Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithm for Constructing Related Spanning Directed Forests of Minimum Weight

Vasily Buslov

Abstract

An algorithm is proposed for constructing directed spanning forests of the minimum weight, in which the maximum possible degree of affinity between the minimum forests is preserved when the number of trees changes. The correctness of the algorithm is checked and its complexity is determined, which does not exceed $ O (N ^ 3) $ for dense graphs. The result of the algorithm is a set of related spanning minimal forests consisting of $ k $ trees for all admissible $ k $.

Algorithm for Constructing Related Spanning Directed Forests of Minimum Weight

Abstract

An algorithm is proposed for constructing directed spanning forests of the minimum weight, in which the maximum possible degree of affinity between the minimum forests is preserved when the number of trees changes. The correctness of the algorithm is checked and its complexity is determined, which does not exceed for dense graphs. The result of the algorithm is a set of related spanning minimal forests consisting of trees for all admissible .

Paper Structure

This paper contains 17 sections, 2 theorems, 18 equations, 3 figures.

Table of Contents

  1. Basic notations and definitions
  2. Objects and properties used
  3. Arc replacement operation
  4. Different minima on subsets of the set of vertices
  5. Constructing related forests from pseudo-related ones
  6. Pseudo-related and related forests
  7. Descendants among pseudo-descendants
  8. Place of the Chu-Liu-Edmonds algorithm
  9. Bypassing the stage of pseudo-descendants to descendants
  10. Constructing descendants bypassing the stage of pseudo-descendants
  11. Finding a connected component and an absorbing tree
  12. Algorithm for constructing minimal descendants
  13. Using simplified notation
  14. Description of the algorithm
  15. Algorithm complexity
  16. ...and 2 more sections

Key Result

Theorem 1

Let $F\in\tilde{\cal F}^{k+1}$, $y\in{\cal K}_F$, $G\in\tilde{\cal F}^{k}\cap {\cal P}^F_y$. Let also ${\cal D}$ be the set of vertices of that connected component of the induced subgraph $G|_{{\cal V}T^F_y}$ that contains vertex $y$. Then graph $F^G_{\uparrow{\cal D}}$ is a minimal descendant of $F

Figures (3)

  • Figure 1: Arcs of $F$ are depicted, coming from the vertices of its two trees with roots at $y$ and $x$.
  • Figure 2: Arcs of $G$ (which is a pseudo-descendant of $F$) coming from vertices of the set ${\cal V}T^F_y\cup {\cal V}T^G_x$ are depicted. ${\cal D}$ is the set of vertices of connected component of the induced subgraph $G|_{{\cal V}T^F_y}$, which includes the vertex $y$. Arcs of the forest $H$ are the arcs of $G$ coming from the vertices of the set ${\cal V}T^F_y$.
  • Figure 3: Arcs of the forest $R=F^G_{\uparrow{\cal D}}$ (which is a descendant of $F$) coming from the vertices ${\cal V}T^F_y\cup {\cal V}T^G_x$ are depicted. Arcs coming from vertices of $\overline{\cal D}$ coincide for forests $R$ and $F$.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof