Showing papers on "Interval tree published in 1972"

A top-down algorithm for constructing nearly optimal lexicographic trees

Abstract: Publisher Summary This chapter focuses on the top-down algorithm for constructing nearly optimal lexicographic tree, wherein, a lexicographic tree is a binary search tree. The binary search tree has been proposed as a data structure for lists of names which must be both searched and updated frequently. A binary search tree is a rooted, ordered tree such that the out-degree of every node is two foran internal nodes or zero for a leaf. If there are N internal nodes, there are 2N edges and N+ 1 leaves. Each internal node is associated with one name in a set of lexicographically ordered names, A 1 < A 2 < ···

Minimal redundancy decoding method and means

Abstract: Electronically decoding variable length minimal redundancy binary input codes into characters in a character set. Code flexibility is provided by easily supporting (1) different transmitted codes for the same outputted character set, and (2) the same transmitted code for different output character sets. Tree structured codes are used for translating minimal redundancy codes into characters, which are well-known in the prior art of Huffman codes, in which the sink vertices in a binary tree correspond to characters in a character set. The set of path vectors in the binary tree is the minimal redundancy encoding for the characters represented by the sinks. A bit sequence store T has its bit positions set to represent the binary tree, in which the bit positions are set to correspond to the vertices in the tree when it is scanned in left list order; and a 1 setting represents an inner vertex and a 0 setting represents a sink vertex in the binary tree. Store T is easily reset to permit easy changes in the binary tree represented therein, in order to support (1) and (2) above. The path vectors in the tree are transmitted as the coded characters are decoded using the correct tree set into store T.

A comment on the double-chained tree

Abstract: In 1963, Sussenguth [8] suggested that a file should be organized as a double-chained tree for searching and updating. Patt [5] then obtained the optimum double-chained tree under the assumption that no key may prefix another and that all terminal nodes (items of information) have equal probabilities of being searched. Stanfel [6, 7] explored the relation between the double-chained tree and variable length code [3] and solved a special integer programming problem which corresponds to the case of equal probabilities of terminal nodes and a finite set of available symbols for keys with different costs.

A note on optimal doubly-chained trees

Abstract: This note concerns the recent paper by Hu [2] dealing with doubly-chained trees of the type introduced by Sussenguth [9]. In the second part of the paper, Hu deals with the problem of constructing an optimum weighted doubly-chained tree, under the standard assumption that no key is allowed to prefix another, as found, for example, in Patt [6]. He states that for the weights and elements of Figure 1, the optimum doubly-chained tree with all nodes reachable in less than five steps is the one shown in Figure 2. This tree is obtained from the optimal binary tree constructed in Hu and Tan [3], using the technique found in Knuth [5]. But the doubly-chained tree constructed by this method comes from an artificially restricted class of doubly-chained trees, and is thus not optimal in the sense usually defined in the literature [6-9] and described below. This fact is recognized by Hu [1], but was not explicitly stated in the paper in question.

Minimum Search Tree Structures for Data Partitioned into Pages

Abstract: The doubly chained tree structure is an efficient device for organizing a file that must be searched and updated frequently. This paper considers a variation to the doubly chained tree which is more representative of a file in which the data is partitioned into blocks or pages. In this model the cost of searching the next node is 1 if the node is reachable from the same root (i.e., on the same page), and is k if the next node is the next root (i. e., on a different page). An algorithm for constructing doubly chained trees having any number of terminal nodes N, with minimal average search length, is developed. A simple closed expression is derived for the minimum average search time as a function of the number of terminal nodes N and the parameter k. A proof of optimality is also included.

Hybrid trees: A data structure for lists of keys

Abstract: A generalization of the binary tree and the Trie, called the hybrid tree, is defined in which a node can correspond to a key, to any substring in a key, or to a character of the alphabet from which the keys are formed. An algorithm is given for constructing an optimal hybrid tree and it is shown that the search time of an optimal hybrid tree is never larger than the search time of the Trie or the optimal binary tree representing the same set of keys.