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Quota Trees

Tad White

Abstract

We introduce the notion of quota trees in directed graphs. Given a nonnegative integer ``quota'' for each vertex of a directed multigraph $G$, a quota tree is an immersed rooted tree which hits each vertex of $G$ the prescribed number of times. When the quotas are all one, the tree is actually embedded and we recover the usual notion of a spanning arborescence (directed spanning tree). The usual algorithms which produce spanning arborescences with various properties typically have (sometimes more complicated) ``quota'' analogues. Our original motivation for studying quota trees was the problem of characterizing the sizes of the Myhill-Nerode equivalence classes in a connected deterministic finite-state automaton recognizing a given regular language. We show that the obstruction to realizing a given set of M-N class sizes is precisely the existence of a suitable quota tree. In this paper we develop the basic theory of quota trees. We give necessary and sufficient conditions for the existence of a quota tree (or forest) over a given directed graph with specified quotas, solving the M-N class size problem as a special case. We discuss some potential applications of quota trees and forests, and connect them to the $k$ lightest paths problem. We give two proofs of the main theorem: one based on an algorithmic loop invariant, and one based on direct enumeration of quota trees. For the latter, we use Lagrange inversion to derive a formula which vastly generalizes both the matrix-tree theorem and Cayley's formula for counting labeled trees. We give an efficient algorithm to sample uniformly from the set of forests with given quotas, as well as a generalization of Edmonds' algorithm for computing a minimum-weight quota forest.

Quota Trees

Abstract

We introduce the notion of quota trees in directed graphs. Given a nonnegative integer ``quota'' for each vertex of a directed multigraph , a quota tree is an immersed rooted tree which hits each vertex of the prescribed number of times. When the quotas are all one, the tree is actually embedded and we recover the usual notion of a spanning arborescence (directed spanning tree). The usual algorithms which produce spanning arborescences with various properties typically have (sometimes more complicated) ``quota'' analogues. Our original motivation for studying quota trees was the problem of characterizing the sizes of the Myhill-Nerode equivalence classes in a connected deterministic finite-state automaton recognizing a given regular language. We show that the obstruction to realizing a given set of M-N class sizes is precisely the existence of a suitable quota tree. In this paper we develop the basic theory of quota trees. We give necessary and sufficient conditions for the existence of a quota tree (or forest) over a given directed graph with specified quotas, solving the M-N class size problem as a special case. We discuss some potential applications of quota trees and forests, and connect them to the lightest paths problem. We give two proofs of the main theorem: one based on an algorithmic loop invariant, and one based on direct enumeration of quota trees. For the latter, we use Lagrange inversion to derive a formula which vastly generalizes both the matrix-tree theorem and Cayley's formula for counting labeled trees. We give an efficient algorithm to sample uniformly from the set of forests with given quotas, as well as a generalization of Edmonds' algorithm for computing a minimum-weight quota forest.
Paper Structure (11 sections, 14 theorems, 48 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 11 sections, 14 theorems, 48 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Motivation and definitions
  2. Examples
  3. Quota trees
  4. Quota search
  5. The Enough Arrows theorem
  6. Applications
  7. Counting quota trees
  8. Uniformly generating quota forests
  9. Minimum-Weight Quota Forests
  10. Extension to minimum-weight quota forests
  11. Further work

Key Result

Theorem 1

A triple $(G,q,s)$ admits an exact quota forest iff it admits an at-most quota forest and $q(v)\ge s(v)$ for all $v\in V(G)$.

Figures (6)

  • Figure 1: A digraph (a) with quotas and a single-vertex start portfolio; (b) is a valid quota tree, while (c) is not.
  • Figure 2: Expanding the Fibonacci DFA to a larger connected DFA via quota search. (a) The original DFA, with quotas $(3,2,3)$; (b) the expanded DFA. The red edges form a quota tree, guaranteeing connectivity; the green edges are a random completion to a DFA.
  • Figure 3: A directed, edge-weighted graph for which we will compute a minimum quota tree. Quotas are indicated on the vertices; edge weights range from $1$ to $4$, indicated by line thickness.
  • Figure 4: The multigraph $G[x]$ representing the minimal-weight inventory satisfying the edge and node conditions. Edge $e$ is labeled with the inventory value $x_e$ Two clusters violate the subset constraint; these are sources in the strong-component graph of the inventory.
  • Figure 5: The quotient graph $G'$ obtained by collapsing the subsets $S$, and reweighting and duplicating the edges out of the collapsed nodes. New weights are shown according to the legend in Figure \ref{['fig:MQTexample1']}. The label $x_e/c_e$ on an edge $e$ indicates that $c_e$ copies of that edge are available, of which $x_e$ copies are used in the minimal inventory.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Theorem 1: exact vs. at most solvability
  • proof
  • Theorem 2: enough arrows
  • proof
  • Theorem 3
  • Theorem 4: universality of quota-based DFA expansion
  • Theorem 5: counting quota forests
  • Theorem 6: matrix interpretation
  • Theorem 7: matrix-forest
  • Corollary 8: enough arrows
  • ...and 8 more