# Quota Trees

Tad White  IDA Center for Computing Sciences, 17100 Science Drive, Bowie, MD
20715-4300  tad(at)super(dot)org

(Date: July 6, 2017)

###### 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.

###### Key words and phrases:

graph traversal, graph search, automata, DFA, regular languages, Myhill-
Nerode, private information retrieval, graph immersions, arborescences,
spanning trees, Edmonds’ algorithm, lightest paths, matrix-tree, random trees,
Cayley formula, Lagrange inversion, Narayana numbers, combinatorial
reciprocity

###### 2010 Mathematics Subject Classification:

05C30, 05C85, 68R10

##  1\. Motivation and definitions

A recently proposed scheme in the area of private information retrieval [6]
rests in part on the ability to construct arbitrarily complex deterministic
finite automata (DFAs) recognizing a regular language $\mathcal{L}$. While the
theory of simplifying, or minimizing, a finite-state automaton is well known,
the inverse problem of “complicating” a DFA leads to interesting questions
about the structure of the set of DFAs recognizing $\mathcal{L}$.

The Myhill-Nerode theorem implies the existence of a unique minimal DFA
$\mathcal{D}_{\mathcal{L}}$ which recognizes $\mathcal{L}$.
$\mathcal{D}_{\mathcal{L}}$ is a quotient of any connected111Think of a DFA as
a graph $G$ having an initial node and labeled edges coming out of each node;
the DFA is connected if any node in $G$ can be reached from the initial node.
DFA $\mathcal{D}$ recognizing $\mathcal{L}$; that is, the states of
$\mathcal{D}$ can be grouped into equivalence classes, with one class for each
state of $\mathcal{D}_{\mathcal{L}}$, such that the transitions in
$\mathcal{D}$ are coherent with respect to these classes. So in order to
understand the set of connected DFAs which can recognize $\mathcal{L}$, one
wants to know what sizes these equivalence classes can take, and to have an
effective algorithm for constructing a connected DFA with given class sizes.
(Connectedness is the key issue here; if $\mathcal{D}$ is allowed to have
unreachable nodes, then there is no constraint on the sizes other than
positivity.)

The problem turns out to reduce to a very natural graph search
problem.222Throughout this paper, we will often use the term “graph” to mean
what is usually called a directed multigraph; that is, edges are directed, and
both loops and multiple edges are allowed. We will frequently encounter
directed trees, with edges directed away from the root; these are typically
called (out-)arborescences in the literature. Accordingly, forests of out-
directed trees should be called something like silvations. But we will
stubbornly just use “trees” and “forests.” In particular, it turns out that a
connected DFA with specified Myhill-Nerode class sizes exists iff one can
construct a directed tree $T$, together with an immersion $f:T\to G$, such
that the sizes of the vertex preimages match the desired equivalence class
sizes; we call $T$ a “quota tree.” When it exists, a suitable $T$ can be found
via a simple modification of standard graph traversal in which vertices are
visited multiple times, according to the class sizes; $T$ records the
traversal just as an ordinary graph search is recorded via a spanning tree.
$T$ can then be extended (in many ways) to a DFA by adding missing
transitions.

It is easy to interpret this type of graph search in applications other than
automata; the theory expresses itself most naturally in a broader context. In
section 2 we describe some scenarios in which quota trees arise naturally;
these illustrate some quota versions of standard spanning tree optimization
problems. In section 3 we formally define quota trees and forests and state
the main results. Section 4 introduces the corresponding variant of graph
search, called quota search. In section 5 we prove the “enough arrows”
theorem, which gives necessary and sufficient conditions for the existence of
quota trees (or forests) with specified quotas. In section 6 we discuss some
applications, particularly DFAs and the $k$ lightest path problem. In section
7 we address the problem of enumerating quota trees; our primary tool is the
multivariate Lagrange inversion formula. The results of this section give a
much more precise version of the “enough arrows” theorem, which vastly
generalizes both the matrix-tree theorem and Cayley’s formula for counting
labeled trees. In section 8 we strengthen the enumeration results to sample
uniformly from the set of trees (or forests) with given quotas. In section 9
we give an algorithm for finding minimal-weight quota forests. Finally, in
section 10, we identify a few areas for further research.

##  2\. Examples

Before giving formal definitions, we present a few scenarios in which quota
trees arise naturally, so that the reader can choose a comfortable motivating
context for the remainder of the paper.

#### A coupon game

A dealer has a supply of coupon books of various types; all books of a given
type are identical. Each coupon in a book allows you to buy another coupon
book at a particular price. (For example, it might be that in an $A$ book,
coupon 1 is for another $A$ book at $5, coupons 2 and 3 are for $B$ books at
$2 and $3 respectively, and coupon 4 is good for a free $D$ book.) You’re
given one or more coupon books to start with; you win if you can collect all
of the dealer’s coupon books (and you’d like to do so as cheaply as possible.)
You know how many books of each type the dealer has, and what coupons are in
what types of book. Is it possible to collect all the coupons? If so, in how
many different ways, and what is the minimum cost?

#### Network configuration

You have a supply of network devices of various types; all devices of a given
type are identical. Each type has a single input port and several output
ports, each of which can talk to a specific type of device. (For example, an
$A$ device might have an $A$ port, two $B$ ports and a $D$ port, a $B$ device
might have no output ports, and so on.) You would like to connect all of your
devices together so that a message from one particular device can then be
propagated to all of the other devices. Is this possible? If so, in how many
ways, and what configuration minimizes the number of intermediate devices on
each path? When there is only one device of each type, this is a spanning tree
problem.

####  $k$ lightest paths

Given a directed graph $G$, with nonnegative weights on the edges, and an
integer $k\geq 1$, compute the $k$ lightest paths from one or more given
source nodes to each vertex in $G$. This can be interpreted as a minimum quota
tree problem.

#### Tree coloring

Many tree-coloring problems are naturally interpreted as quota-tree questions.
For example, suppose we have $n$ colors, and a subset $S_{i}\subset[n]$ for
each $i\in[n]$. How many ways can we color a rooted tree such that $q_{i}$
nodes have color $i$, and such that the children of a color-$i$ node get
distinct colors selected from $S_{i}$? (For example, for two colors, if there
are no restrictions we get Narayana numbers; if blue nodes can only have red
children we get Motzkin coefficients. See section 7 for more examples.)

##  3\. Quota trees

By a directed multigraph we will mean a tuple $(V,E,i,t)$ where $V$ is a set
of vertices, $E$ is a set of edges, and $i:E\to V$ and $t:E\to V$ return the
initial and terminal vertices of each edge. Edges are oriented; we say an edge
$e$ goes from $i(e)$ to $t(e)$. We may abuse notation by writing $v\to w$ to
mean there is an edge in $G$ from $v$ to $w$, but as $G$ is a multigraph there
may be other edges as well. In particular, loops are allowed (that is, one may
have $t(e)=i(e)$ for some edges $e$) and the edges from $v$ to $w$ are all
distinguishable (or “labeled”) as they are distinct elements of $E$.)

A mapping $f:G\to H$ of multigraphs sends vertices to vertices, edges to
edges, and respects $i$ and $t$: thus, if $e$ is an edge from $v$ to $w$ in
$G$, then $f(e)$ is an edge in $H$ from $f(v)$ to $f(w)$.

Define the instar (resp. outstar) of a vertex $v$ to be the set of incoming
(resp. outgoing) edges at $v$:

$\mathord{\to}v=\\{e\mid t(e)=v\\};\qquad v\mathord{\to}=\\{e\mid i(e)=v\\}$

We say a map $f:G\to H$ is an out-immersion, or simply an immersion, if it
maps $v\mathord{\to}$ injectively into $f(v)\mathord{\to}$. We define a cusp
in $G$ under $f$ to be a pair of edges $e_{1}\neq e_{2}\in v\mathord{\to}$
with $f(e_{1})=f(e_{2})$; thus $f$ is an immersion iff $G$ has no cusps under
$f$.

A quota is a nonnegative-valued function $q:V(G)\to\mathbf{Z}$. A quota tree
with root $*\in V(G)$ and quota $q$ is an immersion $f:T\to G$ where
$(T,\tilde{*})$ is a rooted tree, $f(\tilde{*})=*$, and $|f^{-1}(v)|=q(v)$ for
all $v\in V(G)$. Note that if $q(v)\leq 1$ for all $v\in V(G)$, then the map
$f$ is actually an embedding, and if $q(v)$ is identically $1$, the image
$f(T)$ is a (rooted, directed) spanning tree of $G$.

Finally, a quota forest with start portfolio $s:V(G)\to\mathbf{Z}$ is a
(disjoint) union of quota trees $F=\\{T_{v,i}\mid v\in V(G),1\leq i\leq
s(v)\\}$ such that $T_{v,i}$ is rooted at $v$. The forest also immerses into
$G$; the quota it achieves is the sum of the quotas of the component forests.
Note that we treat all the roots as distinguishable: if a forest contains two
or more non-isomorphic quota trees with roots mapping to the same vertex of
$G$, permuting those trees gives a different quota forest. We will refer to a
forest with quota $q$ and start portfolio $s$ as a $(G,q,s)$-forest (or
$(G,q,s)$-tree, if $||s||_{1}=1$).

A graph with quotas, together with both an example and a nonexample of quota
trees, appears in Figure 1.

Figure 1. A digraph (a) with quotas and a single-vertex start portfolio; (b) is a valid quota tree, while (c) is not. $2$$2$$2$ |  |
---|---|---
(a) |  (b) |  (c)

### Out-coverings

One can think of a spanning tree $G$ as “living in” $G$, but a more natural
home for a quota tree is actually a covering space of $G$, which we will now
describe. We will say a map $\pi:G\to H$ is an out-covering if
$\pi(v\mathord{\to})$ maps bijectively onto $\pi(v)\mathord{\to}$ for all $v$
in $V(G)$. In this situation, given an (out-)immersed tree $f:T\to H$ with
root $w\in H$, and a preimage $v\in\pi^{-1}(w)$, there is a unique lift
$\tilde{f}:T\to G$ with root $v$; the (right) inverse of the operation
$f\mapsto\tilde{f}$ is given by $f\mapsto\pi\circ f$.

As with topological spaces, we can define a universal out-cover by considering
paths from a distinguished vertex. A (finite directed) path in $G$ from $*$ is
a sequence $\\{e_{i}\mid 1\leq i\leq l\\}$ of directed edges of $G$, with
$i(e_{1})={*}$ and $t(e_{i})=i(e_{i+1})$. We define the universal out-cover of
$(G,{*})$ to be the directed graph $(\tilde{G},\tilde{*})$ whose vertices are
the finite directed paths from $*$, having an edge (labeled $e_{i}$) from
$e_{1}\cdots e_{i-1}$ to $e_{1}\cdots e_{i}$. It’s easy to see that
$\tilde{G}$ is a (generally infinite) rooted tree, in which the root
$\tilde{*}$ corresponds to the length-zero path in $G$ from $*$. The natural
map $\pi:\tilde{G}\to G$ taking a directed path to its endpoint in $G$ is an
immersion. Note that the in-degree of each vertex $\tilde{v}\in\tilde{G}$ is
one; the out-degree of $\tilde{v}$ is the same as the out-degree of
$\pi(\tilde{v})$. (In particular, if $G$ is a DFA over an alphabet $\Sigma$,
then $\tilde{G}$ is a regular tree, directed outward from the root, with each
vertex having out-degree $|\Sigma|$.)

With this setup, it is easy to see that if $f:(T,t)\to(G,{*})$ is an immersion
of a rooted directed tree into $G$, then $f$ can be lifted uniquely to a map
$\tilde{f}:(T,t)\to(\tilde{G},\tilde{*})$ such that
$f=\pi\circ\tilde{f}$.333There is a larger “universal cover” that appears in
the literature (see for example [11]), based on paths whose edges which need
not be coherently oriented. This is essentially the topological universal
cover of $G$ (see [13]), constructed by ignoring orientations, which also has
the same universal lifting property for immersions of rooted directed trees.
However, the universal out-cover is the smallest space which has this
property, and so is the “natural” home for quota trees. We note that Yamashita
and Kaneda, in their study of computing in anonymous networks, referred to the
universal out-cover $(\tilde{G},\tilde{*})$ as the view of $\tilde{*}$ within
the topological universal cover (see [18].) The map $\tilde{f}$ is injective,
so we can view $T$ as sitting inside of $\tilde{G}$.

##  4\. Quota search

The problems in section 2, as well as the original problem of computing
possible Myhill-Nerode class sizes, correspond to a variant of graph search in
which we are given a positive “quota” $q(v)$ for each $v\in V(G)$, and we wish
to visit each vertex $v$ exactly $q(v)$ times. (When $q(v)=1$ for all $v$,
this is ordinary graph traversal.)444Setting $q(v)=0$ for any particular $v$
is legal; it essentially amounts to working in the induced graph $G-\\{v\\}$.
We refer to this goal as quota search.

We assume familiarity with standard graph traversal as described, for example,
in [3, ch. 22], to which we will make some modifications. Given a directed
graph $G$ and a set $S$ of start vertices, generic graph search keeps track of
discovered but unprocessed vertices in a generic priority queue. As we will be
dealing with multigraphs, and visiting vertices multiple times, we will need
to be more careful to distinguish between an edge from $u$ to $v$ and the pair
$(u,v)$ itself; indeed, it is much easier to describe quota search succinctly
by considering edges rather than vertices. So our algorithm encodes the search
forest $F$ via a predecessor function $\pi:E(F)\to E(F)$, rather than the more
usual $\pi:V(G)\to V(G)$. Accordingly, we replace the usual VisitVertex
procedure with an analogous UseEdge, which inserts an edge taken from the
queue into the search forest.

Recall that the quota forest $F$ does not actually live in $G$, so we must
distinguish between an edge $\tilde{e}$ in $F$ and its image $e=f(\tilde{e})$
under the immersion $f:F\to G$, whose construction will be implicit. An edge
$\tilde{e}$ in the queue should be thought of as living in the universal cover
$\tilde{G}$, not in $G$.

Instead of coloring vertices Black or White according to whether or not they
have been visited, we keep track of the number of remaining visits to a vertex
$v$. Thus, Black and White correspond to quotas of $0$ and $1$ respectively.

In ordinary graph search, we gain nothing by repeating a search from the same
start point, but allowing repeated starts even from the same node can be
useful if we need to arrive at vertices multiple times. As described in
section 4, we replace the set $S$ of start vertices with a nonnegative start
portfolio $s:V(G)\to\mathbf{Z}$; $s(v)$ is the number of times a search can be
started from a given vertex. (Thus, if we’re doing a single search from one
particular vertex $w$, we set $s(v)=1$ if $v=w$ and $0$ otherwise.)

Finally, it is useful to distinguish two natural variants of search. In the
“exact” version of quota search, we want our search forest to contain exactly
$s(v)$ trees with root $v$. (This corresponds, in the coupon-game scenario, to
requiring that every coupon in the initial collection be used up.) In the “at-
most” version, the search forest may contain at most $s(v)$ trees with root
$v$; that is, we don’t need to use all of our coupons. The two versions are
closely related:

######  Theorem 1 (exact vs. at most solvability).

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

###### Proof.

Since an exact forest actually solves the at-most problem, and clearly
requires $q(v)\geq s(v)$ for all $v\in V(G)$, one direction is trivial. On the
other hand, if we have an at-most quota forest $F$ with fewer than $s(v)$
trees rooted at lifts of $v$, we can simply cut off some of the $q(v)$
occurrences in $F$ of lifts of $v$ from their parents, making them roots of
new trees. This works as long as $q(v)\geq s(v)$. ∎

Both the exact and at-most versions of quota search can be handled with a
single meta-algorithm. In both cases we initialize $Q$ with (sentinel edges
corresponding to) the start portfolio. In order to implement ExactQuotaSearch,
we simply arrange for QueueExtract to return the start portfolio first; to
implement AtMostQuotaSearch, we drop that restriction, in which case the
number of new trees created, and what their roots are, will depend on the
particular queue extraction algorithm.

We capture the resulting generic quota search meta-algorithm as Algorithm 1.
It succeeds if it ends with all quotas reduced to zero. The “enough arrows”
theorem will characterize triples $(G,q,s)$ such that a quota forest exists
(in which case the algorithm is guaranteed to succeed for any specialization
of QueueExtract.)

Algorithm 1 Generic quota search

function NewEdge($\tilde{e}$,$e^{\prime}$)

$e^{\prime}\in E(G)$; $\tilde{e}\in E(F)$ with $f(t(\tilde{e}))=i(e^{\prime})$

a new edge $\tilde{e}^{\prime}$ with $f(\tilde{e}^{\prime})=e^{\prime}$,
$\pi(\tilde{e}^{\prime})=\tilde{e}$

end function

5:

function NewSentinelEdge($v$)

$v\in V(G)$ $\triangleright$ $v$ will be the root of a tree in $F$

a new sentinel edge $\tilde{e}$ with $f(\tilde{e})=NULL$, $f(t(\tilde{e}))=v$

end function

10:

procedure UseEdge($\tilde{e}$)

an edge $\tilde{e}$ such that $v=f(t(\tilde{e}))$ satisfies $q(v)>0$

Add $\tilde{e}$ to $F$; this updates $F$ and $q(v)$ and adds
$t(\tilde{e})\mathord{\to}$ to $Q$

$F=F\cup\\{\tilde{e}\\}$

15: $q(v)\leftarrow q(v)-1$

for $e^{\prime}\in v\mathord{\to}$ do $\triangleright$ in practice, skip
$e^{\prime}$ if $q(t(e^{\prime}))$ is already zero

QueueInsert$(Q,\textsc{NewEdge}(\tilde{e},e^{\prime}))$

end for

end procedure

20:

function GenericQuotaSearch($G$,$q$,$s$)

$G$ a directed graph; $q$ and $s$ are nonnegative functions on $V(G)$

quota forest $F$, predecessor map $\pi:E(F)\to E(F)$, and immersion $f:F\to G$

$Q,F\leftarrow\emptyset$

25: for $v\in V(G),k\in\\{1,\ldots s(v)\\}$ do

$\textsc{QueueInsert}(Q,\textsc{NewSentinelEdge}(v))$

end for

while $Q$ is nonempty do

30: $\tilde{e}\leftarrow\textsc{QueueExtract}(Q)$

if $q(f(t(\tilde{e})))>0$ then

$\textsc{UseEdge}(\tilde{e})$

end if

end while

35: return $\pi,f$ unless all $q(v)$’s are zero $\triangleright$ else fail

end function

#### Algorithm success and achievable parameters

Whenever UseEdge is called, $q(v)$ is the number of remaining required visits
to $v$. Thus the algorithm succeeds (i.e. visits all vertices the required
number of times) if and only if, upon termination, $q(v)=0$ for all $v\in
V(G)$. It turns out that, in contrast with ordinary graph search, success is
not possible for all pairs $(q,s)$. We will call $(G,q,s)$ achievable if some
(and, it turns out, any) quota search in $G$ with start portfolio $s$ achieves
the quotas $q$. (It is easy to see that achievability does not depend on
whether we are talking about “exact” or “at most” quota search.) The “enough
arrows” theorem in the next section precisely characterizes the achievable
parameters.

#### Quota search viewed in $\tilde{G}$

One way to think about quota search is that we replace each vertex $v$ with a
supply of copies of itself; when we “visit” $v$, we’re actually visiting a
fresh copy. When the start portfolio is [a single copy of] a single vertex
$*$, this allows us to describe quota search as occurring in the forward
universal cover $\tilde{G}$ of $G$. Specifically, we do ordinary graph search
in $(\tilde{G},\tilde{*})$, but only visit a vertex $\tilde{v}$ provided
$q(v)>0$, where $v=\pi(\tilde{v})$, in which case we decrement $q(v)$.
Finally, if the start portfolio $s$ is a multiset of vertices, we effectively
work in the disjoint union of $s(v)$ copies of $(\tilde{G},\tilde{v})$ for all
$v$. Whether the search trees are built sequentially, or at the same time, is
controlled by the order in which QueueExtract selects edges for consideration.

#### Optimization problems

As with ordinary graph search, the versatility of this meta-algorithm comes
from the variety of ways of choosing which element to extract from $Q$ at each
step. By specializing $Q$ to be a FIFO queue, a LIFO stack, or a more general
priority queue results in quota-search we obtain quota variants of algorithms
such as breadth-first search, depth-first search, or Dijkstra’s algorithm.

If we are optimizing an objective function which depends only on the forest
$F$, but not the particular traversal of $F$, then the data associated with an
edge $\tilde{e}$ in the queue $Q$ may only depend on the unique path to
$\tilde{e}$ in $F$; we will call such data intrinsic. For example, if the
edges of $G$ have weights, it is natural to consider the “minimum quota
forest” problem, a generalization of the minimum spanning tree problem in
which we wish to minimize the sum of the weights of the edges in a quota
forest with the given start portfolio and quotas. In this case we take the key
for an edge $\tilde{e}$ in $Q$ to be the weight of its image $e=f(\tilde{e})$
in $G$. Similarly, a quota version of Dijkstra’s algorithm is obtained by
taking the key to be the sum of the weights in the path to $\tilde{e}$ in the
search forest; see section 6. In both cases the keys are intrinsic.

It may be tempting, knowing that a vertex will be visited $q(v)$ times, to
assign the $k$-th visit to a vertex a cost which depends on $k$. However, this
is not intrinsic: different traversals of the same forest could then result in
different tree costs. But it would be perfectly legal to assign edge
$\tilde{e}$ a cost which depends on the number of visits to $t(\tilde{e})$ (or
any other nodes) on the path in $F$ to $\tilde{e}$.

Of course, not all graph optimization problems are solvable via graph search.
For instance, a very natural problem is to find a minimum-weight quota tree
(or forest) given weights on the edges of $G$; here we must emphasize that we
really mean quota arborescence (or branching.) When the quotas are at most
$1$, this is just the minimum arborescence (or branching) problem. An
algorithm for solving this problem has been given by Edmonds [4] and others.
Rather than accreting a tree via graph search, it iterates through a sequence
of putative solutions. Edmonds’ algorithm adapts beautifully to find minimum
quota trees (and, in particular, find the minimum-cost solution to the coupon
collecting problem in section 2.) We discuss minimum-weight quota trees in
section 9.

#### Relaxation

Many natural priority queue keys have a property which allows us to maintain a
smaller queue. As noted previously, an intrinsic cost associated to an edge
$\tilde{e}$ in $Q$ is some function $c(\tilde{p})$ of the unique path
$\tilde{p}=\tilde{e}_{1}\cdots\tilde{e}_{k}=\tilde{e}$ in the quota forest
from the root to $\tilde{e}$. We say $c$ is append-monotonic if key order is
invariant under appending a common path: that is, if we have two paths
$\tilde{p}_{1}$ and $\tilde{p}_{2}$ satisfying $c(\tilde{p}_{1})\leq
c(\tilde{p}_{2})$, and both ending at lifts of a common vertex $v$, then for
any path $p_{3}$ in $G$ starting at $v$, then

$c(\tilde{p}_{1}\tilde{p}_{3})\leq c(\tilde{p}_{2}\tilde{p}_{3}).$

If $f$ is append-monotonic, we know the best extensions of paths will be
extensions of best paths. So we can just keep track of the $q(v)$ best paths
to each vertex $v$. This is the quota-search analogue of what is called
relaxation in ordinary graph search (see [3, Ch. 24]): namely, when we arrive
at a previously seen vertex via a new path, we can keep the better of the two
paths and discard the other. In generic quota search, we might handle this
with a min-max queue of size $q(v)$ at each node $v$, in which case a generic
implementation of QueueExtract via two stages of binary heaps would take $\lg
V+\lg q(v)$ operations.

#### Complexity analysis

In the generic version of quota search, we visit each vertex $v$ $q(v)$ times,
doing one queue extraction and $|v\mathord{\to}|$ insertions. So the number of
insertions and extractions (and space) required is $\sum_{v}q(v)Adj(v)$ where
$Adj(v)=|v\mathord{\to}|+1$. When $q(v)=1$ for all $v$, and $Q$ is a simple
queue or stack (so that insertions and extractions can be done in constant
time), note that this reduces to $O(V+E)$, the complexity of ordinary graph
search.

If $Q$ is a priority queue, this leads to a complexity of

$O\left(\sum_{v}q(v)Adj(v)(\lg\sum_{v}q(v)Adj(v))\right)$

operations if binary heaps are used. If the queue keys are append-monotonic,
we can apply relaxation as above, reducing the work to

$O\left(\sum_{v}q(v)Adj(v)(\lg V+\lg q(v))\right).$

(This reduces to $O(E\lg V)$ when the quotas are identically 1.) As usual,
more sophisticated heap structures can provide further asymptotic improvement.

##  5\. The Enough Arrows theorem

In this section, we identify two conditions which the data $(G,q,s)$ must
satisfy in order for quota search to succeed; one is global, the other is
local. We show that these conditions are in fact sufficient: there exists a
quota forest meeting the specified quotas if and only if these conditions
hold. (In section 7 we will give an independent proof based on direct
enumeration of quota forests.)

Global:

$(G,q,s)$ is _connected_ if, for every node $v$ with $q(v)>0$, there exists a
node $u$ with $s(u)>0$ and a path in $G$ from $u$ to $v$. Note this only
depends on the support of $q$ and $s$.

Local:

$(G,q,s)$ has _enough arrows_ if the inequality

(1) $s(w)+\mathbf{in}(w)\geq q(w)$

holds for each $w\in V(G)$, where $\mathbf{in}(w):=\sum_{v}q(v)m_{vw}$.

We remark that the enough arrows condition can be written as

$\mathbf{s}+\mathbf{q}M\geq\mathbf{q},$

where $\mathbf{q}$ and $\mathbf{s}$ are the vectors of values of $q$ and $s$
respectively, and $M$ is the adjacency matrix of $G$.

Connectivity is clearly necessary in order to achieve even one visit to every
vertex with positive quota. To see why having enough arrows is necessary, note
that each visit to node $w$ arises either by starting at $w$, or by following
an edge from another node $v$. We visit node $v$ $q(v)$ times; each time, we
have $m_{vw}$ edges we can potentially follow to node $w$. Thus the maximum
number of arrivals at node $w$ is the left-hand side of (1), which must be at
least $q(w)$.

A note on terminology: especially in the context of automata, directed graphs
are typically drawn with arrows representing both transitions and initial
states. The left-hand side of the inequality (1) counts the maximum number of
arrows that can be drawn into each class (see figure 1); the right-hand side
represents the number of targets that need to be hit by these arrows.

######  Theorem 2 (enough arrows).

With the notation above, generic at-most quota search in $G$ with start
portfolio $s$ will achieve the quotas $q$ if and only if $(G,q,s)$ is
connected and has enough arrows.

###### Proof.

We have already argued the necessity of these conditions. The converse is
essentially by induction on $\sum_{v,w}q(v)m_{vw}$, and will follow from the
fact that connectivity and having enough arrows are invariant under the main
loop. Connectivity is automatically preserved.

So suppose we have enough arrows entering the main loop. At each iteration,
the queue $Q$ represents an effective “at most” start portfolio; so let $s(v)$
denote the number of edges $e$ in $Q$ with $t(e)=v$. Before the QueueExtract,
we have $s(v)>0$; it decreases by one with the extraction. We consider two
cases:

Case 1: $q(v)=0$. In this case inequality in the $v$-th coordinate of (1)
continues to hold since the right-hand-side is zero; all other coordinates in
the inequality are unchanged. So (1) is preserved in this case.

Case 2: $q(v)>0$. In this case VisitVertex adds, for each $w$, $m_{vw}$ edges
$v\to w$ into $Q$, and decrements $q(v)$. Thus the increase in $s$ and the
decrease in the sum on the left-hand side of (1) exactly cancel out.

Hence both connectedness and having enough arrows are preserved. At the end of
the algorithm, there are no edges left in $Q$; (1) implies
$\textbf{0}=\mathbf{s}\geq\mathbf{q}\geq\textbf{0}$, that is, we have reduced
all the quotas to zero, and the algorithm has succeeded. ∎

#### Remarks

We revisit the special case of ordinary graph search of a directed graph $G$
from a particular vertex $*$. Assume all vertices are reachable from $*$. We
have $q(v)=1$ for all $v\in V(G)$. But, by connectivity, each vertex in $G$
must either have an edge coming into it, or must be the start vertex $*$.
Thus, in this special case, having enough arrows is a consequence of
connectivity, explaining why the issue does not become apparent for ordinary
graph traversal.

The enough arrows theorem has a very similar flavor to the following theorem
[15, Theorem 5.6.1] characterizing directed graphs with Eulerian circuits;
namely, a global connectivity condition and a local degree condition. We state
it here since we’ll need it in section 9.

######  Theorem 3.

A digraph without isolated vertices is Eulerian if and only if it is connected
and balanced (i.e. $\textrm{indeg}(v)=\textrm{outdeg}(v)$ for all vertices
$v$.)

##  6\. Applications

### DFA expansion and Myhill-Nerode class sizes

A deterministic finite-state automaton, or DFA, is a tuple
$\mathcal{D}=(S,\Sigma,\delta,i,a)$, where $S$ is a finite set of states,
$\Sigma$ is an alphabet, $\delta:S\times\Sigma\to S$ is the transition map,
$i\in S$ is the initial state, and $a\subset S$ are the accept states. It is
useful to think of a DFA as a directed multigraph over $S$; for each $i\in S$
and $s\in\Sigma$ there is a directed edge from $i$ to $\delta(i,s)$ with label
$s$.

The transition map $\delta$ has a unique extension to a map
$\delta:S\times\Sigma^{*}\to S$ satisfying

$\delta(s,w_{1}w_{2})=\delta(\delta(s,w_{1}),w_{2})$

for all states $s$ and strings $w_{1}$, $w_{2}\in\Sigma^{*}$. 666That is,
$\delta$ defines a semigroup action of $\Sigma^{*}$ on $S$. ($\delta(s,w)$
just starts at $s$ and then applies the unary operators specified by the
symbols in $w$.) The automaton $\mathcal{D}$ accepts a string $w$ iff
$\delta(i,w)\in a$; that is, the path defined by $w$, starting at the initial
state, ends at an accept state. The automaton is called connected if the
extension $\delta:S\times\Sigma^{*}\to S$ is onto; that is, all states are
reachable by some path from the initial state. The set of strings accepted by
$\mathcal{D}$ is called the language recognized by $\mathcal{D}$. For the
purposes of this paper, a language to be regular iff it is recognized by some
DFA.

Given a regular language $\mathcal{L}$, the Myhill-Nerode theorem [10, Ch. 3]
implies that there is a unique minimal DFA $\mathcal{D}_{\mathcal{L}}$ which
recognizes $\mathcal{L}$. Furthermore, if $\mathcal{D}$ is any connected DFA
recognizing $\mathcal{L}$, then there is a quotient map
$\phi:\mathcal{D}\to\mathcal{D}_{\mathcal{L}}$ which is a homomorphism in the
sense of universal algebra [1]. That is, $\phi$ maps each state of
$\mathcal{D}$ to a state of $\mathcal{D}_{\mathcal{L}}$, such that transitions
are preserved:

(2) $\delta(\phi(v),s)=\phi(\delta(v,s))\textrm{\ for $v\in\mathcal{D}$,
$s\in\Sigma$}$

Not surprisingly, $\mathcal{D}_{\mathcal{L}}$ is connected (for if it had
unreachable states, those could be omitted to yield a smaller automaton
recognizing $\mathcal{L}$.)

As in [6], we might want to be able to construct and count larger DFAS
recognizing $\mathcal{L}$. We can use the enough arrows theorem to effectively
characterize the possible sizes of the Myhill-Nerode equivalence classes in a
connected DFA recognizing a language $\mathcal{L}$. If $\mathcal{D}$ is
connected, all of its states are reachable by a graph search from the initial
state of $\mathcal{D}$. The Myhill-Nerode theorem implies that the
corresponding graph search tree in $\mathcal{D}$ corresponds to a quota search
in $\mathcal{D}_{\mathcal{L}}$, with the quota for each state in
$\mathcal{D}_{\mathcal{L}}$ being the Myhill-Nerode equivalence class size.
Therefore, the graph of $\mathcal{D}_{\mathcal{L}}$, with these quotas and the
start portfolio consisting of just the initial state, must satisfy (1).

Furthermore, the converse direction of the theorem implies that _any_
collection of class sizes having enough arrows is achievable, since the
connectedness of the minimal DFA $\mathcal{D}_{\mathcal{L}}$ is automatic. The
quota search tree that witnesses the connectivity of $\mathcal{D}$ represents
a construction of part of the transition map $\delta$ for $\mathcal{D}$, but
there will be transitions that need assigning. The remaining transitions can
be assigned completely arbitrarily, subject to the homomorphism constraint
(2). This not only characterizes the sizes of the Myhill-Nerode classes that
can arise in a DFA recognizing $\mathcal{L}$, it yields an efficient algorithm
for constructing all DFAs realizing those sizes, when the “enough arrows”
condition holds. We refer to this process as quota-based DFA expansion.

We emphasize that satisfying the connectivity and enough arrows conditions
does _not_ guarantee connectivity of a given extension structure. In
particular, it is not true that if $\mathcal{D}$ is a connected DFA, and
$\mathcal{D}^{\prime}\to\mathcal{D}$ is a quotient map with preimage sizes
satisfying (1), then $\mathcal{D}^{\prime}$ is connected. But the existence of
some connected $\mathcal{D}^{\prime}$ is guaranteed.

#### Example: the Fibonacci language

At the top of Figure 2 is the minimal DFA $\mathcal{D}_{\mathcal{L}}$
recognizing the “Fibonacci language” $\mathcal{L}$ of strings over
$\Sigma=\\{a,b\\}$ without two consecutive $b$’s. We expand this DFA to obtain
one with Myhill-Nerode class sizes $3$, $2$ and $3$ respectively, which
satisfies the “enough arrows” condition (1) since

$(1\ 0\ 0)+(3\ 2\ 3)\begin{pmatrix}1&1&0\\\ 1&0&1\\\ 0&0&2\end{pmatrix}=(6\ 3\
8)\geq(3\ 2\ 3).$

Select an initial node for $\mathcal{D}$ which maps down to the initial node
of $\mathcal{D}_{\mathcal{L}}$, and do a quota search; the red arrows in the
lower diagram in Figure 2 show the results of a (breadth-first) quota search.
This leaves some remaining transitions which can be filled in arbitrarily, as
long as they map to the correct class. One set of choices for these arrows is
shown in green.

The enough arrows theorem allows us to precisely characterize the possible
class size vectors $(x,y,z)$. (1) requires

$(1+x+y,x,y+2z)\geq(x,y,z)$

coordinatewise; the first and last of these are vacuous (in general, nodes
with self-loops give a vacuous constraint). So the necessary and sufficient
condition for $(x,y,z)$ to be the Myhill-Nerode class sizes of an automaton
recognizing $\mathcal{L}$ is simply that $x\geq y\geq 1$.

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. (a) | $3$$2$$3$ababa,b
---|---
(b) | ababababababa,ba,b

Quota-based DFA construction achieves the goal of effectively generating
random connected DFAs recognizing $\mathcal{L}$, with specified Myhill-Nerode
equivalence class sizes, in such a way that any connected DFA can in principle
be produced. The Myhill-Nerode theorem guarantees that any connected DFA
$\mathcal{D}$ recognizing $\mathcal{L}$ has a quotient map down to the
(connected) minimal DFA $\mathcal{D}_{\mathcal{L}}$. The connectivity of
$\mathcal{D}$ is witnessed by some search tree $\mathcal{T}$ in the universal
path cover of $\mathcal{D}$. If we randomize QueueExtract so that it returns a
randomly selected element of $Q$, we guarantee that quota search can return
$\mathcal{T}$ as the search forest. (At this point, connectivity of
$\mathcal{D}$ implies that the Myhill-Nerode class sizes must satisfy the
“enough arrows” condition.) By assigning the remaining transitions randomly,
we guarantee that quota-based DFA expansion can produce $\mathcal{D}$. This
proves the following theorem:

######  Theorem 4 (universality of quota-based DFA expansion).

Let $\mathcal{L}$ be a regular language, and let $\mathcal{D}$ be a connected
DFA recognizing $\mathcal{L}$. Then:

  * •

$\mathcal{D}$ can be constructed from the minimal DFA
$\mathcal{D}_{\mathcal{L}}$ by quota-based DFA expansion;

  * •

the Myhill-Nerode equivalence class sizes of $\mathcal{D}$ must satisfy the
“enough arrows” condition, with the start portfolio being one copy of the
initial state of $\mathcal{D}$.

We remark that even when $\mathcal{T}$ is chosen uniformly from the set of
quota trees achieving the given M-N class sizes, the resulting DFA is not
sampled uniformly from the connected DFAs with these class sizes, as different
DFAs admit different numbers of spanning trees. In principle this method could
be combined with standard methods such as Markov Chain Monte Carlo. Efficient
uniform DFA generation is a topic for further research.

###  $k$ shortest paths

It turns out that quota search very naturally solves the problem of
constructing a tree which contains the $k$ shortest (or lightest, if edges are
weighted) paths from a source node (or portfolio) to vertices in a graph
$G$.777See [5] for efficient algorithms and numerous references. Numerous
clever methods have been developed in connection with this problem; these are
no doubt also applicable in the more general context of quota search. For
example, when the edge weights are nonnegative, a solution is to use Dijkstra
quota search (DQS) with all quotas initialized to $k$.

For simplicity we assume that the source portfolio is a single vertex $\ast$,
so we’re building a tree $T$. Viewed as operating in the universal path cover
$\tilde{G}$, for each encounter with a vertex $v=f(t(e))$, DQS keeps track of
the distance from $\tilde{\ast}$ to a corresponding lift $\tilde{v}$ in
$\tilde{G}$. The edge $\tilde{e}$ de-queued at each step extends a path
$\tilde{p}$ in $T$ to a path $\tilde{p}\tilde{e}$ which minimizes this
distance; $\tilde{e}$ is added to $T$ if $q(v)>0$. The point now is that if we
have a path to $v$ which is among the $k$ lightest, then we may assume all
initial subpaths are among the lightest $k$ paths to their corresponding
endpoints, and are in $T$ by construction. Thus, by setting the quota at every
vertex to $k$, we are guaranteed that the quota tree consists of a set of $k$
lightest paths to all vertices.

DQS also solves the network configuration problem in section 2, although since
we are minimizing the number of edges in paths rather than their weighted
lengths, breadth-first quota search gives a simpler solution. As remarked
earlier, the coupon problem described in section 2 is an example of the
minimum quota arborescence problem; its solution requires an analogue of
Edmonds’ algorithm [4], which we will discuss in section 9.

##  7\. Counting quota trees

The enumeration of spanning trees is well understood. The most fundamental
result, the matrix-tree theorem, expresses the number of (directed) spanning
trees of a graph $G$ as a principal minor of the Laplacian of $G$. As a
special case, one obtains Cayley’s classical formula that the complete graph
$K_{n}$ has $n^{n-2}$ spanning trees with a specified root. These turn out to
be special cases of a more general result for quota trees.

As usual, $G$ is a directed (multi)graph, possibly with loops, having $m_{ij}$
distinct edges from vertex $i$ to vertex $j$. Let $q:V(G)\to\mathbf{Z}$ be a
quota function, $s:V(G)\to\mathbf{Z}$ a start portfolio, and $M=(m_{ij})$ the
adjacency matrix of $G$. The following symbol is indispensable in expressing
counts of quota trees.

Given a directed multigraph $G$ with $n\times n$ adjacency matrix
$M=(m_{ij})$, and $n$-long vectors $\mathbf{a}=(a_{i})$ and
$\mathbf{b}=(b_{i})$, define the quota symbol

(3) $\left\\{\begin{array}[]{c}\mathbf{a}\\\
\mathbf{b}\end{array}\right\\}_{G}:=\det
M(\mathbf{a},\mathbf{b})\prod_{i}\binom{a_{i}}{b_{i}}(a_{i})^{-1},$

where the binomial coefficient $\binom{n}{k}$ is zero unless $0\leq k\leq n$,
the matrix
$M(\mathbf{a},\mathbf{b})=\mathrm{diag}(\mathbf{a})-M\mathrm{diag}(\mathbf{b})$,
and for any index $i$ with $a_{i}=0$ we omit the factor $a_{i}^{-1}$ and
delete the corresponding row and column of $M(\mathbf{a},\mathbf{b})$. (We
remark that loops in $G$ do not affect $M(\mathbf{a},\mathbf{b})$ but do
affect the binomial coefficients.)

######  Theorem 5 (counting quota forests).

Let $G$, $q$ and $s$ be as above. As in the enough arrows condition (1), set
$\mathbf{in}_{j}=\sum_{i}q_{i}m_{ij}=\mathbf{q}M$ where $M=(m_{ij})$ is the
adjacency matrix of $G$. Then the number of quota forests with quota
$\mathbf{q}$ and start portfolio exactly $\mathbf{s}$ is given by

$\left\\{\begin{array}[]{c}\mathbf{in}\\\
\mathbf{q}-\mathbf{s}\end{array}\right\\}_{G}.$

The determinant arising in this theorem has a natural combinatorial
interpretation, which we will need. It represents the (weighted) counts of
spanning forests of the subgraph of $G$ determined by the support of $q$,
relative to the start portfolio $s$. In particular, the determinant is nonzero
precisely when the triple $(G,q,s)$ is connected. To state this precisely,
given weights on the edges and vertices of a graph, define the weight of a
tree to be the weight of its root times the product of the weights of the
edges it contains, and the weight of a forest to be the product of the weights
of its component trees.

######  Theorem 6 (matrix interpretation).

$\det M(\mathbf{in},\mathbf{q}-\mathbf{s})$ is the sum of the weights of all
spanning forests of $G$, where vertex $i$ has weight $s_{i}$, and an edge
$i\to j$ has weight $q_{j}-s_{j}$.

This result follows immediately from the following “matrix-forest theorem,”
which is equivalent to (but much more symmetric than) the usual “matrix-tree
theorem” [15, Thm. 5.6.4]:

######  Theorem 7 (matrix-forest).

Let $G$ be a directed multigraph with adjacency matrix $M=(m_{ij})$. Define
the Laplacian $\Delta G$ to be $\mathrm{diag}(\mathbf{in})-M$, where
$\mathbf{in}_{j}=\sum_{i}m_{ij}$. Then for indeterminates
$\mathbf{s}=(s_{i})$, $\det(\mathrm{diag}(\mathbf{s})+\Delta G)$ is the sum of
the weights of all spanning forests of $G$, where the weight of an edge $i\to
j$ is $m_{ij}$ and the weight of a vertex is $s_{i}$.

In particular, for any subset $I$ of the vertices, let $s_{I}$ denote the
monomial $\prod_{i\in I}s_{i}$. Then the coefficient
$[s_{I}]\det(\mathrm{diag}(\mathbf{s})+\Delta G)$ is the sum of the (edge)
weights of all spanning forests of $G$ with root set equal to $I$.

######  Corollary 8 (enough arrows).

A triple $(G,q,s)$ admits an exact quota forest if and only if $q(v)\geq s(v)$
for each $v$, $(G,q,s)$ is connected, and the enough arrows condition holds at
each vertex.

###### Proof.

The $i$-th binomial coefficient in Theorem 5 is nonzero precisely when the
local “enough arrows” condition holds at the $i$-th vertex and $q_{i}\geq
s_{i}$. By Theorem 7, the determinant in Theorem 5 is nonzero precisely when
there exists at least one spanning forest of (the support of $q$ in) $G$ whose
roots are contained in the support of $s$; that is, when $(G,q,s)$ is
connected. The enough arrows theorem now follows immediately from Theorem 5. ∎

We remark that the quota symbol simultaneously generalizes binomial
coefficients and spanning trees. When the graph $G$ has one vertex and no
edges, the quota symbol is a single binomial coefficient. On the other hand,
for general $G$, when the quotas are all $1$, the quota symbol
$\left\\{\begin{array}[]{c}\mathbf{in}\\\
\mathbf{q}-\mathbf{s}\end{array}\right\\}_{G}$ counts spanning forests (or
trees) of $G$. So it is not surprising that the symbol satisfies a recurrence
which reduces in the former case to Pascal’s rule, and in the latter case to
the recurrence for counting spanning trees by deleting and contracting an
edge:

$\\#T(G)=\\#T(G\setminus e)+\\#T(G/e).$

While we won’t need it here, we state the recurrence for completeness; its
proof is implicit in the proof of Theorem 13.

######  Theorem 9 (quota symbol recurrence).

The quota symbol (3) can be computed recursively as follows:

$\left\\{\begin{array}[]{c}\mathbf{a}\\\
\mathbf{b}\end{array}\right\\}_{G}=\left\\{\begin{array}[]{ll}0,&\textrm{unless
$\mathbf{0}\leq\mathbf{b}\leq\mathbf{a}$ and $\mathbf{a}\geq\mathbf{b}M$, in
which case:}\\\ 1,&\textrm{if $\mathbf{b}=\mathbf{0}$; else}\\\ 0,&\textrm{if
$\mathbf{a}=\mathbf{b}M$;}\end{array}\right.$

otherwise

$\left\\{\begin{array}[]{c}\mathbf{a}\\\
\mathbf{b}\end{array}\right\\}_{G}=\left\\{\begin{array}[]{c}\mathbf{a}-\delta_{i}\\\
\mathbf{b}-\delta_{i}\end{array}\right\\}_{G}+\left\\{\begin{array}[]{c}\mathbf{a}-\delta_{i}\\\
\mathbf{b}\end{array}\right\\}_{G}$

where $\delta_{i}$ is the vector with a $1$ in the $i$-th position and $0$
elsewhere, and $i$ is an index such that $a_{i}>(\mathbf{b}M)_{i}$.

Corresponding to the two variants of quota search we have described, one might
also ask for the number of at-most quota forests. (Of course the answers agree
for trees, but sometimes one or the other expression is easier to evaluate.)

######  Corollary 10 (counting at most quota forests).

Fix $G$, $\mathbf{q}$ and $\mathbf{s}$ as in Theorem 5. The number of
$(G,\mathbf{q},\mathbf{s}^{\prime})$-quota trees with a start portfolio
$\mathbf{s}^{\prime}$ satisfying $\mathbf{s}^{\prime}\leq\mathbf{s}$
coordinatewise is given by

$\left\\{\begin{array}[]{c}\mathbf{in}+\mathbf{s}\\\
\mathbf{q}\end{array}\right\\}_{G}.$

Before proving these results, we pause to consider some special cases. We will
let $Q_{=}(G,q,s)$ denote the number of quota forests of $G$ with quota $q$
and start portfolio exactly $s$; similarly $Q_{\leq}(G,q,s)$ will count quota
forests with start portfolio at most $s$.

#### Example: one-vertex graphs

Let $R_{k}$ denote the $k$-leaf rose, having a single vertex and $k$ loops; in
this case, quotas and start portfolios are just scalars $q$ and $s$. By
Theorem 5 and Corollary 10, we find

$Q_{=}(R_{k},q,s)=\left\\{\begin{array}[]{ll}[q=s]&\textrm{if\ }kq=0;\\\
\frac{s}{q}\binom{kq}{q-s}&\textrm{otherwise;}\end{array}\right.$
$Q_{\leq}(R_{k},q,s)=\left\\{\begin{array}[]{ll}[q=0]&\textrm{if\ }kq+s=0;\\\
\frac{s}{kq+s}\binom{kq+s}{q}&\textrm{otherwise.}\end{array}\right.$

It’s useful to check the at-most counts. For $k=0$,
$Q_{\leq}(R_{0},q,s)=\binom{s}{q}$ as expected, since we just select which of
the $s$ possible starts get chosen.
$Q_{\leq}(R_{1},q,s)=\frac{s}{q+s}\binom{q+s}{q}=\binom{q+s-1}{q}$, as it
counts the number of $s$-tuples $(T_{1},\ldots,T_{s})$ where each $T_{i}$ is a
(possibly empty) directed path graph and the total number of nodes is $k$. For
$k=2$ each $T_{i}$ is now a binary tree, and
$Q_{\leq}(R_{2},q,s)=\frac{s}{2q+s}\binom{2q+s}{q}$ is equal to the entry
$C(q+s-1,q)$ in the so-called Catalan triangle [12, A009766]; when $s=1$ this
just counts binary trees on $n$ nodes: $1,1,2,5,14,\ldots$. (For higher $k$ we
get $k$-th level Catalan numbers; see [12, A069269].)

We remark that the ordinary $q$-generating function for $Q_{\leq}(R_{k},q,s)$
is a hypergeometric series:

$\displaystyle\sum_{q}Q_{\leq}(R_{k},q,s)z^{q}$ $\displaystyle=$
$\displaystyle 1+\sum_{q\geq 1}\frac{s}{kq+s}\binom{kq+s}{q}z^{q}$
$\displaystyle=$
$\displaystyle{}_{k}F_{k-1}\left(\begin{array}[]{c}\frac{s}{k},\frac{s+1}{k},\ldots,\frac{s+k-1}{k}\\\
\frac{s+1}{k-1},\frac{s+2}{k-1},\ldots,\frac{s+k-1}{k-1}\end{array}\bigg{|}\frac{k^{k}}{(k-1)^{k-1}}z\right).$

We also note the following relationship, which falls in the category of
“combinatorial reciprocity laws” as described by Stanley [14]. When we
formally substitute $-s$ in the expression for $Q_{\leq}(R_{k},q,s)$, we
obtain $\frac{-s}{kq-s}\binom{kq-s}{q}$. When $kq\leq s$, it turns out that
this counts (up to an alternating sign) the number of ways to select a set of
$q$ disjoint copies of the path graph $P_{k}$ in the cycle graph $C_{s}$. When
$k=1$, this reduces to the usual binomial coefficient reciprocity law, namely
that $\binom{-s}{q}$ and $\binom{s}{q}$ count selections of $q$ objects from
$s$ objects respectively with and without replacement. For general $k$, this
gives a combinatorial reciprocity law for higher-order Catalan triangles.

#### Example: quota trees over $K_{n}$

It is natural to consider a start portfolio consisting of a single vertex, as
this situation arises in the context of spanning trees as well as
deterministic automata. We view the complete graph $G=K_{n}$ as a directed
graph with adjacency matrix $J_{n}-I_{n}$ (where $J_{n}$ is as usual the
matrix of all $1$’s.) We remark that quota trees over $K_{n}$ can be viewed as
trees colored with $n$ colors, having $q_{i}$ nodes of color $i$, such that a
node cannot share a color with either its parent or any of its siblings. In
the special case of a constant quota $q$ at each vertex, we get an especially
nice answer: the number of quota trees over $K_{n}$, with a given start vertex
and constant quota $q$ at each vertex, is

$\binom{(n-1)q}{q}^{n}\frac{n^{n-2}}{(n-1)^{n-1}((n-2)q+1)}.$

Taking $q=1$ yields $n^{n-2}$, so we recover as a special case Cayley’s
formula for the number of spanning trees of $K_{n}$ with a specified root.

#### Example: quota trees over $K_{n}$ with loops

Loops don’t enter into spanning trees, but are relevant to quota forests. We
remark that loops do not affect the determinant in the definition of the quota
symbol (3), but they do affect the rest of the terms. As an example, let
$K_{n}^{\circ}$ be the graph $K_{n}$ with a loop added at each vertex, so that
the adjacency matrix is the all-ones matrix. Its quota trees correspond to
tree-colorings as in the preceding example, except that a node is now allowed
to share a color with its parent. When the quota is a constant $q$ at each
root, the number of quota trees starting at any fixed root works out to be

$\binom{nq}{q}^{n}\frac{1}{n(q(n-1)+1)}.$

For $n=2$, the number of quota trees with quotas $(i,j)$ and start portfolio
at most $(1,0)$ is given by

$\displaystyle\left\\{\begin{matrix}i+j+1&i+j\\\
i&j\end{matrix}\right\\}_{K_{2}^{\circ}}:$
$\begin{array}[]{c|cccccc}i\diagdown j&0&1&2&3&4&5\\\ \hline\cr
0&1&0&0&0&0&0\\\ 1&1&1&1&1&1&1\\\ 2&1&3&6&10&15&21\\\ 3&1&6&20&50&105&196\\\
4&1&10&50&175&490&1176\\\ 5&1&15&105&490&1764&5292\\\ \end{array}$

Up to indexing, these are the Narayana numbers ([12, A001263], [15, ex.
6.36]); they appear in numerous contexts (e.g. Dyck paths counted by length
and peak, antichains in the poset $2*(k-1)*(n-k)$, the $h$-vector of the dual
simplicial complex to the associahedron $A_{n}$, etc.)

Notice that the diagonals in the preceding table add up to the Catalan
numbers; this is a special case of a very general fact. Let $\pi:\tilde{G}\to
G$ be a (not necessarily universal) out-covering, $q$ and $\tilde{q}$ quotas
on $G$ and $\tilde{G}$ respectively, and $s$ and $\tilde{s}$ start portfolios
such that $s(v)=\sum_{\tilde{v}\in\pi^{-1}(v)}\tilde{s}(\tilde{v}).$ By the
discussion in section 3, given a $(G,q,s)$ quota forest $F$, once we lift the
root of each tree to an arbitrary preimage in $\tilde{G}$, this determines a
unique lift of $F$. Thus, counting quota trees in $\tilde{G}$ refines the
counting of quota forests in $G$ in the sense that

$Q_{=}(G,q,s)=\sum_{\tilde{q}}Q_{=}(\tilde{G},\tilde{q},\tilde{s}),$

where the sum ranges over all (achievable) quotas $\tilde{q}$ such that

$\sum_{\tilde{v}\in\pi^{-1}(v)}\tilde{q}(\tilde{v})=q(v)$

for all $v\in V(G)$.

Returning to the current example, since $K_{2}^{\circ}$ has constant outdegree
2, one can construct an out-covering $K_{2}^{\circ}\to R_{2}$. So the number
of quota trees in $K_{2}^{\circ}$ where the quota has $l_{1}$-norm $n$ is the
number of quota trees in $R_{2}$ with quota $n$, which we have already seen is
given by a Catalan number.

More generally, there are five essentially different ways to write down a
strongly connected rooted two-vertex graphs with outdegree 2. In each case,
the diagonal sums of the quota tree counts are Catalan numbers, but the quotas
reflect different interesting properties of the binary trees. All five cases
appear as different entries in Sloane [12]; we list these as the first five
rows of Table 1, which collects a number of two-vertex graphs whose quota tree
counts have already been studied in other contexts.

$\begin{array}[]{c|l}G&\textrm{Corresponding entry in Sloane
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}\pgfsys<EMAIL_ADDRESS>related to generalized Catalan sequences}\\\ \hline\cr\end{array}$ Table 1.
Some two-vertex graphs whose quota tree counts appear, possibly re-indexed, in
Sloane’s Encyclopedia of Integer Sequences. In each case, the start portfolio
is one copy of each filled-in vertex.

#### Example: quota forests over $K_{n}$ with symmetric roots

It is even more symmetrical to count quota forests over $K_{n}$, where we take
both $q$ and $s$ to be constant over all vertices. The quota tree count is

$\binom{(n-1)q}{q-s}^{n}\frac{(nq-s)^{n-1}s}{(n-1)^{n-1}q^{n}}.$

In particular, if $q=s$, the count is exactly one, reflecting the fact that
each tree in the forest is an isolated node.

#### Example: path graphs

The path graph $P_{n}$ has only a single spanning tree from any root; however,
quota trees are much more interesting. Intuitively, we have $n$ parallel
semitransparent panes of glass; at each one, a laser beam can pass through,
reflect, both, or neither. When we fire a beam into one pane, the trajectory
is then a tree immersing into $P_{n}$, whose quotas count the number of times
each pane is encountered. If all quotas are $q$, and the beam is initially
fired into one of the outer panes, the number of quota trees works out to

$\left(\frac{1}{2}\binom{2q}{q}\right)^{n-2}=a_{q}^{n-2},$

where $a_{q}=(1,3,10,35,126,\cdots)$ is sequence A001700 in Sloane. When we
fire the laser into any one of the internal panes, the answer works out to
$c_{q}a_{q}^{n-3}$, where $c_{q}=\binom{2q+1}{q}/(2q+1)$ is the $q$-th Catalan
number.

#### Example: cycle graphs

With the notation of the preceding example, the cycle graph $C_{n}$ has

$\binom{2q}{q}^{n}\frac{n}{2^{n-1}(q+1)}=\frac{2n\,a_{q}^{n}}{q+1}$

quota trees from any fixed root, when all vertex quotas are set to $q$.

#### Proof of Theorem 5

The strategy is to write down a functional equation jointly satisfied by the
generating functions for quota trees rooted at all vertices of $G$, and solve
it using the multivariate Lagrange inversion formula.888Problem 3.3.42 in [9]
is very similar; however, it counts trees rather than forests, and omits the
immersion condition. Following [9], let $R$ be a ring with unity,
$R[[\boldsymbol{\lambda}]]_{1}$ the set of formal power series in
$\boldsymbol{\lambda}=(\lambda_{1},\ldots,\lambda_{n})$ over $R$ with
invertible constant term, and $R((\boldsymbol{\lambda}))$ the ring of formal
Laurent series over $R$.

######  Theorem 11 (Multivariate Lagrange).

[9, Th. 1.2.9]
Suppose $\mathbf{w}=(w_{1}(\mathbf{t}),\ldots,w_{n}(\mathbf{t}))$ jointly
satisfy the functional equations
$w_{i}(\mathbf{t})=t_{i}\phi_{i}(\mathbf{w})$, where
$\mathbf{t}=(t_{1},\ldots,t_{n})$. Let $f(\boldsymbol{\lambda})\in
R((\boldsymbol{\lambda}))$ and
$\boldsymbol{\phi}=(\phi_{1}(\boldsymbol{\lambda}),\ldots,\phi_{n}(\boldsymbol{\lambda}))$,
where $\phi_{i}\in R[[\boldsymbol{\lambda}]]_{1}$. Then

$f(\mathbf{w}(\mathbf{t}))=\sum_{\mathbf{q}}\mathbf{t}^{\mathbf{q}}[\boldsymbol{\lambda}^{\mathbf{q}}]\left\\{f(\boldsymbol{\lambda})\boldsymbol{\phi}^{\mathbf{q}}(\boldsymbol{\lambda})\left\|\delta_{ij}-\frac{\lambda_{j}}{\phi_{i}(\boldsymbol{\lambda})}\frac{\partial\phi_{i}(\boldsymbol{\lambda})}{\partial\lambda_{j}}\right\|\right\\}.$

Given a graph $G$ with $n$ vertices, we will take $w_{i}(\mathbf{t})$ to be
the generating function

$w_{i}(\mathbf{t})=\sum_{T}\mathbf{t}^{\mathbf{q}(T)}=\sum_{T}t_{1}^{q_{1}(T)}\cdots
t_{n}^{q_{n}(T)},$

where $T$ ranges over all quota trees rooted at vertex $i$, and $q_{j}(T)$ is
the number of occurrences of vertex $j$ in $T$. The first observation is that
the $w_{i}$’s jointly satisfy the functional equation

(5) $w_{i}(\mathbf{t})=t_{i}\prod_{j}(1+w_{j}(\mathbf{t}))^{m_{ij}}$