Some new classes of balanced trees, defined by very simple balance criteria, are introduced. Those trees can be maintained by partial rebuilding at lower update cost than previously used weight-balanced trees. The used balance criteria also allow us to maintain a balanced tree without any balance information stored in the nodes.

We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information stored in the nodes. Thus, in the case when amortized bounds are sufficient, there is no need for sophisticated balance criteria. The maintenance algorithms use partial rebuilding. This is important for certain applications and has previously been used with weight-balanced trees. We show that the amortized cost incurred by general balanced trees is lower than what has been shown for weight-balanced trees. � 1999 Academic Press 1.

Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

Two chapters of this thesis analyze expert consulting problems via game theoretic models; the first points out a close connection between the problem of consulting a set of experts and the problem of searching. The last chapter presents a solution to the dictionary problem of supporting and update (Insert and Delete) operations on a set of key values. The first chapter shows...

The hierarchical edge bundle method clusters the graph edges to better understand and analyze graphs, but its effectiveness relies critically on the quality of the hierarchical organization of its nodes and edges. This paper proposes a novel graph visualization approach that extracts the community structure of a network and organizes it into a more balanced and meaningful hierarchy so that its edge bundle rendering better indicates its structure. Results on several data sets demonstrate that this approach clarifies realworld communication, collaboration and competition network structure and reveals information missed in previous visualizations.

Till current date in majority books on algorithm and research papers, they talk about height of a binary tree in terms like height balanced binary tree. In this paper the notion of width of a binary tree has been introduced and later the recursive algorithm based on the traversal techniques of the binary tree is given. Later the iterative version of algorithm using the notion of stack is introduced. The width of a binary tree is defined based on the number of nodes at every level. The highest of all is the width of a binary tree. The same concept can be applied to the general tree.

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