A GENERIC APPROACH FOR SPARSE PATH PROBLEMS
Marc Pouly
Cork Constraint Computation Center, University College Cork, Ireland
Keywords:
Sparse path problems, Valuation algebras, Local computation, Tree-decomposition methods.
Abstract:
This paper shows how sparse path problems can be solved by tree-decomposition techniques. We analyse the
properties of closure matrices and prove that they satisfy the axioms of a valuation algebra, which is known
to be sufficient for the application of generic tree-decomposition methods. Given a sparse path problem where
only a subset of queries are required, we continually compute path weights of smaller graph regions and
deduce the total paths from these results. The decisive complexity factor is no more the total number of graph
nodes but the induced treewidth of the path problem.
1 INTRODUCTION
In recent years, a large number of formalisms for au-
tomated inference have been proposed. Typical ex-
amples are: probability potentials from Bayesian net-
works, Dempster-Shafer theory, different constraint
systems and logics, Gaussian potentials and density
functions, relational algebra, possibilistic formalisms
and many more. Inference based on these formalisms
is a computationally hard task which motivated the
introduction of tree-decomposition methods. But it
also turned out that they all share some common
algebraic properties which are pooled in the valua-
tion algebra framework (Shenoy and Shafer, 1990;
Kohlas, 2003). Based on this framework, a collec-
tion of generic tree-decomposition methods has been
derived. Thus, instead of re-inventing such inference
procedures for each different formalism, it is suffi-
cient to verify a small axiomatic system to gain access
to efficient generic procedures and implementations.
This is known as the local computation framework.
In parallel to these developments, tree-decomposition
methods were successfully applied for the solution
of sparse linear systems. For equation systems over
fields, it has been shown that these approaches, which
aim at the minimization of fill-ins in matrices, are
subject to the valuation algebra framework, and that
the according tree-decomposition procedures are spe-
cializations of the generic local computation methods
(Kohlas, 2003; Pouly and Kohlas, 2010). However,
many important applications can be reduced to the so-
called algebraic path problem which requires to solve
a fixpoint equation system with values from a semir-
ing. Essentially, there are two approacheswhich focus
on solving sparse fixpoint systems over semirings by
tree-decomposition techniques: Similar to the inverse
matrix in case of linear systems over fields, the quasi-
inverse matrix provides a solution to a semiring fix-
point system. Such quasi-inverse matrices always ex-
ist for closed semirings (Lehmann, 1976) and can be
computed by the well-known Floyd-Warshall-Kleene
algorithm. (Radhakrishnan et al., 1992) combined this
insight with LDU decomposition for semiring matri-
ces (Backhouse and Carr´e, 1975) to obtain a tree-
decomposition algorithmfor sparse fixpoint equations
over closed and idempotent semirings. Again, this ap-
proach is covered by the local computation frame-
work with the fixpoint equations satisfying the val-
uation algebra axioms. Alternatively, (Chaudhuri and
Zaroliagis, 1997) proposed a second method for the
particular problem of computing shortest distances. In
this paper, we will identify the algebraic requirements
of this second method and show that it complies with
the valuation algebra framework. This enables the ap-
plication of existing, generic inference procedures for
the solution of sparse path problems. Moreover, we
generalize this idea from shortest distances to a wider
class of semirings called Kleene algebras which fur-
ther includes other graph related path problems as for
example the computation of maximum capacities or
reliabilities, and also many other problems that are
not directly related to graphs but which can neverthe-
less be reduced to a path problem. We refer to (Rote,
1990) for an extensive listing of such examples.
197
Pouly M. (2010).
A GENERIC APPROACH FOR SPARSE PATH PROBLEMS.
In Proceedings of the 2nd International Conference on Agents and Artificial Intelligence - Artificial Intelligence, pages 197-202
DOI: 10.5220/0002702701970202
Copyright
c
SciTePress
2 VALUATION ALGEBRAS
The basic elements of a valuation algebra are so-
called valuations. Intuitively, a valuation can be re-
garded as a representation of knowledge about the
possible values of a set r of variables. It can be said
that each valuation φ refers to a finite set of variables
d(φ) ⊆ r called its domain. Further, let D be the power
set of r and Φ a set of valuations with their domains
in D. We assume the following operations in (Φ,D):
• Labeling: Φ → D; φ 7→ d(φ),
• Combination: Φ× Φ → Φ; (φ,ψ) 7→ φ ⊗ ψ,
• Projection: Φ×D → Φ; (φ,x) 7→ φ
↓x
for x ⊆ d(φ).
We further impose the following axioms on Φ and D:
1. Commutative Semigroup:Combination is associa-
tive and commutative.
2. Labeling: For φ, ψ ∈ Φ, d(φ ⊗ ψ) = d(φ) ∪ d(ψ).
3. Projection: For φ ∈ Φ and x ⊆ d(φ), d(φ
↓x
) = x.
4. Transitivity: For φ ∈ Φ and x ⊆ y ⊆ d(φ),
(φ
↓y
)
↓x
= φ
↓x
.
5. Combination: For φ, ψ ∈ Φ with d(φ) = x, d(ψ) =
y and z ∈ D such that x ⊆ z ⊆ x∪ y,
(φ⊗ ψ)
↓z
= φ⊗ ψ
↓z∩y
;
6. Domain: For φ ∈ Φ with d(φ) = x, φ
↓x
= φ.
7. Idempotency: Forφ ∈ Φ and x ⊆ d(φ), φ⊗φ
↓x
= φ.
These axioms require natural properties regarding
knowledge modelling. The first axiom indicates that
if knowledge comes in pieces, the sequence does not
influence their combination. The labeling axiom tells
us that the combination of valuations gives knowledge
over the union of the involved variables. Transitivity
says that projection can be performed in several steps,
and the combination axiom states that we either com-
bine a new piece to the already given knowledge and
focus afterwards to the desired domain, or we first cut
the uninteresting parts of the new knowledge out and
combine it afterwards. The domain axiom expresses
that trivial projection has no effect and finally, idem-
potency states that combining a piece of knowledge
with a part of itself gives nothing new.
Definition 1. A system (Φ,D) satisfying the axioms
1 to 6 is called a valuation algebra. If axiom 7 holds,
then it is called an idempotent valuation algebra.
A listing of formalisms that adopt the structure of
a valuation algebra was already given in the intro-
duction. Among them, the valuation algebras of re-
lations and crisp constraints are idempotent. We refer
to (Pouly, 2008; Kohlas, 2003) for further examples.
2.1 Generic Inference Problem
The computational interest in valuation algebras is
stated by the inference problem. Given a set of val-
uations {φ
1
,. .. , φ
n
} ⊆ Φ called knowledgebase and a
set of queries x = {x
1
,. .. , x
s
}, the inference problem
consists in computing
φ
↓x
i
= (φ
1
⊗ ··· ⊗ φ
n
)
↓x
i
(1)
for 1 ≤ i ≤ s. For example, if the knowledgebase
consists of the CPTs from a Bayesian network, then
the inference problem reflects the computation of
marginals from the join probability distribution. If the
knowledgebase models a constraint system, then the
inference problem with the empty query corresponds
to satisfiability, if the knowledgebase contains rela-
tions, then the inference problem mirrors query an-
swering in relational databases.
The complexity of combination and projection
generally depends on the size of the factor domains
and often shows an exponential behaviour. Accord-
ing to axiom 2 and 3, the domains of valuations grow
under combination and shrink under projection. Effi-
cient inference algorithms must therefore confine in
some way the size of intermediate results, which can
be achieved by alternating the operations of combina-
tion and projection. This is the promise of local com-
putation. The valuation algebra axioms are sufficient
for the definition of general local computation pro-
cedures which solve the inference problem indepen-
dently of the underlying formalism. These algorithms
include fusion (Shenoy, 1992) and bucket elimination
(Dechter, 1999) for inference problems with a single
query, and the Shenoy-Shafer architecture (Shafer and
Shenoy, 1988) for multiple queries. If (some weaker
condition of) axiom 7 is present, other local computa-
tion architectures with a more efficient scheduling of
computations can be derived (Lauritzen and Spiegel-
halter, 1988; Jensen et al., 1990; Kohlas, 2003).
3 LOCAL COMPUTATION
Local computation methods are usually described as
message-passing schemes on covering join trees (also
called tree-decomposition):
Definition 2. A join tree is a labeled tree (V,E,λ,D)
whose labeling function λ : V → D satisfies the run-
ning intersection property, i.e. for two nodes v
1
,v
2
∈
V, if X ∈ λ(v
1
) ∩ λ(v
2
), then X is contained in every
node label on the unique path between v
1
and v
2
. A
join tree covers the inference problem if
• for each factor φ
i
, there is a node v ∈ V such that
d(φ
i
) ⊆ λ(v),
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198
• for each query x
i
, there is a node v ∈ V such that
x
i
⊆ λ(v).
A detailled description of all local computation
methods can be found in (Pouly, 2008). Here, we only
cite the main theorem that all multi-query procedures
have in common:
Theorem 1. At the end of the message-passing, each
node i contains φ
↓λ(i)
.
If the query x
j
is covered by some node i ∈ V,
we obtain the query answer as a consequence of the
transitivity axiom by one last projection
φ
↓x
j
=
φ
↓λ(i)
↓x
j
. (2)
In local computation methods, all computations
take place in the join tree nodes. The domains of in-
termediate factors, which determine the complexity
of combination and projection, are therefore bounded
by the largest node label in the join tree. This mea-
sure is called treewidth. In other words, the smaller
the treewidth, the more efficient is local computa-
tion. Finding a covering join tree with a minimum
treewidth is NP-complete (Arnborg et al., 1987). But
there are good heuristics (Lehmann, 2001).
4 ALGEBRAIC PATH PROBLEM
The algebraic path problem aims at the unification of
various path problems in terms of the solution of a
generic fixpoint equation with valuesfrom a semiring.
Definition 3. A tuple hE,+, × , 0, 1i with binary oper-
ations + and × is called semiring if:
• + and × are associative and + is commutative;
• for a,b,c ∈ E: a× (b + c) = a × b + a× c;
• for a,b,c ∈ E: (a+ b) × c = a × c + b × c;
• + and × have neutral elements 0 and 1;
• a× 0 = 0× a = 0 for all a ∈ E.
If the semiring is idempotent, satisfying a+ a = a for
all a ∈ E, then the following relation is a partial order:
a ≤ b if, and only if a+ b = b. (3)
Typical examples of idempotent semirings are the
Boolean semiring h{0,1},max,min, 0, 1i, the trop-
ical semiring hN ∪ {0,∞},min,+, ∞,0i, the arctic
semiring hR ∪ {−∞},max,+, −∞, 0i, the probabilis-
tic semiring h[0,1],max,·,0,1i and the bottleneck
semiring hR ∪ {+∞,−∞}, max,min,−∞,∞i.
For n ∈ N we next consider the set of n × n ma-
trices M ∈ M (E, n) with values from an idempotent
semiring E. This set forms itself an idempotent semir-
ing. We define the power sequence of matrices:
M
(r)
= I + M + M
2
+ ... + M
r
. (4)
If we interpret M as the adjacency matrix of a graph
with edge weights from the tropical semiring, then
M
(r)
corresponds to the shortest distances containing
at most r edges. Consequently, we obtain the shortest
distances between each pair of graph nodes by
M
r≥0
M
r
= I + M + M
2
+ ... (5)
Observe that this infinite sum is not always defined.
But if the above sequence converges with a suitable
notion of topology (Gondran and Minoux, 2008), it
can be shown that its limit M
∗
always satisfies
M
∗
= MM
∗
+ I = M
∗
M + I. (6)
This motivates the following definition.
Definition 4. The algebraic path problem consists in
solving the fixpoint equation X = MX +I = XM+I.
There may be no solution, one solution or in-
finitely many solutions to this equation. However, for
computational purposes it is often convenientto avoid
this difficulty by ensuring the existence of a solu-
tion axiomatically. Such semirings are called closed
semirings (Lehmann, 1976) and they provide for each
a ∈ E an element a
∗
∈ E such that a
∗
= aa
∗
+ 1 =
a
∗
a+1. Moreover,given a matrix M with values from
a closed semiring, it is possible to compute M
∗
induc-
tively from the values of the underlying semiring:
• For n = 1 we define
a
∗
=
a
∗
.
• For n > 1 we decompose the matrix M into sub-
matrices B,C, D, E such that B and E are square
and define
B C
D E
∗
=
B
∗
+ B
∗
CF
∗
DB
∗
B
∗
CF
∗
F
∗
DB
∗
F
∗
(7)
where F = E + DB
∗
C.
M
∗
as a result of this construction is a solution to
the algebraic path problem (Lehmann, 1976). In other
words, matrices over closed semirings themselves
form a closed semiring. The proof of this statement
affords the Floyd-Warshall-Kleene algorithm which
performs this task in time O(n
3
). But to derive a val-
uation algebra, we first introduce an even more struc-
tured class of semirings called Kleene algebras which
are closed and idempotent semiring with an additional
monotonicity property:
Definition 5. A tuple hE, +, ×, ∗, 0, 1i with an unary
operation ∗ is called Kleene algebra if:
1. hE,+,×,0,1i is an idempotent semiring;
A GENERIC APPROACH FOR SPARSE PATH PROBLEMS
199
2. a
∗
= 1+ a
∗
a = 1+ aa
∗
for a ∈ E;
3. ax ≤ x implies that a
∗
x ≤ x for a,x ∈ E;
4. xa ≤ x implies that xa
∗
≤ x for a,x ∈ E.
For example, the Boolean semiring is a Kleene al-
gebra with 0
∗
= 1
∗
= 1. The tropical semiring of non-
negative integers is a Kleene algebra with a
∗
= 0 for
all a ∈ N ∪ {0,∞}. The arctic semiring is a Kleene al-
gebra with a
∗
= 0 for a ≥ 0 and a
∗
= −∞ otherwise.
The probabilistic semiring is a Kleene algebra with
a
∗
= 1 for a ∈ [0,1], and the bottleneck semiring is a
Kleene algebra with a
∗
= ∞ for all a ∈ R ∪ {−∞,∞}.
Kleene algebras have many interesting properties.
Most important among them are the closure proper-
ties: a ≤ a
∗
, a
∗∗
= a
∗
and a ≤ b implies that a
∗
≤ b
∗
,
but also 1 = 1
∗
≤ a
∗
and
(a+ b)
∗
= (a
∗
+ b)
∗
= (a
∗
+ b
∗
) (8)
for all a,b ∈ E. We refer to (Kozen, 1994) for the
proofs of these and other elementary properties.
Since Kleene algebras are closed semirings, we
may compute M
∗
of a matrix M with values from a
Kleene algebra by the same algorithm and the result
again satisfies monotonicity (Conway, 1971). This
proves that matrices over Kleene algebras themselves
form a Kleene algebra, and as a further implication of
the monotonicity law, it confirms that computing M
∗
over the tropical semiring indeed gives shortest dis-
tances. Accordingly, the arctic semiring gives maxi-
mum capacities, the probabilistic semiring maximum
reliabilities and the Boolean semiring connectivities.
5 KLEENE VALUATIONS
We prove in this section that closures matrices satisfy
the valuation algebra axioms. Let r = {X
1
,. .. , X
n
}
be a finite set of variables and D its powerset. For a
Kleene algebra hE,+,×,∗,0,1i and s ∈ D, we con-
sider labeled matrices M : s × s → E and refer to
d(M) = s as their domain. We then write M (E,s) for
the set of all labeled matrices with domain s ∈ D and
also define the set of all closures of labeled matrices
with domain in D as Φ = {M
∗
|M ∈ M (E,s) and s ∈
D}. We next introduce some operations in Φ start-
ing with the projection. For M
∗
∈ Φ, t ⊆ d(M
∗
) and
X,Y ∈ t,
(M
∗
)
↓t
(X,Y) = M
∗
(X,Y). (9)
This simply corresponds to the restriction of the ma-
trix M
∗
to the variables in t. It is easy to prove that
Φ is closed under projection, i.e. the restriction of a
closure matrix again results in a closure matrix. Intu-
itively, considering a subgraph of a graph with short-
est distances still contains shortest distances. Clearly,
the projection operator satisfies transitivity. We next
introduce the direct sum of labeled matrices: Let
M
1
∈ M (E,s) and M
2
∈ M (E,t) with s∩ t =
/
0 and
X,Y ∈ s∪t, we define
(M
1
⊕ M
2
)(X,Y) =
M
1
(X,Y) if X,Y ∈ s,
M
2
(X,Y) if X,Y ∈ t,
0 otherwise.
It follows from the inductive definition of M
∗
that the
closure operation distributes over the direct sum, i.e.
(M
1
⊕ M
2
)
∗
= M
∗
1
⊕ M
∗
2
. (10)
This allows us to define an operation of vacuous ex-
tension for M ∈ M (E,s) and s ⊆ t by M
↑t
= M ⊕ I
t−s
.
The application of the closure operation and vacu-
ous extension are interchangeable. It follows directly
from (10) and 1
∗
= 1 that
M
↑t
∗
=
M ⊕ I
t−s
∗
=
M
∗
⊕ I
t−s
=
M
∗
↑t
.
(11)
Thus, Φ is also closed under vacuous extension. Fi-
nally, we introduce a very intuitive combination rule
for elements in Φ. Imagine that we have two closure
matrices which express shortest distances in two pos-
sibly overlapping regions of a large graph. Then, the
shortest distance matrix for the unified region is found
by vacuously extending both matrices to their union
domain, taking the component-wise minimum which
corresponds to semiring addition and computing the
new shortest distances. Thus, for M
∗
1
,M
∗
2
∈ Φ with
d(M
∗
1
) = s and d(M
∗
2
) = t we define
M
∗
1
⊗ M
∗
2
=
(M
∗
1
)
↑s∪t
+ (M
∗
2
)
↑s∪t
∗
. (12)
We directly conclude from this definition that Φ is
closed under combination and also that combina-
tion is commutative. Proving associativity is more in-
volved but follows from (11) and (8). Furthermore,
this definition of combination also fulfills the combi-
nation axiom:
Lemma 1. If M
∗
1
,M
∗
2
∈ Φ with d(M
∗
1
) = s, d(M
∗
2
) = t
and s ⊆ z ⊆ s∪ t we have
(M
∗
1
⊗ M
∗
2
)
↓z
= M
∗
1
⊗ (M
∗
2
)
↓z∩t
. (13)
Decompose M
∗
1
⊗ M
∗
2
= ((M
∗
1
)
↑s∪t
+ (M
∗
2
)
↑s∪t
)
∗
with respect to z and t − z. The statement then follows
from (7). Finally, we observe that closure matrices
fulfill idempotency. For M
∗
∈ Φ with s ⊆ t = d(M
∗
),
M
∗
⊗ (M
∗
)
↓s
=
h
M
∗
+ ((M
∗
)
↓s
)
↑t
i
∗
= M
∗∗
= M
∗
.
This follows from the idempotency of addition and
the properties 1 ≤ a
∗
and a
∗∗
= a
∗
. Altogether, we
proved the following theorem:
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200
Theorem 2. Closures of Kleene valued matrices with
labeling, projection (9) and combination (12) satisfy
the axioms of a valuation algebra.
Every Kleene algebra therefore induces an idem-
potent valuation algebra of matrix closures. For
the particular case of the tropical semiring of non-
negative integers, this corresponds to the result of
(Chaudhuri and Zaroliagis, 1997) that implicitly used
the above properties by referring to Bellmann’s prin-
ciple of optimality. The final section shows how lo-
cal computation with valuation algebras of matrix clo-
sures are used for the solution of path problems.
6 SOLVING PATH PROBLEMS
Considering decompositions of large graphs or net-
works is very natural. A typical example is a road map
of Europe that is decomposed into smaller road maps
for each country. Thus, we assume an adjacency ma-
trix M of a large graph which is decomposed into a set
of matrices {M
1
,. .. , M
n
} taking values from a Kleene
algebra. These matrices correspond to the adjacency
matrices of some smaller graph regions. Represent-
ing the nodes of the total graph by the variable set
s = d(M) we obtain M = M
↑s
1
+ ··· + M
↑s
n
and using
the properties (8) and (11)
M
∗
=
M
↑s
1
+ ··· + M
↑s
n
∗
= M
∗
1
⊗ ··· ⊗ M
∗
n
.
Assume next that we are interested in some specific
path weights M
∗
(X,Y) for variables X,Y ∈ s. These
pairs of variables form the query set x and the task
consists in computing
(M
∗
)
↓{X,Y}
= (M
∗
1
⊗ ... ⊗ M
∗
n
)
↓{X,Y}
(14)
for each query {X,Y} ∈ x. This defines an inference
problem according to Section 2.1 which can be solved
by local computation. We first construct a join tree
covering all knowledgebase factors M
∗
i
and all queries
{X,Y} ∈ x and then execute a multi-query local com-
putation architecture. At the end of the message-
passing, the query answers are obtained from Equa-
tion (2). During the local computation process, two
messages are sent along each edge, and each mes-
sage is combined to some node content. For a join
tree (V,E,λ, D) we thus perform 2(|V| − 1) combi-
nations and 2(|V| − 1) projections. Combination is
surely the more costly operation since projection only
corresponds to matrix restriction. Using the algorithm
of Floyd-Warshall-Kleene we obtain the closure M
∗
of a matrix M in time O(|d(M)|
3
). Moreover, the
largest matrix domain that occurs during the local
computation process is bounded by the treewidth ω.
We obtain O(|V|ω
3
) for the time complexity of com-
puting Equation (14) since only matrices are stored,
the space complexity is O(|V|ω
2
). However, there is
an important issue regarding the treewidth complex-
ity. When dealing with path problems, people are of-
ten interested in a large number of paths. These query
sets may either be structured (e.g. single-source prob-
lems), or they may be arbitrary sets of queries. But
since the join tree must cover all queries, the treewidth
may grow significantly if a large number of queries
is present. In the worst case, when all possible path
weights have to be computed, the join tree consists of
a single node containing all variables. Except for such
extreme cases, this problem can be addressed by ex-
ploiting idempotency.If the valuation algebra is idem-
potent, it hold that
φ =
m
O
i=1
φ
↓λ(i)
. (15)
This was shown by (Kohlas, 2003). The following
lemma can be derived from this important result:
Lemma 2. Let (v
1
,. .. , v
k
) be a path in the join tree
from node v
1
to node v
k
with v
i
∈ V for 1 ≤ i ≤ k. We
then have for an idempotent valuation algebra
φ
↓λ(v
1
)∪...∪λ(v
k
)
=
k
O
i=1
φ
↓λ(v
i
)
. (16)
For an uncovered query {X,Y} we always find two
nodes v
1
,v
k
∈ V such that X ∈ λ(v
1
) and Y ∈ λ(v
k
).
For the path (v
1
,. .. , v
k
) connecting the two nodes, it
follows from the transitivity axiom that
φ
↓λ(v
1
)∪λ(v
k
)
=
k
O
i=1
φ
↓λ(v
i
)
!
↓λ(v
1
)∪λ(v
k
)
,
from which we obtain the query answer by one last
projection to {X,Y}. Computing this formula is yet
too expensive. But a clever application of the combi-
nation axiom uncovers the following algorithm:
1. initialize η := φ
↓λ(v
1
)
;
2. repeat for i = 1. .. k − 1
η := φ
↓λ(v
i+1
)
⊗ η
↓λ(v
1
)∪(λ(v
i
)∩λ(v
i+1
))
3. return η
↓{X,Y}
.
Theorem 3. The algorithm outputs φ
↓{X,Y}
for
{X,Y} ⊆ λ(v
1
) ∪ λ(v
k
) and v
1
,v
k
∈ V.
The domain of η will never exceed the union of
two node labels. Therefore, its time complexity is
driven by the double of the treewidth. Also, the num-
ber of combinations is bounded by the largest path
in the tree with has at most |V| nodes. We obtain for
A GENERIC APPROACH FOR SPARSE PATH PROBLEMS
201
the time complexity O(|V| · (2ω)
3
) = O(|V| · ω
3
). To
sum it up, given a factorized path problem and some
query set, we only consider the knowledgebase for the
construction of the join tree and ignore the query set.
This gives us the smallest treewidth that can be found
for this inference problem. After local computation,
each node v ∈ V contains φ
↓λ(v)
according to Theorem
1. For each query {X,Y} ∈ x, we search two nodes
v
1
,v
k
∈ V that cover this query {X,Y} ⊆ λ(v
1
)∪λ(v
2
)
and identify the path between them. Then, the above
query answering algorithm is executed. Doing so, all
queries in x can be computed with a total time com-
plexity of O(|x| · |V| · ω
3
). It is clear that for the com-
plete query set, we have |x| = |V|
2
/2 which makes
the time complexity of this algorithm worse than the
direct computation of M
∗
. However, by storing inter-
mediate results in the above algorithm, it is possible
to reduce the complexity of the all-pairs problem to
O(|V|
2
· ω
3
) which corresponds to the construction of
an optimum path tree (Pouly and Kohlas, 2010). Thus,
for extremely sparse path problems this approach may
still be worthwhile. If however only some smaller
subset of queries is required, the performance of this
algorithm is equal to other sparse matrix techniques
which proved their worth in many applications.
7 CONCLUSIONS
We have shown in this article that closure matrices
over Kleene algebras always induce a valuation alge-
bra. This uncovers many new and important instances
of the local computation framework which can now
be studied in this more general setting. It further gives
a general and efficient algorithm for the solution of
sparse path problems when either only a subset of
all queries are of interest, or if a high sparsity rate
is present. There is no need to specify the query set
in advance. The propagated join tree can thus be con-
sidered as the result of a pre-compilation, upon which
queries can later be answered in a dynamic way. This
approach does not assume any structure in the query
set which makes it more generally applicable than
other path algorithms. Finally, the query answering al-
gorithm is only based on the properties of idempotent
valuation algebras and can thus be applied to other
formalisms than matrices over Kleene algebras. This
however still deserves closer investigation.
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