Narrow Results

In this paper a new data structure named M-heaps is proposed. This data structure is a modification of the well known binary heap data structure. The new structure supports insertion in constant time and deletion in O(log n) time. Finally a generalization of the data structure to d – ary M-heaps is presented. This structure has similar time-bounds for insertion and deletion.

A language $L$ is said to be dense if every word in the universe is an infix of some word in $L$. This notion has been generalized from the infix operation to arbitrary word operations $\varrho$ in place of the infix operation ($\varrho$-dense, with infix-dense being the standard notion of dense). It is shown here that it is decidable, for a language $L$ accepted by a one-way nondeterministic reversal-bounded pushdown automaton, whether $L$ is infix-dense. However, it becomes undecidable for both deterministic pushdown automata (with no reversal-bound), and for nondeterministic one-counter automata. When examining suffix-density, it is undecidable for more restricted families such as deterministic one-counter automata that make three reversals on the counter, but it is decidable with less reversals. Other decidability results are also presented on dense languages, and contrasted with a marked version called $\varrho$-marked-density. Also, new languages are demonstrated to be outside various deterministic language families after applying different deletion operations from smaller families. Lastly, bounded-dense languages are defined and examined.

We show here that on a red-black tree with only two pointers — left pointing to the left child and right pointing to the right child — for each node, some standard operations such as searching, deletion, and insertion can be performed in O(log n) time, in the worst case, using only O(1) auxiliary space. No procedures do these fundamental operations more efficiently in time and space. The time and the space bounds of our algorithms are both optimal. The result is achieved by applying the pointer altering technique on red-black trees. The usefulness of the technique is further illustrated in this paper. Efficient implementations of the algorithms are also addressed here.

2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any .

Duchenne and Becker muscular dystrophies are allelic X-linked disorders resulting from defects in the gene coding for the dystrophin muscle protein. The dystrophin gene is more than 2300kb in size and consists of 79 exons. This large size and complexity presents a challenge to direct identification of point mutations and small deletions that cannot be identified by multiplex deletion testing or Southern blotting. One approach to this problem is to analyse the expression of ectopic dystrophin mRNA transcripts. Although the dystrophin gene transcript is distributed only over approximately 0.1% of the genome, analysis of such ectopic lymphocyte dystrophin transcripts can shed light on the pathogenic events at the transcriptional level.