(rn,n) = min(i1 log(') n 5 m / n } . However, by adding packets to their algorithm we derive an O(m log(/3(m, n ) ) ) algorithm.3. Using the first result, we derive an O(n(t(m,n) + n log n)) algorithm for finding maximum weighted matching in general graphs. This settles an open problem posed by Tarjan [16, p.123], where the weaker bound of O ( n m log(n2/rn)) was conjectured.
4.Using the first result, we derive an O(t(rn, n) + n log n) algorithm for finding optimum branching in directed graphs. This answers affirmatively a question by Tarjan 115, p.341 (whether an o(m logn) algorithm exists). ~
We describe algorithms and data structures for maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We give a fully dynamic planarity testing algorithm that maintains a graph subject to edge insertions and deletions and that allows queries that test whether the graph is currently planar, or whether a potential new edge would violate planarity, in O(n 1Â2 ) amortized time per update or query. We give fully dynamic algorithms for maintaining the connected components, the best swap and the minimum spanning forest of a planar graph in O(log n) worst-case time per insertion and O(log 2 n) per deletion. Finally, we give fully dynamic algorithms for maintaining the 2-edge-connected components of a planar graph in O(log n) amortized time per insertion and O(log 2 n) per deletion. All of the data structures, except for the one that answers planarity queries, handle only insertions that keep the graph planar. All our algorithms improve previous bounds. The improvements are based upon a new type of sparsification combined with several properties of separators in planar graphs. ]
We provide an efficient algorithm for computing the candidate keys of a relational database schema. The algorithm exploits the 'arrangement' of attributes in the functional dependencies to determine which attributes are essential and useful for determining the keys and which attributes should not be considered. A more generalized algorithm using attribute graphs is then provided which allows a uniform and simplified solution to find all possible keys of a relational database schema when the attribute graph of Functional Dependencies (FDs) is not strongly connected. .
Minimality.No proper subset of K should have the above property.K is considered a superkey if it satisfies the uniqueness but not the minimality property. Those attributes of R that participate in a key are called prime attributes. If a relation has more than one key, each key is referred to as a candidate key of R. Functional dependencies (FDs) represent the interrelationship among attributes of a relation. A functional dependency is denoted by X
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.