Frobenius networks: routing,gossiping and diameter

Alison Thomson

The University of Melbourne

Outline

• Motivation• Routing and gossiping• Frobenius graphs• Frobenius Double loop graphs:

• Classification• Labelled lattices• Diameter and forwarding indices

Motivation

• Cayley graphs are excellent network architectures forapplications such as parallel computing

• Vertices are processors• Edges are communication links• Cayley graphs are vertex transitive so “local algorithms work

globally” [B. Alspach]• Some important parameters for an efficient network:

• Degree (“port constraint”)• Diameter• Minimum gossip time• Edge (arc) forwarding index

A popular example

• A double loop graph is a degree 4 circulant• DLn(a, b) = Cay(Zn, S) S = {±a,±b}• Geometric representation: shortest path tile:

• Each vertex is a numbered square cell• Each generator corresponds to a different direction in the

plane• E.g. DL29(1, 12)• Vertex set Z29

• Edges have “label” 1 or 12• Contains a shortest path from [0] to every other vertex

Example: DL29(1, 12)

Gossiping model

• Each vertex u has a (distinct) message Mu which must be sentto all other vertices

• During each time step:• an arc may carry at most one message• a vertex may exchange messages with all of its neighbours

(all port)• an edge may be used simultaneously in both directions (full

duplex)• Minimum gossip time τ(Γ) is the minimum number of time steps

required

Routing and forwarding index

• A routing is a (fixed) collection of paths in a graph• In a 1-all routing, a path is assigned from a single "source" vertex

to every other vertex• In an all-all routing, a path is assigned between every ordered

pair of vertices• For a given routing, the load on an edge is the number of paths

which use the edge.• We will consider all-all routings, a "worst-case" traffic scenario• For a graph Γ, the edge forwarding index π(Γ) is the minimum

maximum load taken over every possible all-all routing

Optimal routing and gossiping

• Edge forwarding index:• [Heydemann, Meyer, Sotteau, 1989] π(Γ) is minimised iff

there exists a routing for which the load of every edge isequal

• [Solé, 1994] Orbital Regular graphs have optimal π.• [Fang, Li, Praeger, 1998] Frobenius graphs have optimal π.

• Minimum gossip time:• [Zhou, to appear] First-kind Frobenius graphs have optimal τ

Frobenius Groups

• A Frobenius group L acting on a set V satisfies:• L is transitive on V but not regular• The only element of L fixing more than one point of V is the

identity• L = G ⋊ H where• G is the Frobenius kernel and is regular on V

• H is the Frobenius complement and is the stabiliser of a point ofV

First-kind Frobenius Graphs (I)

• [Fang, Li, Praeger, 1998]• Let L = G ⋊ H be a Frobenius group• A first-kind Frobenius graph is a Cayley graph Cay(G, S)

• Vertex set is G (the Frobenius kernel)

• Connection set is S = aH , for some a ∈ G, where• H acts on G by conjugation• 〈aH〉 = G

• Either |H| is even or a is an involution

First-kind Frobenius Graphs (II)

• Equivalent definition:• Cay(G, S) is a first-kind Frobenius graph if:

• G has a subgroup of automorphisms, H ≤ Aut(G), suchthat:• H is regular on S• H is semiregular on G \ {1}

• [Bermond, Kodate, Pérennes, 1996] A complete rotation ofCay(G, S) is a group automorphism of G which cyclicallypermutes the elements of S

• A first-kind Frobenius graph is rotational (i.e. admits a completerotation) iff H is cyclic

Example: DL29(1, 12)

DL29(1, 12) = Cay(Z29, S) where S = {±[1],±[12]}

• Aut(Z29) = (Z∗

29, ∗), the group of invertible elements modulo 29

• Z∗

29acts on Z29 via multiplication

• Define H, a subgroup of Z∗

29, where H = 〈[12]〉

• [12]2 = −[1], [12]3 = −[12], [12]4 = [1]

• H is cyclic of order 4, and H = S (as sets)• H is semiregular on Z29 \ {[0]}

Type

• Let Γ = Cay(G, S), be a first-kind Frobenius graph of diameter D

• Define δi to be the number of H-orbits at distance i from theidentity

• Γ has type -(δ1, . . . , δD)

• E.g. DL29(1, 12) has type-(1, 2, 3, 1)

Calculating π and τ

• Let Γ = Cay(G, S), be a first-kind Frobenius graph of diameter Dand type -(δ1, . . . , δD)

• [Fang, Li, Praeger, 1998] Γ has edge-fowarding index :

π(Γ) = 2∑D

i=1iδi

• [Zhou, to appear] Γ has minimum gossip time:τ(Γ) = (|G| − 1)/|S|

• [Zhou, to appear] Explicit construction of optimal routing andgossiping schemes

• Based on a family of symmetric spanning trees• Defining these trees requires knowledge of the type

Routing trees

• In any Cayley graph construction of an all-all routing is simple:• Define a 1-all routing tree rooted at the 0 vertex• Extend this to an all-all routing by applying automorphisms• This set of trees defines a (single) path between every pair

of vertices

E.g. Routing tree for DL29(1, 12)

Routing in Frobenius graphs

• In a first-kind Frobenius graph:• We can define a routing tree rooted at 0 which is H-invariant• Autormorphisms in H leaves the routing tree unchanged• Equal load on every edge in the corresponding all-all routing

(hence π is optimal)• Gossip optimally by transmitting the message Mu along the

tree rooted at u

Classification Theorem

• THEOREM: Let n be an odd positive integer. Then the followingare equivalent:• Every prime factor of n is congruent to 1 modulo 4• There exists [h] ∈ Zn such that [h2] = [−1]

• DLn(1, h) is a rotational first-kind Frobenius double loopgraph

• The automorphism [x] 7→ [hx] is a complete rotation

• We have good (though non-deterministic) algorithms to findsquare roots modulo n (e.g. Tonelli’s algorithm)

Idea of Proof

• We seek n, S and H such that Cay(Zn, S) is first-kind Frobenius• i.e. we seek H ∈ Z

∗

n such that H is:• regular on S• semiregular on Zn \ {[0]}

• For a connected circulant, we can assume w.l.o.g. that [1] ∈ S[Boesch & Tindell, 1985]

• Since [1] ∈ S we have H = S

• H regular on S ⇒ H is cyclic. i.e. H = {[h], [h2], [h3], [h4] = [1]}• Since [−1] ∈ S, we have h2 ≡ −1 (mod n)

• Solutions exist ⇐⇒ Every prime factor of n is congruent to 1modulo 4

• H is semiregular on S ⇐⇒ [h̄][x] = [x] has only trivial solutions

• ⇐⇒ [h̄ − 1] has an inverse modulo n for all [h̄] ∈ H \ {[1]}• Proof via elementary number theory

Multiple Solutions

We can have different Frobenius graphs for the same n...

DL65(1, 18) DL65(1, 8)

Multiple Solutions

• Square roots modulo n may not be unique• We can have distinct solutions g, h to the congruence

x2 + 1 ≡ 0 (mod n), such that g 6≡ ±h (mod n).• E.g. DL65(1, 8) and DL65(1, 18)

• THEOREM: Let n = pα1

1pα2

2. . . pαi

i be a positive integer such thatpj

αj ≡ 1 (mod 4) for each j. Then there exist exactly 2i−1 nonisomorphic first-kind Frobenius double loop graphs on n vertices.

• Idea of proof:• Count solutions via elementary number theory• Double loop graphs are 4 − CI-graphs [Sun, 1998] ⇒ each

solution produces a distinct graph

Infinite families (I)

“Optimal" double loop graphs (Yebra. et. al. 1985)

Infinite families (II)

“Windmill" double loop graphs

Properties

Windmill family:

• DLn(1, 2q), where n = 4q2 + 1

• First-kind Frobenius graphs with diameter 2q − 1

• Type-(1, 2, . . . , q − 1, q, q − 1, . . . , 2, 1)

• π = 2q3

Optimal family [Yebra et. al.1985]:• Double loop graphs with maximum number of vertices n for a

given diameter D

• DLn(1, 2D + 1), where n = 2D2 + 2D + 1

• First-kind Frobenius graphs with diameter D

• Type-(1, 2, . . . , D)

• π = 1

3D(D + 1)(2D + 1) [Narayanan et. al. 2001]

Labelled integer lattice

• Approach introduced in [Zerovnik, Pisanski, 1993]

• Γ = DLn(a, b) can be used to periodically label Z2

• Labelling function l : Z2 → Zn, where l(x, y) = [xa + yb]

• Distance in Γ relates to l1-distance in Z2

• To find d([u], [v]):• Fix a point in Z

2 with label [u]• Find the minimum l1 distance to a point labelled [v]

• The zero lattice X0 is the set of points with label [0]

• Can calculate all distances w.r.t. X0 (because Γ is vertextransitive)

Zero lattice for DL23(1, 6)

E.g. d([6], [14]) = d([0], [8]) = 3

Fundamental parallelogram

• Let {b1,b2} be a basis for the zero lattice of Γ = DLn(a, b)

• {b1,b2} defines a fundamental parallelogram

• Corners have label [0]

• Each vertex [v] appears exactly once as a label

• d([0], [v]) = l1-distance from point labelled [v] to a nearest corner

• [Zerovnik, Pisanski, 1993] Formula for diameter D in terms of a“packed” basis {b1,b2}:

THEOREM: If ||b1 + b2|| = ||b1 − b2|| > max{||b1||, ||b2||} and both||b1|| and ||b2|| are odd, then Γ has diameter D = ⌊||b1 + b2||/2⌋ − 1.Otherwise, D = ⌊||b1 + b2||/2⌋.

• Idea: vertices at distance D from [0] are in the centre of thefundamental parallelogram

Parallelogram for DL23(1, 6)

Basis vectors: b1 = (5, 3) and b2 = (−1, 4)

Some more number theory

• Recall: Frobenius double loop graphs have the form DLn(1, h)

where [h2] = [−1]

• n admits a primitive representation as a sum of two squares ifn = s2 + t2, and gcd(s, t) = 1

• There is a 1-1 correspondence between primitiverepresentations and Frobenius graphs DLn(1, h)

• E.g. 65 = 82 + 12 and 65 = 72 + 42.

• Given n = s2 + t2, we have ±[h] = ±[s][t]−1

• Given h such that [h2] = [−1], we get s as a remainder fromapplying the Euclidean algorithm to (n, h) (Smith’s Algorithm)

Zero lattice for DL29(1, 12)

A basis for the zero lattice is {(5, 2), (−2, 5)}. Note that 29 = 52 + 22.

Diameter Theorem

THEOREM: Let Γ = DLn(1, h) be a first-kind Frobenius double loopgraph. Then:

• The fundamental parallelogram for Γ is a square• A packed basis for the zero lattice is {(s, t), (−t, s)}, where

n = s2 + t2, s > t, and [s + th] = [0]

• Γ has diameter D = s − 1

Proof: Use some basic lattice theory, then apply diameter formula from[Zerovnik, Pisanski, 1993]

Different tiles for DL23(1, 6)

In general, the shortest path tile for a double loop graph is not unique.

Shortest path tile is unique

LEMMA: The shortest path tile for a Frobenius double loop graph isunique

• Idea of Proof:• Work in the fundamental square• Assume there are two different corners closest to the same

point• This contradicts either n = s2 + t2 or D = s − 1

COROLLARY: A given point of Z2 is closest to exactly one point of the

zero lattice• Solution to a (very) special case of the Discrete Voronoi problem

We only need 2 parameters to describe the tile shape...

Tile shape: DL85(1, 38)

A formula for π

THEOREM : Let Γ = DLn(1, h) be a Frobenius double loop graph, andlet s and t be positive integers such that n = s2 + t2, s > t and[s + th] = [0]. Then

• Γ has type-(1, 2, . . . , r − 1, r, µ, µ − 1, . . . , 2, 1) where• r = 1

2(s + t − 1)

• µ = 1

2(s − t − 1)

• Γ has forwarding indices

π(Γ) = 2~π(Γ) = 2~πm(Γ) = πm(Γ) = 1

12(3s3 + 2t3 + 3st2 − 3s − 2t)

Bounds on π and D

• Range of values for s and t lead to bounds on π and D.

THEOREM: Let Γ = DLn(1, h) be a Frobenius double loop graph. Thediameter of Γ satisfies:

⌈√

n/2⌉ − 1 ≤ D ≤ ⌊√

n − 1⌋ − 1

The edge-fowarding index of Γ satisfies:

⌈

1

12

√

n

2(4n − 5)

⌉

≤ π(Γ) ≤⌊(

1

4(n − 1)

3

2

)⌋

• Amongst all first-kind Frobenius graphs on n vertices, D and πare minimised simultaneously

• Lower bounds achieved by “optimal” infinite family• Upper bounds achieved by “windmill” infinite family

Infinite families (I)

“Optimal" double loop graphs (Yebra. et. al. 1985, Narayanan et. al.2001)

Infinite families (II)

“Windmill" double loop graphs

Thank you!

• X. G. Fang, C. H. Li, and C. E. Praeger, On orbital regular graphsand Frobenius graphs, Discrete Math. 182 (1998), 85–99

• L. Narayanan, J. Opatrny, and D. Sotteau, All-to-all opticalrouting in chordal rings of degree 4. Algorithmica 31 (2001), no.2, 155–178

• A. Thomson, S. Zhou, Frobenius circulant graphs of valency four• Y. Wang, X. G. Fang, and D. F. Hsu, On the edge-forwarding

indices of Frobenius graphs, Acta Mathematica Sinica, EnglishSeries, 22 (2006), no. 6, 1735–1744

• J.L.A. Yebra, M.A. Fiol, P. Morillo, and I. Alegre, The diameter ofundirected graphs associated to plane tessellations, Ars Comb B20 (1985), 159–171

• J. Žerovnik, T, Pisanski, Computing the diameter in multiple-loopnetworks, J. Algorithms, 13(2):226-243, 1993

• S. Zhou, A class of arc-transitive Cayley graphs as models forinterconnection networks, to appear