Variable length tree structures having minimum average search time

ABSTRACT

Sussenguth suggests in a paper (1963) that a file should be organized as a doubly-chained tree structure if it is necessary both to search and to update frequently. Such a structure provides a compromise between the fast search/slow update characteristics of binary searching and the slow search/fast update characteristics of serial searching. His method, however, contains the limiting restriction that all terminal nodes lie on the same level of the tree. This paper considers the effect of relaxing this restriction. First, trees which have the property that a priori the filial set of each node is well defined are studied. It is proved that coding the nodes within each filial set with respect to the number of terminal nodes reachable from each is necessary and sufficient to guarantee minimum average search time. Then the more general case (that is, where the entire structure of the tree is changeable) is treated. A procedure is developed for constructing a tree with a minimum average search time. A simple closed expression for this minimum average search time is obtained as a function of the number of terminal nodes. The storage capacity required to implement the doubly-chained tree structure on a digital computer is also determined. Finally, the total cost of the structure, using Sussenguth's cost criterion, is computed. It is shown that significant improvements in both the average search time and the total cost can be obtained by relaxing Sussenguth's restriction that all terminal nodes lie on the same level of the tree.

REFERENCES

• 1: Edward H. Sussenguth, Jr., Use of tree structures for processing files, Communications of the ACM, v.6 n.5, p.272-279, May 1963 [doi>10.1145/366552.366600]
• 2: HARDY, G. H., LITTLEWOOD, J. E., AND POLYA, G. Inequalities. Cambridge U. Press, Cambridge, England, 1934, pp 261-262.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE
      I.2.8 Problem Solving, Control Methods, and Search
          Subjects: Graph and tree search strategies

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Sorting and searching

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Trees

General Terms:
Design, Performance, Theory

Keywords:
double chaining, file searching, information retrieval, tree stuctures