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A gentle tutorial and a structured reformulation of Bock's algorithm for minimum directed spanning trees

Yuxi Wang, Jungyeul Park

Abstract

This paper presents a gentle tutorial and a structured reformulation of Bock's 1971 Algol procedure for constructing minimum directed spanning trees. Our aim is to make the original algorithm readable and reproducible for modern readers, while highlighting its relevance as an exact decoder for nonprojective graph based dependency parsing. We restate the minimum arborescence objective in Bock's notation and provide a complete line by line execution trace of the original ten node example, extending the partial trace given in the source paper from initialization to termination. We then introduce a structured reformulation that makes explicit the procedure's phase structure, maintained state, and control flow, while preserving the logic of the original method. As a further illustration, we include a worked example adapted from {jurafsky-martin-2026-book} for dependency parsing, showing how a maximum weight arborescence problem is reduced to Bock's minimum cost formulation by a standard affine transformation and traced under the same state variables.

A gentle tutorial and a structured reformulation of Bock's algorithm for minimum directed spanning trees

Abstract

This paper presents a gentle tutorial and a structured reformulation of Bock's 1971 Algol procedure for constructing minimum directed spanning trees. Our aim is to make the original algorithm readable and reproducible for modern readers, while highlighting its relevance as an exact decoder for nonprojective graph based dependency parsing. We restate the minimum arborescence objective in Bock's notation and provide a complete line by line execution trace of the original ten node example, extending the partial trace given in the source paper from initialization to termination. We then introduce a structured reformulation that makes explicit the procedure's phase structure, maintained state, and control flow, while preserving the logic of the original method. As a further illustration, we include a worked example adapted from {jurafsky-martin-2026-book} for dependency parsing, showing how a maximum weight arborescence problem is reduced to Bock's minimum cost formulation by a standard affine transformation and traced under the same state variables.

Paper Structure

This paper contains 128 sections, 1 theorem, 137 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. The problem
  3. Minimum directed spanning tree rooted at an origin
  4. A compact integer programming view
  5. Why Bock introduces dual variables
  6. Complementary slackness viewpoint
  7. The algorithm
  8. Input and state variables
  9. Original Algol procedure
  10. Description
  11. A step-by-step reading of Bock's original algorithm
  12. Tracing Bock’s algorithm on the original $10\times10$ matrix
  13. Input
  14. State variables
  15. Initialization (L2)
  16. ...and 113 more sections

Key Result

Theorem 1

For any cost matrix $C$ and root $j_0$, the programs bock and bock_wang are behaviorally equivalent. Starting from aligned initial states, they remain aligned throughout execution, perform the same sequence of entering-edge selections and dual increments, maintain the same component partition of the

Figures (1)

  • Figure 1: Jurafsky--Martin example used to illustrate Bock's algorithm

Theorems & Definitions (2)

  • Theorem 1: Behavioral equivalence
  • proof