Operator Calculus on Generalized Zeon Algebras:

Theory and Application to Multi-Constrained

Path Problems

Rene Schott∗, G. Stacey Staples†

November 12, 2012

Abstract

Classical approaches to routing problems invariably require construc-tion of trees and the use of heuristics to prevent combinatorial explosion.The operator calculus approach presented herein, however, allows suchexplicit tree constructions to be avoided. Introduced here is the notionof generalized zeon algebras and their associated operator calculus. Theinherent combinatorial properties of generalized zeons make them usefulfor routing problems by implicitly pruning the underlying tree structures.Moreover, through the use of generalized idempotent algebras, max-minoperators can be implemented for non-additive weights. As an applica-tion, an operator calculus approach to multi-constrained path problemsis described.

Keywords: shortest paths; message routing; operator calculus; semi-group algebras.

1 Introduction

Operator calculus (OC) methods on graphs have been developed in a number ofearlier works by Schott and Staples [10, 11, 12, 13]. The principal idea under-lying the approach is the association of graphs with algebraic structures whoseproperties reveal information about the associated graphs. By constructing the“nilpotent adjacency matrix” associated with a finite graph, information aboutself-avoiding structures (paths, cycles, trails, etc.) in the graph are revealed bycomputing powers of the matrix.

In the operator calculus approach, graded semigroup algebras are generatedby “null-square” elements such that properties of the algebra “sieve out” paths.In other words, cycles are removed from consideration automatically.∗Universite de Lorraine, IECN, UMR 7502, Vandoeuvre-les-Nancy, F-54506, France, email:

schott@loria.fr†Department of Mathematics and Statistics, Southern Illinois University Edwardsville,

Edwardsville, IL 62026-1653,USA, email: sstaple@siue.edu

1

Consider a directed graph G = (V,E) on n vertices such that associatedwith each edge (vi, vj) ∈ E is a vector weight wij = (wij1, . . . , wijm) ∈ Rm.The point w∗ = (w∗1 , . . . , w

∗m) ∈ X ⊂ Rm is referred to as a Pareto minimum of

X if there does not exist w = (w1, . . . , wm) ∈ X such that

(∀i) [wi ≤ w∗i ], and (1.1)(∃j) [wj < w∗j ]. (1.2)

Equivalently, one says that w∗ is nondominated from below.Defining the weight of a path in an edge-weighted graph as the sum of vector

weights of arcs contained in the path, a Pareto path is then a path whose weightis a Pareto minimum.

For the case m = 1, Dijkstra’s algorithm finds all single source minimumpaths in a directed graph on n vertices with nonnegative edge weights in O(n2)time [4]. The Bellman-Ford algorithm finds single source minimal paths indigraphs with arbitrary edge weights and runs in O(n |E|) time [1, 7].

In the more general case m ≥ 1, Corley and Moon [2] presented an algorithmfor finding all Pareto paths with computational complexity O(mn2n−3 +mnn).

The aim of the current work is to find Pareto paths satisfying multiple con-straints. Given a vector c = (c1, . . . , cm) ∈ Rm, a path is deemed feasible if itsvector weight w = (w1, . . . , wm) satisfies

(∀i) [wi ≤ ci].

Letting Pf denote the collection of feasible paths having fixed source v0and fixed target v∞, the goal is to find a path in Pf whose weight is a Paretominimum. The operator calculus approach described herein can be applied tosieve out the collection of feasible paths and recover all single-source Paretopaths remaining.

2 Theory: Generalized zeon algebras

Zeon algebras are commutative algebras generated by collections of null-squares,ζi : 1 ≤ i ≤ n with ζi

2 = 0 for each i. Their combinatorial properties makethem useful for a variety of counting properties, as seen in a number of previousworks by the current authors ([10],[11], [13]) . By choosing sufficiently largesets of generators, they can be generalized to algebras whose generators arenilpotent of arbitrary index. The resulting generalized zeon algebras are suitablefor a number of combinatorial applications, including multi-constrained routingproblems.

Combinatorial properties of zeons have also been utilized by the authors todevelop a “zeon-Berezin” operator calculus having applications in free proba-bility theory [9]. Other zeon-related work includes the papers of Feinsilver [5]and Feinsilver and McSorley [6].

Definition 2.1. The n-particle zeon algebra, denoted by C`nnil, is defined asthe real abelian algebra generated by the collection ζi (1 ≤ i ≤ n) along with

2

the scalar 1 = ζ0 subject to the following multiplication rules:

ζi ζj = ζj ζi for i 6= j, and (2.1)

ζi2 = 0 for 1 ≤ i ≤ n. (2.2)

A general element u ∈ C`nnil can be expanded as

u =∑I∈2[n]

uI ζI , (2.3)

where I ∈ 2[n] is a subset of [n] = 1, 2, . . . , n used as a multi-index, uI ∈ R,and ζI =

∏ι∈I

ζι.

Remark 2.2. The zeon algebra C`nnil can be realized as a commutative subalge-bra of the Grassmann algebra

∧V over a 2n-dimensional vector space V with

orthonormal basis γi by defining ζi = γiγn+i for each 1 ≤ i ≤ n.A canonical basis element ζI is referred to as a blade. The number of elements

in the multi-index I is referred to as the grade of the blade ζI .The next lemma shows that it is possible to construct elements with arbitrary

index of nilpotency within a zeon algebra of sufficiently high dimension.

Lemma 2.3. Let ζi : 1 ≤ i ≤ n be the null-square generators of C`nnil. Then,for any permutation σ ∈ Sn and positive integers ` ≤ k ≤ n, k∑

j=1

ζσ(j)

`

= `!∑

I⊆σ(1),...,σ(k)|I|=`

ζI . (2.4)

Moreover, if ` > k, then k∑j=1

ζσ(j)

`

= 0. (2.5)

Proof. Since the generators commute, the multinomial theorem applies, withonly square-free terms surviving.

Definition 2.4. For positive integer n, let s = (s1, . . . , sn) ∈ Nn be an n-tupleof positive integers. Then, the zeon algebra of signature s (or s-zeon algebra),denoted C`snil, is the real abelian algebra generated by the collection νi (1 ≤i ≤ n) along with the scalar 1 = ν0 subject to the following multiplication rules:

νi νj = νj νi for i 6= j, and (2.6)νisi = 0 for 1 ≤ i ≤ n. (2.7)

For convenience, the following multi-exponent notation is adopted:

νx :=n∏i=1

νxii := νx11 · · · νxnn . (2.8)

3

Letting S = (x1, . . . , xn) : 0 ≤ xi ≤ si ⊂ N0n, a general element u ∈ C`snil

can be expanded asu =

∑x∈S

ux νx , (2.9)

where ux ∈ R for each multi-exponent x.

Since the components of signature vectors and arbitrary multi-exponents arenonnegative integers, the 1-norm of such a vector x is simply the sum of thecomponents; that is,

‖x‖1 =n∑i=1

|xi| =n∑i=1

xi. (2.10)

Lemma 2.5. The algebra C`snil is isomorphic to a subalgebra of the zeon algebraC`‖s‖1

nil.

Proof. For k ∈ 1, . . . , n, let p(k) denote the kth partial sum

p(k) =k∑j=1

sj , (2.11)

and define p(0) = 0. In light of Lemma 2.3, the desired isomorphism C`snil →C`‖s‖1

nil is obtained from the mapping

νi 7→si∑j=1

ζj+p(i−1). (2.12)

Finally, note that the s-zeon algebra is naturally graded according to

C`snil =‖s‖1⊕k=0

⟨C`snil

⟩k, (2.13)

where the grade-k part of the algebra is defined by⟨C`snil

⟩k

= span (νx : ‖x‖1 = k) . (2.14)

The notation 〈·〉k extends naturally to elements of C`snil.

3 Operator calculus in generalized zeon algebras

The motivation for development of s-zeon operator calculus is based on poly-nomial operator calculus. To begin, raising and lowering operators are defined

4

naturally in terms of polynomial differentiation and integration operators on ba-sis zeons regarded as polynomials in commuting variables. In this formulation,the generators νi of C`snil are regarded as variables in the polynomial sense.

For any generalized zeon algebra with n generators, let ei : 1 ≤ i ≤ ndenote standard unit vectors of the form ei := (0, . . . , 1︸︷︷︸

ith pos.

, . . . , 0). Arbitrary

multi-exponents are then expressed in the form x = (x1, . . . , xn) =n∑i=1

xiei.

Definition 3.1. Let s ∈ N0n be an arbitrary zeon signature. For 1 ≤ j ≤ n,

define the jth s-zeon differentiation operator ∂/∂νj on C`snil by linear extensionof

∂

∂νjνx =

νx−ej if xj ≥ 1,0 otherwise.

(3.1)

Definition 3.2. The s-zeon integrals are defined by∫νxdνj =

νx+ej if xj < sj − 1,0 otherwise.

(3.2)

These polynomial operators induce combinatorial raising and lowering oper-ators by which s-zeon monomials (blades) are raised from grade k to grade k+1or lowered from grade k to grade k−1. These raising and lowering operators canalso be regarded as creation and annihilation operators in the sense of quantummechanics.

Definition 3.3. For each 1 ≤ j ≤ n, define the jth raising operator Rj by linearextension of

Rj νx =

∫νx dνj = νxνj . (3.3)

Define the jth lowering operator Dj by linear extension of

Dj νx =

∂

∂νjνx. (3.4)

Definition 3.4. The jth zeon number operator Λj is defined on the generalizedzeon algebra C`snil by linear extension of

Λj(νx) := xjνx. (3.5)

In particular, for arbitrary multi-exponents x,y and scalars α, β,

Λj(ανx + βνy) = αxjνx + β‖y‖1νy. (3.6)

Definition 3.5. The dual of the jth zeon number operator, denoted Λj?, isdefined on the generalized zeon algebra C`snil by linear extension of

Λj?(νx) :=

(1/xj)νx if xj > 0,0 otherwise.

(3.7)

5

For arbitrary multi-exponents x,y and scalars α, β,

Λj?(ανx + βνy) =α

xjνx +

β

yjνy. (3.8)

An element u ∈ C`snil is said to be scalar-free if its canonical expansion is ofthe form ∑

x6=0

uxνx. (3.9)

Let C`snil? denote the scalar-free subalgebra of C`snil; that is,

C`snil? := u ∈ C`snil : u is scalar free. (3.10)

The zeon occupancy operator Λ and its dual Λ? are Λ =n⊕j=1

Λj and Λ? =

n⊕j=1

Λj?, respectively.

Lemma 3.6. Letting C`snil? denote the scalar-free subspace of C`snil,

ΛΛ?∣∣∣∣C`snil?

= Λ?Λ∣∣∣∣C`snil?

= I. (3.11)

More specifically, ΛΛ? : C`snil → C`snil? is an orthogonal projection.

Proof. Since the components of multi-exponents are nonnegative integers, writ-ing x = (x1, . . . xn) leads to the component sum as the 1-norm; i.e., ‖x‖1 =∑nj=1 xj . Let u =

∑x6=0

uxνx ∈ C`snil and consider

Λ (Λ?u) = Λ

∑x6=0

ux

‖x‖1νx

=∑x 6=0

ux‖x‖1‖x‖1

νx = u. (3.12)

A nearly identical argument shows Λ?(Λu) = u. From the definitions of Λ andΛ?, it is apparent that for nonzero scalar α, Λ(ανx) = 0 if and only if x = 0;i.e., ανx = α. The same can be said of Λ?; i.e., ker Λ = ker Λ? = R. As a result,ker ΛΛ? = ker Λ?Λ = R.

Given a c-dimensional constraint vector s, a total ordering is induced onthe set of multi-exponents x ∈ N0

m by defining x y if and only if ∃k ≥ 1such that xi ≤ yi ∀i ≤ k. When any such total ordering is assigned to themulti-exponents, one is able to define minimal elements of C`snil. This will beuseful in subsequent applications in which minimal elements will be associatedwith optimal solutions.

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Definition 3.7. Fixing a total ordering of the multi-exponents, define aminimal term of u ∈ C`snil by

0u := ux′νx′ , (3.13)

where x′ x for all nonzero multi-exponents in the canonical expansion of u.

The s-zeon algebra will be applied in later sections to sieve out paths satis-fying multiple constraints. In order to retain identifying information about thepaths themselves, another generalization of zeon algebras is considered.

3.1 The path algebra RΩn

For fixed positive integer n, consider the alphabet Σn := ωi : 1 ≤ i ≤ n. Forconvenience, we adopt the following ordered multi-index notation. In particular,letting u = (u1, . . . , uk) for some k, the notation ωu will be used to denote asequence (or word) of distinct symbols of the form

ωu := ωu1ωu2 · · ·ωuk . (3.14)

Appending 0 to the set Σn, multiplication is defined on the words constructedfrom elements of Σn by

ωuωv =

ωu .v if u ∩ v = ∅,0 otherwise,

(3.15)

where u .v denotes sequence concatenation.One thereby obtains the noncommutative semigroup Ωn, whose elements

are the symbol 0 along with all finite words on distinct generators (i.e., finitesequences of distinct symbols from the alphabet Σn). Since there are only ngenerators, it is clear that the maximum multi-index size of semigroup elementsis n. Moreover, these symbols can appear in any order so that the order of the

semigroup isn∑k=0

(n

k

)k! =

n∑k=0

(n)k.

Defining (vector) addition and real scalar multiplication on the semigroupyields the semigroup algebra RΩn of dimension |Ωn|. This semigroup algebrawill be referred to as a path algebra.

Consider the collection of ordered pairs P = (ωi, ωj) ∈ Ωn×Ωn : i 6= j, andnote that |P | = n2−n. Imposing an ordering on P , a bijection f : P → [n2−n]is obtained. Any k-subset of [n2 − n] thereby determines a unique finite wordof Ωn:

ωu = ωu1 · · ·ωuk+1 ↔ f((u1, u2)), . . . , f((uk, uk+1)). (3.16)

In this way, one obtains a semigroup homomorphism φ from the canonicalbasis blades of C`|P |nil onto the words of length two or more in Ωn. In this sense,the semigroup algebra RΩn can be regarded as an extension of a zeon algebra.

7

Remark 3.8. When the pairs of P are unordered, each k-subset of [n(n− 1)/2]determines two finite words of Ωn: ωu, and its reversion ωu; i.e.,

ωu = ωu1 · · ·ωuk+1 ↔ f((u1, u2)), . . . , f((uk, uk+1)) ↔ ωuk+1 · · ·ωu1 = ωu.(3.17)

3.2 Operator calculus on graphs

As discussed in previous work, the nilpotent adjacency matrix of a finite graphcan be used to sieve out the graph’s paths and cycles. The entries of this matrixare the null-square generators of a zeon algebra of appropriate dimension forthe graph. The null-square properties of the algebra naturally remove entriescorresponding to self-intersecting walks from powers of the nilpotent adjacencymatrix.

Of interest in the current work is a method of sieving out paths with multi-dimensional weights (or costs) simultaneously satisfying a number of constraints.Generalized s-zeon generators will be associated with the graph’s edges in sucha way that paths whose weights exceed the constraints are zeroed out by thealgebra’s nilpotent properties.

In particular, extending this matrix construction to C`snil ⊗ RΩn allowsone to enumerate (list) all paths and cycles satisfying multiple constraints in afinite graph by considering powers of the matrix. The associated tree structureunderlying the cycle/path enumeration problem is automatically “pruned” bythe inherent properties of the algebra.

The first step is defining a nilpotent adjacency matrix that preserves path-identifying information.

Definition 3.9. Let G = (V,E) be a graph on n vertices, either simple ordirected with no multiple edges. Let ωi, 1 ≤ i ≤ n denote the null-square,noncommutative generators of RΩn. Define the path-identifying nilpotent adja-cency matrix A associated with G as the n× n matrix

Aij =

ωj if (vi, vj) ∈ E,0 otherwise.

(3.18)

Recalling Dirac notation, the ith row of A is conveniently denoted by 〈vi| Awhile the jth column is denoted by A |vj〉. In this way, A is completely deter-mined by

〈vi|A|vj〉 =

ωj if there exists a directed edge vi → vj in G,

0 otherwise,(3.19)

for all vertex pairs (vi, vj) ∈ E.

Theorem 3.10. Let A be the path-identifying nilpotent adjacency matrix of ann-vertex graph G. For any k > 1 and 1 ≤ i 6= j ≤ n,

ωi⟨vi|Ak|vj

⟩=

∑k-paths w:vi→vj

ωw. (3.20)

8

Moreover, ⟨vi|Ak|vi

⟩=

∑k-cycles w based at vi

ωw. (3.21)

More specifically, when i 6= j, the product of ωi with the entry in row i, columnj of Ak is a sum of basis blades indexed by k-step paths vi → vj in G. Moreover,entries along the main diagonal of Ak are sums of basis blades indexed by thegraph’s k-cycles.

Proof. The result follows from straightforward mathematical induction on kusing properties of the multiplication in RΩn with the observation that theinitial vertex of the walk, vi, is unaccounted for in 〈vi|Ak|vj〉, as seen in (3.18)of the matrix definition. Hence, each term of

⟨vi|Ak|vj

⟩is indexed by the vertex

sequence of a k-walk from vi to vj with no repeated vertices, except possibly viat some intermediate step. Left multiplication by ωi thus sieves out the k-paths.

Considering entries along the main diagonal of Ak, note that the final stepof a k-cycle based at vi returns to vi so that left multiplication by ωi is notrequired for cycle enumeration.

Note that ωi〈vi|Ak is a row vector whose nonzero entries represent all k-pathswith initial vertex vi. Similarly, A|vj〉 is a column vector whose nonzero entriesrepresent 1-paths with terminal vertex vj . Computing the (k + 1)-paths fromvi to vj then requires computing ωi〈vi|AkA|vj〉. Define the indicator functionχ(v`,vj)∈E by

χ(v`,vj)∈E =

1 if (v`, vj) ∈ E,0 otherwise.

(3.22)

Letting λ denote the number of multiplications involved in this computation,

λ =n∑`=1

]k-paths vi → v`χ(v`,vj)∈E ≤ |k-paths with source vi|. (3.23)

It follows immediately that the number of multiplications performed in de-termining ωi〈vi|Ak|vj〉 is bounded above by the number of paths of length atmost k − 1 having initial vertex vi. Hence, the next corollary is obtained.

Corollary 3.11. Given a fixed pair of vertices v0 and v∞, the complexity ofenumerating all k-paths from v0 to v∞ with the path-identifying nilpotent adja-cency matrix is

O(n |paths of length k − 1 or less having initial vertex v0|).

Remark 3.12. The computational complexity stated above is in terms of ba-sis blade multiplications performed within the algebra. Recall that for disjointordered multi-indices p, q, the product ωpωq = ωp.q is given by sequence con-catenation. Hence, some additional polynomial cost is associated with the im-plementation of the algebra multiplication.

9

3.3 Operator calculus approach to multi-constrained paths

The goal now is to extend the path-identifying nilpotent adjacency matrix ap-proach to include weighted edges. In particular, each edge of the graph willbe weighted by a c-tuple of nonnegative integers. In this manner, paths in thegraph will have associated c-dimensional additive weights.

Given vectors x = (x1, . . . , xc) and y = (y1, . . . , yc), the notation x ≤ y istaken to mean the following:

x ≤ y⇔ xi ≤ yi ∀i ∈ 1, . . . , c. (3.24)

The strict inequality x < y is analogously defined.Given a finite graph G in which each edge is weighted with an c-tuple of

nonnegative integers and a constraint vector c = (c1, . . . , cm) ∈ N0m, the multi-

constrained path problem is defined as follows.

Definition 3.13. The MCP (Multi-Constrained Path) problem is to find pathsp from v0 to v∞ in the graph G such that

wt(p) =∑

(vi,vj)∈p

wt((vi, vj)) ≤ (c1, . . . , cm) = c. (3.25)

The goal is to find the set of feasible paths Pf = p = (v0, . . . , v∞) : wt(p) < c,i.e., all paths from v0 to v∞ that satisfy multiple constraints simultaneously.

An important variant of the MCP problem that is of particular interest isthe associated optimization problem.

Definition 3.14. Letting P denote the set of feasible paths from v0 to v∞ in aweighted graph G, the Multi-Constrained Optimal Path problem (MCOP) is tofind a path p = (v0, . . . , v∞) ∈ P from v0 to v∞ such that

wt(p) ≤ wt(q) ∀q ∈ P. (3.26)

Note that for fixed s ∈ Nm, multi-exponents appearing among basis elementsνx ∈ C`snil must satisfy x < s. Given a c-vector s ∈ Nc, multiplication ofarbitrary s-zeon blades consequently satisfies

νxνy =

νx+y if x + y < s,0 otherwise.

(3.27)

In order to apply the OC approach to problems of identifying paths satisfyingmultiple constraints (represented by s), the path-identifying nilpotent adjacencymatrix will be extended by allowing entries from the algebra C`snil ⊗ RΩn. Inthis approach, a path u = (u1, . . . , um) of weight x ∈ N0

m will be represented inC`snil⊗Ωn by an element of the form νx ωu. The concatenation of this path withanother path v = (v1, . . . , v`) of weight y is then represented by the product

(νxωu)(νyωv) =

νx+yωu .v if u .v is a path of multi-weight less than s,0 otherwise.

(3.28)

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3.4 Feasible & optimal paths in c-weighted graphs

Often in routing problems, there are costs associated with each edge of a graph.Connections between nodes may require time, energy, money, etc. Given aninitial vertex v0 and terminal vertex vt in a weighted graph, the collection offeasible paths from v0 to vt refers to all paths whose associated total costs satisfysome predefined constraints. Among these feasible paths, an optimal path canthen be chosen.

Given a vector of constraints c = (c1, . . . , cm) ∈ Nm, properties of the c-zeonalgebra C`cnil can be used to sieve out the feasible paths from the collection ofall paths. The feasible paths can then be ranked and an optimal path chosen.

Definition 3.15. Let c ∈ Nm, and let G = (V,E) be a graph on n verticeswhose edges (vi, vj) are multi-weighted by vectors wij ∈ N0

m. The c-constrainedpath-identifying nilpotent adjacency matrix associated with G is the n×n matrixwith entries in (C`cnil)m ⊗ Ωn determined by

Ψij =

νwijωj if (vi, vj) ∈ E,0 otherwise.

(3.29)

Theorem 3.16. Given a multi-weighted graph G on n vertices with nilpotentmulti-weighted adjacency matrix Ψ, a vector of constraints c = (c1, . . . , ck), anda pair of distinct vertices v0 and v∞, the collection of feasible paths v0 → v∞ inG is given by

ν0ω0

n∑`=1

〈v0|Ψ`|v∞〉 =∑

pathsp:v0→v∞wt(p)<c

νwt(p)ωp. (3.30)

More specifically, feasible paths exist if and only if ν0ω0

∑n`=1〈v0|Ψ`|v∞〉 is

nonzero. For the case v0 = v∞, one has

〈v0|Ψ`|v0〉 =∑

cyclesp:v0→v0wt(p)<c

νwt(p)ωp. (3.31)

Proof. The result follows from Theorem 3.10 in consideration of combinatorialproperties of C`snil.

Corollary 3.17. If ν0ω0

n∑`=1

〈v0|Ψ`|v∞〉 6= 0, then the optimal path p = (v0, . . . , v∞)

exists and is given by

0

(ν0ω0

n∑`=1

〈v0|Ψ`|v∞〉

)= νwt(p)ωp. (3.32)

Proof. By Theorem 3.16, the collection of all feasible paths v0 → v∞ is given by

ν0ω0

n∑`=1

〈v0|Ψ`|v∞〉. By the chosen ordering of paths and definition of 0, the

optimal path is as stated.

11

Figure 3.1: Weighted graph on 70 vertices.

Example 3.18 (Static case). Figure 3.1 depicts a weighted graph on 70 vertices.Each edge is assigned a vector of random positive weights in N4. The graphis not symmetrically weighted; i.e., weights of vertex pairs vary depending ondirection. The constraint vector is c = (50, 70, 70, 90).

A randomly-selected submatrix of the 70× 70 weighted path-identifying ad-jacency matrix associated with the graph is seen in Fig. 3.2. The collection ofadmissible 4-paths from v34 to v47 are then computed using Mathematica, asseen in Fig. 3.3.

Let Pf 0 denote the collection of feasible paths with source v0. That is,

Pf 0 = p = (v0, . . . , v∗) : wt(p) < c, v∗ ∈ V . (3.33)

Following the approach of Corollary 3.11, the number of multiplications re-quired in computing ν0ω0〈v0|Ψ`|v∞〉 is seen to depend on the number of pathsof length `− 1 or less having initial vertex v0 and simultaneously satisfying theconstraints represented by c. The following corollary is obtained as an immedi-ate consequence.

Corollary 3.19. Given a fixed pair of vertices v0 and v∞, the complexity ofcomputing the optimal feasible path from v0 to v∞ via the operator calculusmethod is

O(n |Pf 0|).

3.5 The dynamic multi-constrained path problem

Given a finite set V and positive integer r, an r-dimensional weighting functionϕ is defined on ordered pairs of vertices; i.e., ϕ : V × V → N0

r. A weighted

12

0 0 0 0 Ν

84,2,3,3<Ω88< 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 Ν

82,3,5,5<Ω88< 0 0 0 0

0 0 0 0 0 0 Ν

84,1,4,3<Ω810< 0 0

Ν

84,5,3,2<Ω84< 0 Ν

82,2,4,3<Ω86< 0 0 Ν

83,1,5,3<Ω89< 0 0 0

0 0 0 0 Ν

85,1,1,4<Ω88< 0 0 0 0

0 0 0 Ν

83,5,3,1<Ω87< 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 .

Figure 3.2: An 8 × 8 submatrix of the matrix Ψ associated with the graph ofFig. 3.1.

Paths from 34 to 47 satisfying w £ 850, 70, 70, 90<...

Ν811,11,8,7<

Ω834,10,2,47< + Ν812,9,10,8<

Ω834,10,50,47< + Ν88,10,8,12<

Ω834,38,5,47< + Ν87,12,8,11<

Ω834,38,50,47<

Minimal path term: Ν87,12,8,11<

Ω834,38,50,47<

Minimal path weight: 87, 12, 8, 11<

Minimal Path: 834, 38, 50, 47< .

Figure 3.3: Admissible 4-paths from v34 to v47 in the graph of Fig. 3.1. Minimalweight determined by ordering of multi-exponents.

graph is then defined by the ordered triple G = (V,E, ϕ), where the weight ofan arbitrary edge (vi, vj) ∈ E is the r-vector ϕ((vi, vj)).

Given a fixed collection of vertices V and fixed weighting function ϕ : V ×V → N0

r, a weighted graph process on V is defined as a sequence of weightedfinite graphs (Gt) := ((V,Et, ϕ)). Given a pair of vertices v0, v∞ ∈ V and avector c ∈ Nr of constraints, the problem being considered is to recover the setof feasible paths v0 → v∞ simultaneously satisfying the constraints c.

Note that any graph sequence (Gt) naturally induces an associated sequenceof weighted path-identifying nilpotent adjacency operators (Ψt), where each Ψt

has entries in C`cnil ⊗ Ω|V |.It is important to note that a feasible path p from v0 to v∞ is not necessarily

unique in the dynamic case because the sequence p can be partitioned into stepsoccurring in different frames of the process. As a consequence, scalar coefficientsmust be introduced to represent the path multiplicity within the collection P.

Theorem 3.20. The collection of feasible paths of length k ≥ 1 from v0 → v∞

13

requiring f or fewer frames is given by

ν0ωv0∑

0≤`1,...,`f`1+···+`f=k

〈v0|Ψ1`1 · · ·Ψf

`f |v∞〉 =∑

k-paths p=(v0,...,v∞)∈P

αpνwt(p)ωp

(3.34)where αp is a scalar coefficient representing the multiplicity of the path p in thecollection P.

Proof. Note that k ≥ 1 ensures that at least one of the integers `i is nonzero.For fixed nonnegative integers `1, `2 with `1 + `2 = k,

ν0ωv0〈v0|Ψ1`1Ψ2

`2 |v∞〉 = ν0ωv0∑vj 6=v0

〈v0|Ψ1`1 |vj〉〈vj |Ψ2

`2 |v∞〉

=∑vj 6=v0

∑`1-paths p=(v0,...,vj)∈P

In frame 1

αpνwt(p)ωp

∑

`2-paths q=(vj,...,v∞)∈PIn frame 2

αqνwt(q)ωq

=

∑k-paths p=(v0,...,v∞)∈P`1,`2

αpνwt(p)ωp, (3.35)

where P`1,`2 denotes the collection of feasible paths from v0 to v∞ in which `1steps occur in frame 1 and `2 steps occur in frame 2. Proceeding by induction,the result is established for fixed m-tuple of nonnegative integers (`1, . . . , `m)with `1 + · · · + `m = k. Assuming that for positive integer m0 and fixed non-negative integers `1, . . . , `m0 with `1 + · · · `m0 = k′ ≤ k,

ν0ωv0〈v0|Ψ1`1 · · ·Ψm0

`m0 |v∞〉 =∑

k′-paths p=(v0,...,v∞)∈P`1,...,`m0

αpνwt(p)ωp,

(3.36)it follows that for `m0+1 = k − k′, one has

ν0ωv0〈v0|Ψ1`1 · · ·Ψm0

`m0 Ψm0+1`m0+1 |v∞〉

= ν0ωv0∑vj 6=v0

〈v0|Ψ1`1 · · ·Ψm0

`m0 |vj〉〈vj |Ψm0+1`m0+1 |v∞〉

=∑vj 6=v0

∑k′-paths p=(v0,...,vj)

in P`1,...,`m0

αpνwt(p)ωp

∑

(k−k′)-paths q=(vj,...,v∞)

in P0,...,`m0+1

αqνwt(q)ωq

=

∑k-paths p=(v0,...,v∞)∈P`1,...,`m0+1

αpνwt(p)ωp, (3.37)

Hence, the result is established for positive integer m:

ν0ωv0〈v0|Ψ1`1 · · ·Ψm

`m |v∞〉 =∑

k-paths p=(v0,...,v∞)∈P`1,...,`mαpν

wt(p)ωp. (3.38)

The proof is thus completed by summing over all such m-tuples.

14

Corollary 3.21. The collection of feasible paths of all lengths from initial vertexv0 to terminal vertex v∞ 6= v0 requiring f or fewer frames is recovered from thecanonical expansion of

ν0ωv0∑

0≤`1,...,`f`1+···+`f≤n

〈v0|Ψ1`1 · · ·Ψf

`m |v∞〉 =∑

pathsp:v0→v∞

αpνwt(p)ωp, (3.39)

where αp denotes multiplicity of path p ∈ P.

Proof. First, consider the degenerate case `i = 0 for 1 ≤ i ≤ m. In this case, theproduct of Ψi’s is the identity operator, and v0 6= v∞ gives 0 on the left-handside of the equation. Observing that the maximum path length is n = |V |, therest follows from Theorem 3.20.

Once the collection of feasible paths is obtained, the optimal path can beselected based on an ordering of the multi-exponents.

Corollary 3.22. Given a preferential ordering of multi-exponents, the optimalpath p = (v0, . . . , v∞) from v0 to v∞ in the first f frames of the graph sequence(Gt) is given by

0

ν0ωv0∑

0≤`1,...,`f1≤`1+···+`f≤n

〈v0|Ψ1`1 · · ·Ψf

`f |v∞〉

= αpνwt(p)ωp, (3.40)

provided ν0ωv0〈v0|Ψ1`1 · · ·Ψf

`f |v∞〉 6= 0 for some f-tuple of nonnegative integers`1, . . . , `f, not all of which are zero. Here, αp denotes the multiplicity of pathp ∈ P.

4 Generalized idempotents and max-min oper-ators

Sometimes additive weights are not suitable for the problem being considered.For example, in one version of a store-and-forward satellite constellation usedfor communications, the “sum” of two multi-weight vectors (v1, v2, v3) and(w1, w2, w3) on a coincident pair of links may be defined as

(v1, v2, v3) (w1, w2, w3) = (v1 + w1, v2 + w2,minv3, w3), (4.1)

while also being subject to some constraint. Here, the third component mightrepresent the transmission capacity of a link, such that a minimum capacity isrequired for all links appearing in a feasible path.

15

Definition 4.1. For positive integer n, let C`nidem denote the abelian algebragenerated by the collection εi : 1 ≤ i ≤ n along with the scalar 1 = ε∅ subjectto the following multiplication rules:

εi εj = εj εi for i 6= j, and (4.2)

εi2 = εi for 1 ≤ i ≤ n. (4.3)

Remark 4.2. The algebra C`nidem is constructed within a 2n(n − 1)-particlefermion algebra. Fix n > 0 and consider elements of the form

εi =12

(1 +

(fi + fi

+

2

)(fn2−n+i + f+

n2−n+i

2

)), (4.4)

where fi and f+i denote the ith fermion annihilation and creation operators,

respectively.

The idem-Clifford algebra was first used in [10] for purposes of computinghigher moments of cycle numbers in homogeneous random graphs. As illustratedin the next lemma, it is also useful for computing the maximum and minimumof a pair of integers. First, we consider the extension to the infinite-dimensionalversion of the idem-Clifford algebra.

Definition 4.3. The generalized idempotent algebra C`idem is the infinite di-mensional Abelian algebra defined by the direct sum

C`idem :=∞⊕n=1

C`nidem. (4.5)

The generators εj : 1 ≤ j are pairwise commutative and satisfy εj2 = εj for

all j ≥ 1.

Recalling multi-index notation and the n-set notation, [n] := 1, . . . , n, themax-min operators are now written using generalized idempotents.

Lemma 4.4. Let εi : 1 ≤ i ≤ n denote the idempotent generators of C`nidem.For nonnegative integers s, t ≤ n,

ε[s] ∨ ε[t] := ε[s]ε[t] = ε[maxs,t]. (4.6)

Whence, one can define

ε[s] ∧ ε[t] := ε[s] + ε[t] − ε[s]ε[t] = ε[mins,t]. (4.7)

Proof. Proof is by direct computation.

Identifying ε0 7→ ε∅ := 1 to obtain the multiplicative identity of the algebra,one also obtains the identity with respect to the maximum operator.

Relative to the minimum operator (4.7), one formally defines the elementε[∞] such that

ε[∞] ∧ ε[j] = ε[j]

16

for all j ∈ N.Note that in the finite case, this identity element is given by

ε[∞] := ε[n] ∈ C`nidem.

Remark 4.5. In C`nidem, an alternative formulation of the minimum operator isobtained by

ε[s] ∧ ε[t] := (ε[s]?ε[t]?)?,

where the notation ? indicates the involution defined by the multi-index set-complement

ε[m]? := ε[n]\[m]

for m ≤ n. This extends formally to C`idem by

ε[m]? := εm+1,...,

for m ∈ N0.Letting m denote the number of nonadditive weights assessed by min-max

operators, it is now possible to consider multi-constrained paths by constructinga nilpotent adjacency matrix with entries from C`snil ⊗ C`idem⊗m

⊗ Ωn. Moreto the point, one can define a max-min signature m = (m1,m2) where m1,m2 ≥0 and m1 + m2 = m for generalized idempotents and an associative binaryoperation ∗ on C`snil ⊗ C`idem⊗m

satisfying

(ε[`1] ⊗ · · · ⊗ ε[`m1+m2 ]) ∗ (ε[g1] ⊗ · · · ⊗ ε[gm1+m2 ])

= (ε[`1] ∨ ε[g1])⊗ · · · ⊗ X︸︷︷︸jth

⊗ · · · ⊗ (ε[gm2 ] ∧ ε[gm2 ]) (4.8)

In particular, the jth factor, X, appearing in the tensor product is given by

X =

ε[`j ] ∨ ε[gj ] if 1 ≤ j ≤ m1,

ε[`j ] ∧ ε[gj ] if m1 + 1 ≤ j ≤ m1 + m2.(4.9)

With this in mind, the notation C`snil ⊗ C`m1,m2idem is clear.

Letting a denote the number of additive weights, the multi-exponent notationis now extended to C`snil ⊗ C`m1,m2

idem by

ξ(x1,...,xa,m1,...,mm) = ν(x1,...,xa) ⊗ ε[m1] ⊗ · · · ⊗ ε[mm]. (4.10)

The identity element of C`snil ⊗ C`m1,m2idem is then written as

ξ0 := ν0 ⊗ ε0⊗m1 ⊗ ε[∞]⊗m2 . (4.11)

Define the binary operation on N0a+m1+m2 by

(u v)i :=

ui + vi if 1 ≤ i ≤ a,

maxui, vi if a + 1 ≤ i ≤ a + m1,

minui, vi if a + m1 + 1 ≤ i ≤ a + m1 + m2.

(4.12)

17

The commutative multiplication ∗ on C`snil⊗C`m1,m2idem is defined by linear

extension of

ξu ∗ ξv =

ξuv if u v ∈ C,

0 otherwise.(4.13)

This extends to C`snil ⊗ C`m1,m2idem ⊗ Ωn by

ξvωviξwωvj = (ξv ∗ ξw)ωvi ωvj , (4.14)

which extends inductively to

ξvωp ξwωv` = (ξv ∗ ξw)ωp ωv` =

ξvwωp.v` if v w ∈ C and p ∩ v` = ∅,0 otherwise.

(4.15)for any k-path p = (v0, . . . , vk) of weight v and vertex v` adjacent to vk via anedge of weight w.

The path-identifying nilpotent adjacency matrix can now be defined withentries in C`snil⊗C`m1,m2

idem⊗Ωn. Further, extending ∗ to matrix multiplicationaccording to

〈vi|(A ∗B)|vj〉 =n∑`=1

ai` ∗ b`j , (4.16)

the following corollary to Theorem 3.10 is immediately obtained.

Theorem 4.6. Let G denote a multi-weighted graph on n vertices, and let Cdenote a system of constraints corresponding to generalized zeon signature s andmax-min signature (m1,m2). Let v0 and v∞ denote a pair of distinct vertices,and let Ψ be the multi-weighted nilpotent adjacency matrix for G having entriesin C`snil ⊗ C`m1,m2

idem ⊗ Ωn. The collection of feasible paths v0 → v∞ in G isthen given by

ξ0ω0

n∑`=1

〈v0|Ψ∗`|v∞〉 =∑

pathsp:v0→v∞wt(p)∈C

ξwt(p)ωp. (4.17)

More specifically, feasible paths exist if and only if ξ0ω0

∑n`=1〈v0|Ψ∗`|v∞〉 is

nonzero. For the case v0 = v∞, one has

〈v0|Ψ∗`|v0〉 =∑

cyclesp:v0→v0wt(p)∈C

ξwt(p)ωp. (4.18)

While the Mathematica implementations of the max-min operators (4.6) and(4.7) are accomplished by simply applying the appropriate built-in operators, itis important to note that all necessary operations can be accomplished by usingthe inherent combinatorial properties of the algebras described. Moreover, all ofthese algebras occur as subalgebras of Clifford algebras–lending them a naturalconnection to quantum probability and (by extension) to quantum computing.

18

Figure 4.1: Weighted graph on 150 vertices.

Example 4.7. Figure 4.1 depicts a weighted graph on 150 vertices. Each edgeis assigned a vector of random weights in N0

3.The graph is not symmetrically weighted; i.e., weights of vertex pairs vary

depending on direction. Defining the set

C := (v1, v2, v3) ∈ N03 : (v1 ≤ c1) ∧ (v2 ≤ c2) ∧ (v3 ≥ c3), (4.19)

a weight vector v = (v1, v2, v3) is said to satisfy the constraint c = (c1, c2, c3) ifv ∈ C.

The “sum” of weights taken between two coincident edges is then computedby (4.1), provided v w ∈ C. The constraint vector used in the example isc = (500, 700, 20). That is, the additive sums of the first and second componentvalues are bounded above by 500 and 700, respectively, while the minimumvalue of the third component taken over all links in a path is bounded below by20.

In order to define optimal paths, weights are ordered as follows:

(v1, v2, v3) (w1, w2, w3)⇔ (v1 < w1)∨ [(v1 = w1)∧ (v3 > w3)]∨ [(v1 = w1)∧ (v3 = w3)∧ (v2 ≤ w2)].

(4.20)

A 7 × 7 submatrix of the 150 × 150 weighted path-identifying adjacencymatrix associated with the graph is seen in Fig. 4.2.

19

0 0 0 Ξ

848,7,18<Ω817< 0 0 0

0 0 Ξ

825,15,1<Ω816< 0 0 0 0

0 Ξ

828,5,23<Ω815< 0 0 0 0 0

Ξ

84,45,45<Ω814< 0 0 0 0 0 0

0 0 0 0 0 Ξ

826,17,15<Ω819< 0

0 0 0 0 Ξ

837,44,2<Ω818< 0 0

0 0 0 0 0 0 0 .

Figure 4.2: A 7 × 7 submatrix of the matrix Ψ associated with the graph ofFig. 4.1.

Paths from 50 to 99 with w satisfying 8500, 700, 20<...

Ξ

852,40,27<

Ω850,108,79,99< + Ξ

854,61,30<

Ω850,108,88,99< + Ξ

879,67,29<

Ω850,117,88,99< + Ξ

897,97,40<

Ω850,137,15,99<

Minimal Path Term: Ξ

852,40,27<

Ω850,108,79,99<

Minimal path weight vector: 852, 40, 27<

Minimal Path: 850, 108, 79, 99< .

Figure 4.3: Admissible 3-paths from v50 to v99 in the graph of Fig. 4.1. Minimalweight determined by ordering of multi-exponents.

The collection of admissible 3-paths from v13 to v138 are then computedusing Mathematica, as seen in Fig. 4.3.

The dynamic case of Theorem 3.20 is extended similarly. Note that whenconsidering the collection of feasible paths, p ∈ P only if wt(p) ∈ C.

Theorem 4.8. Let (G` : 1 ≤ `) denote a sequence of multi-weighted graphs onn vertices, and let C denote a system of constraints corresponding to generalizedzeon signature s and max-min signature (m1,m2). Let (Ψ` : 1 ≤ `) be a sequenceof multi-weighted nilpotent adjacency matrices for (G`) having entries in C`snil⊗C`m1,m2

idem ⊗ Ωn. Let v0 and v∞ denote a fixed pair of distinct vertices. Thecollection of feasible paths of length k ≥ 1 from v0 → v∞ requiring f or fewerframes is then given by

ξ0ωv0∑

0≤`1,...,`f`1+···+`f=k

〈v0|Ψ1∗`1 · · ·Ψf

∗`f |v∞〉 =∑

k-paths p=(v0,...,v∞)∈P

αpξwt(p)ωp

(4.21)

20

where αp is a scalar coefficient representing the multiplicity of the path p in thecollection P.

5 Conclusion

The operator calculus approach provides convenient symbolic computationaltools for a broad range of combinatorial problems and practical applications.The methods developed here can be applied directly to multi-constrained qualityof service (QoS) problems as well as problems related to precomputed routingin store-and-forward satellite constellations [3].

References

[1] R. Bellman, On a routing problem, Quarterly of Applied Mathematics, 16(1958),87-90.

[2] H.W. Corley, I.D. Moon, Shortest paths in networks with vector weights,Journal of Optimization Theory and Applications, 46 (1985), 79-86. http://dx.doi.org/10.1007/BF00938761

[3] H. Cruz-Sanchez, R. Schott, Y-Q. Song, G.S. Staples, Operator calculusapproach to minimal paths: Precomputed routing in a store-and-forwardsatellite constellation, Preprint, 2011.

[4] E.W. Dijkstra, A note on two problems in connexion with graphs, Nu-merische Mathematik, 1 (1959), 269271. http://dx.doi.org/10.1007/BF01386390

[5] P. Feinsilver, Zeon algebra, Fock space, and Markov chains, Comm. Stoch.Anal., 2 (2008), 263–275.

[6] P. Feinsilver, J. McSorley, Zeons, permanents, the Johnson scheme, andgeneralized derangements, International Journal of Combinatorics, vol.2011, Article ID 539030, 29 pages, 2011. doi:10.1155/2011/539030

[7] L.R. Ford Jr., D.R. Fulkerson, Flows in Networks, Princeton UniversityPress, 1962.

[8] R. Schott, G.S. Staples, Operator Calculus on Graphs (Theory and Appli-cations in Computer Science), Imperial College Press, London, 2012.

[9] R. Schott, G.S. Staples. Zeons, lattices of partitions, and free probability,Comm. Stoch. Anal., 4 (2010), 311-334.

[10] R. Schott, G.S. Staples, Nilpotent adjacency matrices and random graphs,Ars Combinatoria, 98 (2011), 225-239.

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[11] R. Schott, G.S. Staples, Nilpotent adjacency matrices, random graphs, andquantum random variables, J. Phys. A: Math. Theor., 41 (2008), 155205.

[12] G.S. Staples, Clifford-algebraic random walks on the hypercube, Advancesin Applied Clifford Algebras, 15 (2005), 213-232.

[13] G.S. Staples, A new adjacency matrix for finite graphs, Advances in AppliedClifford Algebras, 18 (2008), 997-991.

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