2–3 tree

Abstract: We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. L ...

Abstract: Consider an ordered, static tree /spl Tscr/ on t nodes where each node has a label from alphabet set /spl Sigma/. Tree /spl Tscr/ may be of arbitrary degree and of arbitrary shape. Say, we wish to support basic navigational operations such as find the parent of node u, the ith child of u, and any child of it with label /spl alpha/. In a seminal work over fifteen years ago, Jacobson (1989) observed that pointer-based tree representations are wasteful in space and introduced the notion of succinct data structures. He studied the special case of unlabeled trees and presented a succinct data structure of 2t + o(t) bits supporting navigational operations in O(1) time. The space used is asymptotically optimal with the information-theoretic lower bound averaged over all trees. This led to a slew of results on succinct data structures for arrays, trees, strings and multisets. Still, for the fundamental problem of structuring labeled trees succinctly, few results, if any, exist even though labeled trees arise frequently in practice, e.g. in the data as in markup text (XML) or in augmented data structures. We present a novel approach to the problem of succinct manipulation of labeled trees by designing what we call the xbw transform of the tree, in the spirit of the well-known Burrows-Wheeler transform for strings. The xbw transform uses path-sorting and grouping to linearize the labeled tree /spl Tscr/ into two coordinated arrays, one capturing the structure and the other the labels. Using the properties of the xbw transform, we (i) derive the first-known (near-)optimal results for succinct representation of labeled trees with O(1) time for navigation operations, (ii) optimally support the powerful subpath search operation for the first time, and (iii) introduce a notion of tree entropy and present linear time algorithms for compressing a given labeled tree up to its entropy beyond the information-theoretic lower bound averaged over all tree inputs. Our xbw transform is simple and likely to spur new results in the theory of tree compression and indexing, and may have some practical impact in XML data processing.

Abstract: Several papers describe linear time algorithms to preprocess a tree, such that one can answer subsequent nearest common ancestor queries in constant time. Here, we survey these algorithms and related results. A common idea used by all the algorithms for the problem is that a solution for complete binary trees is straightforward. Furthermore, for complete binary trees we can easily solve the problem in a distributed way by labeling the nodes of the tree such that from the labels of two nodes alone one can compute the label of their nearest common ancestor. Whether it is possible to distribute the data structure into short labels associated with the nodes is important for several applications such as routing. Therefore, related labeling problems have received a lot of attention recently.Previous optimal algorithms for nearest common ancestor queries work using some mapping from a general tree to a complete binary tree. However, it is not clear how to distribute the data structures obtained using these mappings. We conclude our survey with a new simple algorithm that labels the nodes of a rooted tree such that from the labels of two nodes alone one can compute in constant time the label of their nearest common ancestor. The labels assigned by our algorithm are of size $O(\log n)$ bits where $n$ is the number of nodes in the tree. The algorithm runs in $O(n)$ time.

Abstract: A method, apparatus, and system is described for creating a spatial sieve tree which stores, manages, and manipulates multidimensional data by partitioning the bounds of the nodes of the tree, creating child nodes which each have defined bounds associated with a partitioned portion of their parent node(s) and may be further partitioned into additional levels of child nodes, and determining which level of the tree has the smallest size node in which a data object could wholly fit regardless of the data object's location in coordinate space and the one or more nodes of that determined level that could at least partially contain the data object based on the bounds of the one or more nodes and the data object's location in coordinate space.