Showing papers on "Interval tree published in 1979"
TL;DR: An algorithm is presented which finds a p-median of a tree (for $p > 1$) in time $O(n^2 \cdot p^2 )$.
Abstract: It is shown that the problem of finding a p-median of a network is an $NP$-hard problem even when the network has a simple structure (e.g., planar graph of maximum vertex degree 3). However, results leading to efficient algorithms are presented when the network is a tree: In particular, we first show that a 1-median of a tree is identical to its w-centroid, and obtain Goldman’s $O(n)$ algorithm for finding a 1-median of a tree out of more general considerations. Then, we present an algorithm which finds a p-median of a tree (for $p > 1$) in time $O(n^2 \cdot p^2 )$.
1,333 citations
TL;DR: The purposes of this paper are to cast k-D trees in a database framework, to collect the results on k-d trees that have appeared since the structure was introduced, and to show how the basic data structure can be modified to facilitate implementation in large (and very large) databases.
Abstract: The multidimensional binary search tree (abbreviated k-d tree) is a data structure for storing multikey records. This structure has been used to solve a number of "geometric" problems in statistics and data analysis. The purposes of this paper are to cast k-d trees in a database framework, to collect the results on k-d trees that have appeared since the structure was introduced, and to show how the basic data structure can be modified to facilitate implementation in large (and very large) databases.
463 citations
16 Jul 1979
TL;DR: This paper derives an explicit expression and an asymptotic equivalent for the sth moment about origin of this distribution and compute the average stack size after t units of time during postorder-traversing of a binary tree with n leaves.
Abstract: The height of a tree with n nodes, that is the number of nodes on a maximal simple path starting at the root, is of interest in computing because it represents the maximum size of the stack used in algorithms that traverse the tree. In the classical paper of de Bruijn, Knuth and Rice, there is computed the average height of planted plane trees with n nodes assuming that all n-node trees are equally likely. The first section of this paper is devoted to the computation of the cumulative distribution function of this problem; we give an asymptotic equivalent in terms of familiar functions (Theorem 1). Then we derive an explicit expression and an asymptotic equivalent for the sth moment about origin of this distribution (Theorem 2). In the last section we compute the average stack size after t units of time during postorder-traversing of a binary tree with n leaves. Thereby, in one unit of time, a node is stored in the stack or is removed from the top of the stack.
16 citations
TL;DR: An algorithm which optimizes a weighted binary tree after an insertion or deletion is presented, which is nearly optimal and needs O(n) space.
Abstract: We present an algorithm which optimizes a weighted binary tree after an insertion or deletion. The resulting tree is nearly optimal. The algorithm needs O(n) space. In the case of an insertion the expected number of operations is equal to or less than the height of the tree. All results presented in this paper can also be found in [15].
15 citations
01 May 1979
TL;DR: Algorithms are presented for finding the area, centroid, union, intersection, and complement of binary images, all of which are linear in the number of nodes in the tree.
Abstract: : This paper describes algorithms for computing geometric properties of binary images represented as quadtrees. All the algorithms involve a simple traversal of the tree. Each algorithm, however, performs different operations at the nodes of the tree. Algorithms are presented for finding the area, centroid, union, intersection, and complement of binary images. All the algorithms are linear in the number(s) of nodes in the tree(s).
14 citations
TL;DR: A threshold (binary) image can be represented as a region tree in which each node corresponds to a component of 1's (object) or 0's (background) if regions O and B share a border, then one encloses the other.
14 citations
06 Nov 1979
TL;DR: The problem of performing multiple attribute clustering in a dynamic database is studied and the extended K-d tree method is presented, which uses the basic k-D tree structure after modification as the structure of the directory which organizes the data records in the secondary storage.
Abstract: The problem of performing multiple attribute clustering in a dynamic database is studied. The extended K-d tree method is presented. in an extended K-d tree organization, the basic k-d tree structure after modification is used as the struc ture of the directory which organizes the data records in the secondary storage. The discrimina tor value of each level of the directory determines the partitioning direction of the corresponding attribute subspace. When the record insertion causes the data page to overload, the attribute space will be further partitioned along the direction specified by the corresponding discriminator. For a given query, the number of disk accesses involved is estimated. The design of the discriminator function are then described.
9 citations
TL;DR: A modified max-entropy rule is proposed for constructing nearly optimum binary search tree in the case of ordered keys with given probabilities and the average cost of the trees obtained is shown to be bounded by the entropy of the probability distribution plus a constant not larger than one.
Abstract: A modified max-entropy rule is proposed for constructing nearly optimum binary search tree in the case of ordered keys with given probabilities. The average cost of the trees obtained by this rule is shown to be bounded by the entropy of the probability distribution plus a constant not larger than one. An algorithm for implementing this rule is then suggested and its complexity is investigated in a probabilistic setting.
8 citations
01 May 1979
7 citations
06 Nov 1979
TL;DR: A general and computationally inexpensive algorithmic scheme that falls in a region detection category using a minimal spanning tree as a path of a sequential region grower that traverses the spatially adjacent graph while maintaining the structural organization of an image.
Abstract: This paper presents a general and computationally inexpensive algorithmic scheme that falls in a region detection category. In the scheme, a minimal spanning tree is used as a path of a sequential region grower. The algorithm traverses the spatially adjacent graph while maintaining the structural organization of an image. The graph-theoretical evaluation of heuristics is described and examples of implementation are given.
5 citations