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Arithmetical Binary Decision Tree Traversals

Jinxiong Zhang

TL;DR

A suite of binary tree traversal algorithms that leverage novel representation matrices to flatten the full binary tree structure and embed the aggregated internal node Boolean tests into a single binary vector are presented.

Abstract

This paper introduces a series of methods for traversing binary decision trees using arithmetic operations. We present a suite of binary tree traversal algorithms that leverage novel representation matrices to flatten the full binary tree structure and embed the aggregated internal node Boolean tests into a single binary vector. Our approach, grounded in maximum inner product search, offers new insights into decision tree.

Arithmetical Binary Decision Tree Traversals

TL;DR

A suite of binary tree traversal algorithms that leverage novel representation matrices to flatten the full binary tree structure and embed the aggregated internal node Boolean tests into a single binary vector are presented.

Abstract

This paper introduces a series of methods for traversing binary decision trees using arithmetic operations. We present a suite of binary tree traversal algorithms that leverage novel representation matrices to flatten the full binary tree structure and embed the aggregated internal node Boolean tests into a single binary vector. Our approach, grounded in maximum inner product search, offers new insights into decision tree.
Paper Structure (9 sections, 2 theorems, 23 equations, 3 figures, 7 algorithms)

This paper contains 9 sections, 2 theorems, 23 equations, 3 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Tree Traversal Using Multiplication Operation
  3. Tree Traversal Using Bitvectors
  4. Fuzzy Tree Traversal
  5. Decision Tree Evaluation from Diverse Perspectives
  6. The Column Perspective
  7. The Row Perspective
  8. The Algorithmic Perspective
  9. Discussion

Key Result

Theorem 1

Algorithm alg:matrix identifies the correct exit leaf for every binary decision tree $T(\mathcal{N}, \mathcal{L})$ and input vector $\mathbf{x}$.

Figures (3)

  • Figure 1: Visual representation of a binary decision tree(the left), its left matrix (the middle), and its right matrix (the right), illustrating the transformation from graphical tree structure to matrix form.
  • Figure 2: A fuzzy binary decision tree.
  • Figure 3: A general tree (the left) and its equivalent binary tree (the right) with the same probability distribution over the leaf nodes.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Definition 7
  • Definition 8
  • Definition 9
  • ...and 3 more