Information and Computation
A linear time algorithm for triconnectivity augmentation (extended abstract)
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Finding a smallest augmentation to biconnect a graph
SIAM Journal on Computing
A minimum 3-connectivity augmentation of a graph
Journal of Computer and System Sciences
Graph augmentation and related problems: theory and practice
Graph augmentation and related problems: theory and practice
On the optimal vertex-connectivity augmentation
Journal of Combinatorial Theory Series B
On four-connecting a triconnected graph
Journal of Algorithms
Concurrent threads and optimal parallel minimum spanning trees algorithm
Journal of the ACM (JACM)
Undirected Vertex-Connectivity Structure and Smallest Four-Vertex-Connectivity Augmentation
ISAAC '95 Proceedings of the 6th International Symposium on Algorithms and Computation
Two-Vertex Connectivity Augmentations for Graphs with a Partition Constraint (Extended Abstract)
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
The bridge-connectivity augmentation problem with a partition constraint
Theoretical Computer Science
A novel data structure for biconnectivity, triconnectivity, and k-tree augmentation
CATS '11 Proceedings of the Seventeenth Computing: The Australasian Theory Symposium - Volume 119
A novel data structure for biconnectivity, triconnectivity, and k-tree augmentation
CATS 2011 Proceedings of the Seventeenth Computing on The Australasian Theory Symposium - Volume 119
Optimal design and augmentation of strongly attack-tolerant two-hop clusters in directed networks
Journal of Combinatorial Optimization
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This paper presents a new and simple technique to solve the problem of adding a minimum number of edges to an undirected graph in order to obtain a biconnected, i.e., 2-vertex-connected, resulting graph. Our technique results in a simpler algorithm, which runs in sequential linear time, that is also faster in parallel than the previous result.Previous approaches for the problem require the usage of sorting, and advanced data structures to dynamically maintain either (1) a rooted tree when vertices are collapsing, or (2) the largest two sets among a collection of sets when an element from each of the largest two sets is being deleted.Our algorithm only needs to find a maximum integer among a set of O(n) non-negative integers that are less than n and to compute various simple tree functions, e.g., the number of vertices and a consecutive numbering of the degree-1 vertices in a rooted subtree, on a rooted tree. No sorting routine and dynamic data structure is used in the algorithm. Our simple algorithm implies a linear-time sequential implementation. For parallel implementation, all but the step for finding connected components in our algorithm can be done optimally in O (log n) time on an EREW PRAM, where n is the number of vertices in the input graph. Hence our parallel implementation runs in either O (log n) time using O((n + m) . α(m, n)/log n) processors on a CRCW PRAM, or O(log n) time using O(n + m) processors on an EREW PRAM, where m is the number of edges in the input graph and α is the inverse Ackerman function. The previous best parallel algorithm for solving this problem runs in O(log2n) time using O(n + m) processors on an EREW PRAM.