Showing papers on "Binary tree published in 1979"
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
TL;DR: It is shown that Hu–Tucker type algorithms can be used to find trees, whose leaves preserve a given order, that minimizing certain sums of functions of path length, and also that minimize certain maximum functions of paths length.
Abstract: In this paper we construct binary trees optimal under various criteria. In particular, we show that Hu–Tucker type algorithms can be used to find trees, whose leaves preserve a given order, that minimize certain sums of functions of path length, and also that minimize certain maximum functions of path length. The arguments are based on a new proof of the original Hu–Tucker algorithm.
81 citations
TL;DR: This work introduces D-trees with the following properties: update time after a search is at most proportional to search time, i.e. the overhead for administration is small.
Abstract: We consider search trees under time-varying access probabilities. Let $S = \{ B_1 , \cdots ,B_n \} $ and let $p_i^t $ be the number of accesses to object $B_i $ up to time t, $W^t = \sum {p_i^t } $. We introduce D-trees with the following properties.1) A search for $X = B_i $ at time t takes time $O(\log W^t /p_i^t )$. This is nearly optimal.2) Update time after a search is at most proportional to search time, i.e. the overhead for administration is small.
64 citations
TL;DR: The result that this number is equal to the number of the binary trees with n leaves, using for traversal a maximum size of stack less than or equal to 2k+1−1 is obtained.
Abstract: In this paper we determine the number of binary trees with n leaves which can be evaluated optimally with less than or equal to k registers. Furthermore we obtain the result that this number is equal to the number of the binary trees with n leaves, using for traversal a maximum size of stack less than or equal to 2k+1?1. This fact is only a connection between the numbers of the trees and not between the sets of the trees. We compute also the average number ¯R(n) of registers needed to evaluate a binary tree optimally. We get for all ?>0: $$\bar R(n + 1) = 1d(\sqrt {n)} + C + F(n) + O(n^{ - 0.5 + \varepsilon } )$$ where C = 0.82574... is a constant and F(n) is a function with F(n) = F(4n) for all n>0 and ?0.574
61 citations
01 May 1979
TL;DR: The automatic synthesis of Boolean switching functions by adaptive tree networks is discussed and applications to pattern recognition and optical character recognition problems are described.
Abstract: The automatic synthesis of Boolean switching functions by adaptive tree networks is discussed. The concept of heuristic responsibility, by means of which parts of a tree become specialized to certain subsets of input vectors, is explained. Applications to pattern recognition and optical character recognition (OCR) problems are described.
58 citations
TL;DR: In this paper a fiie is considered to be a collection of records, each of which is an ordered k-tuple of values,Each component of a record is called a key.
18 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: This paper proves optimality properties for such trees by generalizing results of Huffman, Hu and Tucker, Glassey and Karp on t-ary trees and giving a criterion to compare the costs of two generalized Huffman trees with the same collection of outdegrees.
Abstract: We generalize the Huffman construction for t-ary trees to the case where the collection of outdegrees for every internal node is given but the outdegrees are not necessarily a constant. The output trees from this construction are called generalized Huffman trees. We prove optimality properties for such trees by generalizing results of Huffman, Hu and Tucker, Glassey and Karp on t-ary trees. We also give a criterion to compare the costs of two generalized Huffman trees with the same collection of outdegrees. This criterion, when applied to binary trees, considerably strengthens a result of Hu.
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.
TL;DR: In this paper, trees are constructed from ternary relations; the model represents each object of an empirical set by a node of a tree so that a betweenness relation among the nodes in the graph reflects a ternARY relation amongThe objects.
01 Jan 1979
TL;DR: In this article, it was shown that some trees have graceful valuations with additional properties which allow larger trees with graceful values to be constructed from them, and they investigated one such class of trees.
Abstract: Ringel conjectured that every tree has a graceful valuation. While this conjecture remains unsettled, it is apparent, from examples, that some trees have graceful valuations with additional properties which allow larger trees with graceful valuations to be constructed from them. We investigate here one such class of graceful trees.
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.
25 Jun 1979
TL;DR: It is proved that a minimum and complete fundamental representation for hierarchical ADT specification can be established and that the technique provides a straight-forward approach to the specification of arbitrarily complex ADTs (e.g. a relational data base system).
Abstract: We introduce a hierarchical specification technique for abstract data types (ADT) in which the data objects of an ADT are represented by other, simpler ADTs and in which the functional behavior of an ADT is specified in terms of an input/output specification employing applications of the functions of the representing ADTs. By introducing instances of abstract data structures as fundamental representations, the danger is avoided that such representations might imply particular implementations. We will prove that a minimum and complete fundamental representation for hierarchical ADT specification can be established. It will be demonstrated that our technique provides a straight-forward approach to the specification of arbitrarily complex ADTs (e.g. a relational data base system).
03 Oct 1979
TL;DR: This paper shows that the data semantics of queries shuld be described by designating sets of nodes from which v values for attnbutes may be returned to the data consumer, and shows that Bolts is an adequate formalism for conveying the semantics of tree structure processing.
Abstract: This paper defines and demonstrates four philosophies for processing queries on tree structures; shows that the data semantics of queries should be described by designating sets of nodes from which values for attributes may be returned to the data consumer; shows that the data semantics of database processing can be specified totally independent of any machine, file structure or implementation; shows that set theory is a natural and effective vehicle for analyzing the semantics of queries on tree structures; and finally, shows that BOLTS is an adequate formalism for conveying the semantics of tree structure processing.
TL;DR: The binary operation defines an algebraic structure on B that generates a set B of finite trees form a trivial tree (one node) and B contains for every finite tree G exactly one element isomorphic to G.
Abstract: A binary operation on the class of trees is defined that generates a set B of finite trees form a trivial tree (one node) and B contains for every finite tree G exactly one element isomorphic to G. The binary operation defines an algebraic structure on B, and as a consequence the finite tree types are characterized as an initial algebra in the same way as the natural numbers are characterized as an initial algebra by the Peano-Lawvere axiom [2]. Simple and primitive recursion are defined and some applications of the initial algebra characterization are given.
01 Jan 1979
TL;DR: Theorem 1 on identity of optimal solutions of models 1 and 2 would constitute a theoretical basis to establish recurrence formula and define separating number in this article, whereas Theorem 1 is not applicable in this paper.
Abstract: As a continuation of [3], this paper is concerned with L-restricted alphabetic-extended binary trees with cost function and transfer function. The main results obtained here are three theorems (Theorems 2, 3 and 4) on the separating number, whereas Theorem 1 on identity of optimal solutions of models 1 and 2 would constitute a theoretical basis to establish recurrence formula and define separating number.
03 Sep 1979
06 Nov 1979
TL;DR: A new algorithm, called INPROC, is presented, which uses a MARKER field of 2 bits and runs faster than algorithm SELECT, which used a MARKer field of 3 bits for each node in the tree.
Abstract: The process of visiting a mini mal number of nodes to retrieve data satisfying the range condition from a binary search tree is called "selective traversal". Driscall and Lien gave an algorithm SELECT for selective traversal which used a MARKER field of 3 bits for each node in the tree. In this paper we present a new algorithm, called INPROC, which uses a MARKER field of 2 bits and runs faster than algorithm SELECT.
01 Jan 1979
TL;DR: A promising method of rescheduling the event notices for significantly reducing the amount of memory space required is introduced in this thesis, which extends the range of applications of PASCAL to the field of simulation.
Abstract: The Use of PASCAL as a Discrete Simulation Language by Ming-Jang Chang PASCAL Is now widely accepted as a powerful computer programming language that can be efficiently implemented. This thesis extends the range of applications of PASCAL to the field of simulation. The simulations considered are discrete event simulations, that is, simulations in which events occur at discrete, randomly spaced points in time. The advantages of PASCAL for writing discrete event simulation programs include programmer-definable data structures, readability and superior error diagnosis. Two algorithms programmed in PASCAL are presented to maintain the event list, one is based on a doubly linked list which is a commonly used method in contemporary simulation programs and the other employs a specially designed binary tree. An advantage of the latter algorithm over the former is that as the number of event notices increases, the latter will from some point onward require less time than the former to execute. A noteworthy aspect of the tree algorithm is the use of recursion and the concept of a virtual root. This thesis also introduces a promising method of rescheduling the event notices for significantly reducing the amount of memory space required. To illustrate the adaptability of these algorithms, three practical simulation problems are discussed, modeled and programmed.
01 Jun 1979
TL;DR: Pedagogical techniques will be describe d that might be of use to the teacher : the first diagrams the recursive proces s as a tree (what else?) and the secon d allows a "quick and dirty" method for determining the actual order of th e nodes visited for each traversal b y walking around the tree in thre e different modes.
Abstract: In a Data Structures course of the typ e recommended in the ACM's Curriculum '6 8 report [1], algorithms for traversal o f binary trees are generally taught [2 , 3]. Sometimes, the recursiv e definitions of preorder, postorder, an d endorder traversals are initiall y confusing to the student. Tw o pedagogical techniques will be describe d that might be of use to the teacher : the first diagrams the recursive proces s as a tree (what else?) and the secon d allows a "quick and dirty" method fo r determining the actual order of th e nodes visited for each traversal b y walking around the tree in thre e different modes. 1. For illustrative purposes, let u s assume we wish to traverse th e following tree in preorder. Figure 1 which is to recursively : 1. Visit the root nod e 2. Traverse the left subtre e 3. Traverse the right subtree [4 ] Actually I prefer Lewis and Smith' s notation of the process as "NLR-recursive" [5] (N=Node, L=Left , R=Right) since it is more succinc t and easier to remember so that a 'pidgin' language version of th e procedure could be given as follows : NLR (T) IF (T=") RETUR N PRINT ROOT (T) NLR (LEFT(T)) NLR (RIGHT(T)) RETUR N Where T represents a tree or a subtree, the function ROOT extract s the root of a given tree T, and th e functions LEFT and RIGHT return th e left and right subtrees of the give n tree, T. As we traverse the tree i n NLR order, we print the value of eac h node visited. If we invoke NLR with the tree show n in the previous diagram, we can sho w not only the order of evaluation o f each statement in the procedure (th e circled numbers on the branches), bu t also the tree structure of th e recursive process itself : 36