Table Of ContentPartition Parameters for Girth Maximum (m,r) BTUs
Vivek S Nittoor Reiji Suda
Department Of Computer Science, The University Of Tokyo
Japan
Abstract—Thispaperdescribesthecalculation 1.4 Symmetric Permutation Tree and its
3 of the optimal partition parameters such that the properties
1 girth maximum (m,r) Balanced Tanner Unit lies
0 infamilyofBTUsspecifiedbythem usingaseries A m Symmetric permutation tree SPT{m} is defined as
2 a labeled tree with the following properties:
ofproved resultsandthus creates a frameworkfor
n specifying a search problem for finding the girth
a 1. SPT{m} has a single root node labeled 0 .
maximum (m,r) BTU. Several open questions for
J
girth maximum (m,r) BTU have been raised. 2. SPT{m} has m nodes at depth 1 from the root
2
Keywords Permutation Groups, Cycle Index, node.
2
Balanced Tanner Unit, Girth Maximum (m,r)
3. SPT{m} has nodes at depths ranging from 1 to
] BTU
M m , with each node having a labels chosen from
{1,2,...,m} . The root node 0 has m successor
D
1 Introduction nodes. Each node at depth 1 has m−1 successor
s. nodesatdepth2. Each nodeatdepthihasm–i+1
c successornodesatdepthi+1. Eachnodeatdepth
[ We have introduced a family of bi-partite graphs called
Balanced Tanner Units (BTUs) in [1] and have intro- m–1 has 1 successor node at depth m .
2
v duced a family of graphs Φ(β1,β2,...,βr–1) . The goal 4. No successor node in SPT{m} has the same node
3 of this paper is to derive the forms for optimal partitions label as any of its ancestor nodes.
8 β1,β2,...,βr–1 suchthatthegirthmaximum(m,r)BTU
8 lies in Φ(β1,β2,...,βr–1) using a series of mathematical 5. Notwosuccessornodesthatshareacommonparent
6 results that build upon the fundamentals of BTUs intro- node have the same label.
.
2 duced in [1] .
6. The sequence of nodes in the path traversal from
1 We review a few definitions from [1].
the node at depth 1 to the leaf node at depth m in
2
1 SPT{m}representsthepermutationrepresentedby
v: 1.1 Set P2(m) the leaf node.
Xi P (m)referstothesetofpartitionsofm∈Nthatconsist 7. SPT{m} has m! leaf nodes each of which represent
2
an element of the symmetric groupof degree m de-
r of numbers that are greater than or equal to 2.
a noted by Sm .
1.2 Partition Component
1.5 Compatible Permutations
If β ∈ P (m) refers to y q = m then each {q } for
2 j=1 j j 1. Twopermutationsonasetofselementsrepresented
1≤j ≤y is referred to as a partition component of β.
P by (x1x2...xs);xp 6=xq ∀p6=q;1≤p≤s;1≤q ≤
s;p,q ∈ N where 1 ≤ x ≤ s;i ∈ N;1 ≤ i ≤ s and
i
1.3 Definition of Φ(β1,β2,...,βr–1) (y1y2...ys);yp 6= yq ∀p 6= q;1 ≤ p ≤ s;1 ≤ q ≤
s;p,q ∈ N where 1 ≤ y ≤ m;i ∈ N;1 ≤ i ≤ s are
i
Φ(β1,β2,...,βr–1)referstothefamilyofalllabeled(m,r) compatible if and only if x 6=y ∀i∈N;1≤i≤s .
i i
BTUs with compatible permutations p p ...,p ∈
1, 2, r
Sm;pi ∈/ C(p1,p2,...,pi−1) for 1 < i ≤ r that occur 2. A set of r permutations on a set of s elements rep-
in the same order on a complete m symmetric permu- resented by (xi,1xi,2...xi,s);xi,p 6=xi,q ∀p6=q;1≤
tation tree , x1,1 < x2,1 < ... < xr,1 where pj = p ≤ s;1 ≤ q ≤ s;p,q ∈ N where 1 ≤ xi,α ≤ s∀1 ≤
(xj,1xj,2...xj,m);1 ≤ j ≤ r , such that βi−1 is the par- i≤r;1≤α≤s;i,α∈Narecompatibleifandonly
tition between permutations pi–1 and pi for all integer if xi,α 6= xj,α ∀i 6= j;1 ≤ α ≤ s;1 ≤ i ≤ r;1 ≤ j ≤
values of i given by 1<i≤r . r; i,j,α∈N .
1.6 Notation compatible permutations p ,p ,...,p that occur in the
1 2 r
sameorderonacompletemsymmetricpermutationtree.
pi ∈/ C(Im,p2,...,pi–1) : pi is compatible with permuta- Let β ∈ P (m) be the partition represented between p
i 2 i
tions Im,p2,...,pi–1 . and p for 1 ≤ i ≤ r–1 . Hence, any (m,r) BTU is
i+1
isomorphic to an element of Φ(β1,β2,...,βr–1) for some
2 General Approach at Construct- choice of β1,β2,...,βr–1 ∈P2(m) .
ing (m,r) BTU with the maxi-
2.4 Implicit Enumeration Conjecture
mum girth
All non-isomorphic (m,r) BTUs can be enumerated by
2.1 Conjecture 1. An exhaustive enumeration of {β1,β2,...,βr–1}
where β1,β2,...,βr–1 ∈P2(m).
Any non-isomorphic (m,r) BTU can be constructed
2. For each of the above enumeration of
by choosing compatible permutations p ,p ,...,p ∈
1 2 r
S ;p ∈/ C(p ,p ,...,p ) for 1≤j ≤r−1 that occur {β1,β2,...,βr–1} we construct all non-isomorphic
m j+1 1 2 j
in the same order on a m complete symmetric permuta- (m,r)BTUwithβibeingthepartitionbetweenpi–1
andp forallintegervaluesofigivenby2≤i≤r−1
tion tree, x < x < ... < x i.e., x < x for i
1,1 2,1 r,1 j,1 j+1
1≤j ≤r−1,wherep =(x x ...x )for1≤i≤r. represented by the set Φ(β1,β2,...,βr–1) .
i i,1 i,2 i,m
Proof Without loss of generality, any labeled (m,r)
2.5 Definition of Micro-partition
BTU can be represented by a choice of permutations
p ,p ,...,p ∈ S ;p ∈/ C(p ,p ,...,p ) for 1 ≤ j ≤
1 2 r m j+1 1 2 j Given partitions β1,β2,...,βr−1 ∈ P2(m) with βi of
r−1 thatoccur in the same orderon a m complete sym- the form yi p = m;i ∈ N,1 ≤ i ≤ r − 1 , a
metric permutation tree, x < x < ... < x and j=1 i,j
1,1 2,1 r,1 micro-partition of β with respect to β is defined
i+1 i
henceanynon-isomorphic(m,r)BTUcanbeconstructed as x ∈NP∪{0};1≤i<r−1;1≤j ≤y ;1≤z ≤y
i,j,z i i+1
by choosing compatible permutations p1,p2,...,pr ∈ such that yi+1x = p ;1 ≤ j ≤ y ;1 ≤ i < r−1
Sm;pj+1 ∈/ C(p1,p2,...,pj) for 1≤j ≤r−1 that occur and yi x z=1=ip,j,z ;1i≤,j z ≤y ;1i≤i<r−1 .
in the same order on a m complete symmetric permuta- j=1Pi,j,z i+1,z i+1
tion tree, x1,1 <x2,1 <...<xr,1. 2.6PMicro-partition to label mapping
2.2 Conjecture For each micro-partition of βi+1 with respect to βi ,
x ∈ N∪{0};1 ≤ i < r−1;1 ≤ j ≤ y ;1 ≤ z ≤ y
i,j,z i i+1
Any non-isomorphic (m,r) BTU can be constructed we define Micro-partition to label mapping y(x )
i,j,z
by choosing compatible permutations p1,p2,...,pr ∈ as an ordered set of xi,j,z distinct labels.
S ;p ∈/ C(p ,p ,...,p ) for 1 ≤ j ≤ r−1 such that
m j+1 1 2 j
1. For β , the labels from {1,2,...,m} for each par-
x = jfor1 ≤ j ≤ r, where p = (x x ...x ) for 1
j,1 i i,1 i,2 i,m
tition component gets fixed by Ψ(β ).
1≤i≤r. 1
Proof Withoutlossofgenerality,anynon-isomorphic 2. For β2,...,βr−1 each distinct choice of a micro-
(m,r) BTU is isomorphic to a BTU represented by a set partition to label mapping leads to numerous la-
of permutations {p1,p2,...,pr} where p1,p2,...,pr ∈ beled graphs, which in turn would correspond to
Sm;pj+1 ∈/ C(p1,p2,...,pj) for 1 ≤ j ≤ r−1 such that elements in Φ(β1,β2,...,βr−1) .
x = j for 1 ≤ j ≤ r, since it could be brought to the
j,1
form by row exchanges or permutations on depth for the 2.7 Algorithm to enumerate all non-
permutation representation of the labeled (m,r) BTU.
isomorphic (m,r) BTUs for r ≥ 3
2.3 Theorem 1. Enumerate all distinct ordered combinations of
{β1,β2,...,βr–1} where β1,β2,...,βr–1 ∈P2(m) .
Any (m,r) BTU is isomorphic to an element of
Φ(β1,β2,...,βr–1) for some choice of β1,β2,...,βr–1 ∈ 2. aFtoeraeallchuniniqsutaenccoemobfi{nβa1t,ioβn2,s.o.f.,Mβri–c1r}o-,pwaretietniounmseor-f
P (m) .
2 β w.r.t. β for 1≤i≤r–1 .
Proof In general any (m,r) BTU can be diagonal- i+1 i
ized by suitable row and column exchanges, and with- 3. For each instance of a combinations of Micro-
out loss of generality, p = I .We map the positions partitions of β w.r.t. β for 1 ≤ i ≤ r–1 , we
1 m i+1 i
of 1 s in the first column to x ;2 ≤ i ≤ r where enumerate all sets of r–1 Unordered labeled parti-
i,1
p = (x x ...x ) . Thus, map the (m,r) BTU to tions.
i i,1 i,2 i,m
4. For each instance of a set of r–1 Unordered labeled ) {
partitions,weenumerateallpossiblesetsofordered We evaluate the girth of the chosen (m,r) BTU repre-
r–1 Ordered labeled partitions using Reduced Cy- sented by the chosen permutations p ,p ,...,p ∈ S
1 2 r m
cle Enumeration corresponding to each Unordered such that pi ∈/ C(p1,...,pi−1);1<i≤r;
labeled partition enumerated in the previous step. }
...
5. Weconstructp ,p ,...,p correspondingtothe set
2 3 r }
ofr–1Orderedlabeledpartitionsenumeratedinthe
}
previous step. ( p =I ).
1 m }
Choose p ,p ,...,p ∈ S with the best girth from
We shall develop the rationale for the this algorithm in 1 2 r m
the above explorations.
subsequent sections.
2.11 Theorem
2.8 Algorithm α
Algorithm α choosesa (m,r) BTUwiththebestgirth
1
A labeled (m,r) BTU can be characterized by a set of
in Φ(β1,β2,...,βr−1) given β1,β2,...,βr−1 ∈P2(m) .
compatible permutations p ,p ,...,p ∈ S such that
1 2 r m Proof The Algorithm α clearly covers the space
1
pi ∈/ C(p1,...,pi−1);1<i≤r . of all non-isomorphic (m,r) BTUs in the family
p =I ;
1 m Φ(β1,β2,...,βr−1) . Hence, the chosen p1,p2,...,pr ∈
for(each possible p ;p ∈/ C(p ) ) {
2 2 1 S by Algorithm α represents a (m,r) BTU with the
m 1
for(each possible p ;p ∈/ C(p ,p ) ) {
3 3 1 2 best girth in the family Φ(β1,β2,...,βr−1) .
...
for(each possible pr;pr ∈/ C(p1,p2,...,pr−1) ) { 2.12 Ordered labeled Partition
for(each chosen p ,p ,...,p ∈ S such that p ∈/
1 2 r m i
C(p1,...,pi−1);1<i≤r) An ordered labeled Partition of m consists of dis-
We evaluate the girth of the chosen (m,r) BTU;
tinct ordered y subsets B ,B ,...,B of the set A =
1 2 y
}
{1,2,...,m} such that
...
} 1. TheorderedsubsetssatisfytheconditionB ∩B =
i j
} Φ the null set ∀i6=j,1≤i≤y;1≤j ≤y
Choose p ,p ,...,p ∈ S with the best girth from
1 2 r m
the above explorations. 2. Each ordered set Bi ⊂ A with number of distinct
elements p ;1≤i≤y .
i
2.9 Theorem
2.13 Unordered labeled Partition
Algorithm α chooses a (m,r) BTU with the best girth.
Proof Even though the algorithm does not enumerate Given β ∈ P2(m) which refers to yi=1pi = m , an un-
orderedlabeledpartitionofm is a collectionofdistincty
all labeled (m,r) BTUs since we choose p = I , it
1 m P
subsets B ,B ,...,B of the set A = {1,2,...,m} such
clearlycoversthespaceofallnon-isomorphic(m,r)BTUs 1 2 y
that
with a huge amount of duplications since any labeled
(m,r) BTU is an element ofΨ(β1,β2,...,βr−1) for some
1. B ∩B =Φthenullset∀i6=j,1≤i≤y;1≤j ≤y
β1,β2,...,βr−1 ∈P2(m) . The algorithm clearly chooses i j
a (m,r) BTU with the best girth from elements of all 2. Each set B ⊂ A with number of distinct elements
i
possible sets Φ(β1,β2,...,βr−1) . pi;1≤i≤y .
2.10 Algorithm α1 2.14 Unordered labeled Partition Map-
ping Enumeration Problem
Enumerate all distinct possible {β1,β2,...,βr−1} such
that β1,β2,...,βr−1 ∈P2(m) ; Given β ∈ P (m) which refers to y p = m , the
for(each enumeration of β1,β2,...,βr−1 ∈P2(m)) { unordered lab2eled partition mapping Ein=u1mieration prob-
p =I ; P
1 m lem refers to the Enumeration of all distinct y subsets
for(each possible p ;β (p ,p );p ∈/ C(p ) ) {
2 1 1 2 2 1 B ,B ,...,B of the set A={1,2,...,m} such that
1 2 y
for(each possible p ;β (p ,p );p ∈/ C(p ,p ) ) {
3 2 2 3 3 1 2
... 1. B ∩B =Φthenullset∀i6=j,1≤i≤y;1≤j ≤y
i j
for(eachpossiblepr;βr−1(pr,pr−1);pr ∈/ C(p1,p2,...,pr−1) .
2. Each set B ⊂ A with number of distinct elements the number of distinct permutations {p ;β (p ,p ) =
i 2 1 2 1
p ;1≤i≤y. (m),p ∈/ C(p )} is given by
i 2 1
f(β)=(m−1)∗ (m−1)! .
2.15 Ordered labeled Partition Mapping j,distinctpj (p1,j−1)∗Qyi=11,i6=j(p1,i)
For β = (m) ; we obtain (m−1)! distinct permutations
P
Enumeration Problem ona m symmetric permutationtree, using the abovefor-
mula.
Given β ∈ P (m) which refers to y p = m , the
2 i=1 i
ordered labeled partition mapping Enumeration prob-
lem refers to the Enumeration of all Pdistinct ordered y 2.19 Conjecture
subsets B ,B ,...,B of the set A={1,2,...,m} such
1 2 y
Any labeled (m,2) BTU with the first permutation p =
that 1
I can be represented by a set of ordered labeled parti-
m
1. B ∩B =Φ,thenullset∀i6=j,1≤i≤y;1≤j ≤y tions on m elements.
i j
. Proof Since any labeled (m,2) BTU is isomorphic to
Ψ(β)forsome β ∈P (m) ,any labeled(m,2)BTUwith
2
2. Each ordered set Bi ⊂ A with number of distinct the first permutation p1 = Im and second permutation
elements pi;1≤i≤y . p2;p2 ∈/ C(p1);β(p1,p2);p1 = Im can be mapped to an
ordered labeled partition K(β,p ) on m in the following
1
2.16 Partitions → micro-partition → Un- manner. Each ordered subset in K(β,p1) consists of qi
ordered labeled Partitions → Or- elementsofdistinctlabels{li,1,li,2,...,li,yi}suchthatla-
belsl inp andl inp arelocatedatthesamedepth,
dered labeled Partitions → Compat- i,1 1 i,2 2
labels l in p and l in p are located at the same
ible Permutations → labeled (m,r) i,2 1 i,3 2
depth, ..., and finally, labels l in p and l in p are
BTUs i,yi 1 i,1 2
located at the same depth. Thus,
K(β,p ) has the form (l ,...,l ),(l ,...,l ),
Givenasetofpartitionsβ1,β2,...,βr–1 ∈P2(m), wecan 1 1,1 1,y1 2,1 2,y1
...,(l ,...,l ) .
have many possible sets of micro-partitions. For each set y1,1 y1,y1
ofmicro-partitions,wecanhavemanypossibleunordered
labeledpartitions. Foreachsetof unorderedlabeledpar- 2.20 Corollary
titions, we canhave many possible orderedlabeledparti-
tions. For each set of r–1 ordered labeled partitions, we Given an ordered labeled partition on m represented by
can construct a set of compatible permutations K(β,p1) , any circular permutation on ordered subsets
{Im,p2,...,pr–1} which represents a labeled (m,r) yields the same labeled (m,2) BTU.
BTU.
2.21 Corollary
2.17 Multiple levels of enumeration
Given an ordered labeled partition on m represented by
Depth Permutation & Label permutations that preserve K(β,p ) , with first permutation p = I , p gets pre-
1 1 m 2
p1 andp2 andcreatedifferentinstancesofp3 forthesame cisely defined resulting in a unique labeled (m,2) BTU.
1. Enumerationofallpossiblemicro-partitionsofβ
i+1
w.r.t. β . 2.22 Conjecture
i
2. For each micro-partition, enumeration of all possi- Any labeled (m,r) BTU with the first permutation p1 =
blenon-orderedlabeledpartitionscorrespondingto Im can be represented by a set of r–1 ordered labeled
β . partitions on m elements.
i+1
Proof Let the given labeled (m,r) BTU con-
3. For each non-ordered labeled partition, enumerate sists of permutations p = I and p ;p ∈/
1 m i+1 i+1
all cycle orders. C(p ,p ,...,p );p = I ;2 ≤ i ≤ r − 1.Starting
1 2 i 1 m
with p = I , the compatible permutation p can be
1 m 2
2.18 Constrained labeled Partition Map- represented as an ordered labeled partition K1(β1,p1)
on m in the following manner. Each ordered subset
ping Problem
in K (β ,p ) consists of q elements of distinct labels
1 1 1 1
The Permutation Enumeration formulae derived in [1] {l ,l ,...,l } such that labels l in p and
1,1,1 1,1,2 1,1,yi 1,1,1 1
gives us the number of candidate permutations corre- l in p are located at the same depth, labels l
1,1,2 2 1,1,2
sponding to a specified partition. We recallfrom [1] that in p and l in p are located at the same depth, ...,
1 1,1,3 2
andfinally, labels l in p andl in p are located cycle order on a subset B is an ordered subset C with
1,1,y1 1 1,1,1 2 i i
at the same depth. the same elements suchthat O =C ∪C ∪...∪C is an
1 2 y
The compatible permutation p ;p ∈/ ordered labeled partition on m .
i+1 i+1
C(p ,p ,...,p );p = I ;2 ≤ i ≤ r − 1 can be repre-
1 2 i 1 m
sented as an ordered labeled partition Ki(βi,pi) on m in 2.26 Corollary
the following manner. Each ordered subset in K (β ,p )
i i i
consists of q ;1 ≤ j ≤ y elements of distinct labels A defined cycle orderon eachunorderedsubset of anun-
j i
{l ,l ,...,l } such that labels l in p and ordered labeled partition on m , gives us an ordered
i,j,1 i,j,2 i,j,yi i,j,1 i
l in p are located at the same depth, labels l labeled partition on m .
i,j,2 i+1 i,j,2
in p and l in p are located at the same depth, ...,
i i,j,3 i+1
andfinally, labelsl inp andl inp arelocated 2.27 Conjecture
i,j,yi i i,j,1 i+1
at the same depth.
Any labeled (m,r) BTU with the first permutation p =
Thus, any labeled (m,r) BTU with the first permu- 1
I can be represented by a set of r–1 sets of micro-
tation p = I can be represented by a set of r–1 or- m
1 m partitions of β w.r.t. β for 1 ≤ i ≤ r−1 , mapping
dered labeled partitions on m elements represented by i+1 i
from micro-partitions to labels, and defined cycle orders
{K (β ,p )};1≤i≤r−1 .
i i i
for each partition component.
Each K (β ,p ) has the form (l ,l ,...,l ),
i i i i,1,1 i,1,2 i,1,yi Proof Any labeled (m,r) BTU with the first permuta-
(l ,l ,...,l ),..., (l ,l ,...,l ) .
i,2,1 i,2,2 i,yi i,yi,1 i,yi,2 i,yi,yi tion p = I can be represented by a set of r–1 ordered
1 m
labeledpartitionsonm. Eachofthe r–1orderedlabeled
2.23 Conjecture partitionscouldberepresentedby r–1unorderedlabeled
partitions and specific cycle orders for each subset given
A set of r−1 ordered labeled permutations on m rep-
separately.
resented by {K (β ,p )};1 ≤ i ≤ r − 1 corresponds
i i i Eachofthe r–1unorderedlabeledpartitionscouldbe
to a labeled (m,r) BTU, if the resultant permuta-
uniquelyspecifiedby asetofr–1setsofmicro-partitions
tions {p };1 ≤ i ≤ r − 1 are compatible with
i+1 of β w.r.t. β for 1≤i≤r−1 , mapping from micro-
i+1 i
{p ,p ,...,p };1≤i≤r−1 .
1 2 i partitions to labels. Hence, it follows that any labeled
Proof This follows from the fact that permutations
(m,r) BTU with the first permutation p = I can be
{p ,p ,...,p } correspond to a labeled (m,r) BTU if 1 m
1 2 r representedbyasetofr–1setsofmicro-partitionsofβ
i+1
and only if p ∈/ C(p ,p ,...,p ) for 1≤i≤r–1 .
i+1 1 2 i w.r.t. β for1≤i≤r−1,mappingfrommicro-partitions
i
tolabels,anddefinedcycleordersforeachpartitioncom-
2.24 Conjecture ponent.
If the first permutation p of a labeled (m,r) BTU is
1 2.28 Corollary
specified, andgivenset ofr−1orderedlabeledpermuta-
tionsonmrepresentedby{Ki(βi,pi)};1≤i≤r−1such Given a set of permutations p1,p2,...,pi where p1 =Im
that the resultant permutations {pi+1};1≤i≤r−1 are and pi+1 ∈/ C(p1,p2,...,pi), each permutation satisfy-
compatible with {p1,p2,...,pi};1 ≤ i ≤ r−1, then the ingthecondition{pi+1;βi(pi+1,pi),pi+1 ∈/ C(p1,...,pi)}
labeled (m,r) BTU is unique. can be represented by micro-partitions of β w.r.t. β
i+1 i
Proof This follows from the fact that permutations for 1 ≤ i ≤ r−1, i unordered labeled partitions corre-
{p1,p2,...,pr} correspond to a labeled (m,r) BTU if sponding to micro-partitions, and finally i ordered la-
andonlyif pi+1 ∈/ C(p1,p2,...,pi) for 1≤i≤r–1 and beled partitions.
since the first permutation p is specified, we obtain
1
p2 using {K(β1,p1)} . Similarly, we obtain pi+1 from 2.29 Micro-partitions Enumeration
p and {K } for 1 ≤ i ≤ r–1 . We do not sepa-
i (βi,pi) Problem
rately need an explicit specification of β ;1 ≤ i ≤ r−1
i
since it is alreadyimplicitly specifiedin the definitions of Given β ,β ∈ P (m) , the micro-partition enumera-
i+1 i 2
{Ki(βi,pi)};1≤i≤r−1. tion problem refers to enumeration of all possible micro-
partitions of β w.r.t. β .
i+1 i
2.25 Definition Of Cycle Order
2.30 Micro-partitions Mapping Enumer-
Given an unordered labeled partition U = B ∪ B ∪
1 2 ation Problem
...∪B on m such that B ∩B = Φ the empty set for
y i j
i 6= j,1 ≤ i ≤ y;1 ≤ j ≤ y , and each unordered subset Given micro-partitions of β w.r.t. β and
i+1 i
B contains distinct elements of the set {1,2,...,m} , a p ,p ,...,p to enumerate all possible unordered la-
i 1 2 i
beled partitions of m with each subset of labels corre- with the minimum label number fromeachordered
sponding to each of the cycles of β . labeledpartitionsofmcorrespondingto p ∈/ C(p )
i+1 2 1
w.r.t. p .
1
2.31 Conjecture Micro-partitions to Per-
mutation Counting {p3;β2(p3,p2),p3 ∈/ Wedonotloseanynon-isomorphic(m,r)BTUthatcould
beconstructedfromthesetofr−1unorderedlabeledpar-
C(p2,p1 = Im)}
titions of m in the enumerationprocess by imposing this
Given micro-partitions x of β w.r.t. β , the num- constraint.
1,j,z 2 1
ber of ways to map the micro-partitions to unordered
labeled partitions is given by the following expression
p − z−1x 2.34 Micro-partitions Isomorphism The-
yj=11 yz2=1 1,j k=1 1,j,k . orem for (m,r) BTUs
xP1,j,z !
Q PrQoof This follows directly from number of ways
Enumeration of all non-isomorphic elements in
to choose x elements from p − z−1x ele-
1,j,z 1,j k=1 1,j,k Φ(β1,β2,...,βr−1)
ments for 1 ≤ z ≤ y and 1 ≤ j ≤ y which yields
p − z2−1x P1 Allnon-isomorphic(m,r)BTUsinΦ(β1,β2,...,βr−1)
y1 y2 1,j k=1 1,j,k . where β1,β2,...,βr−1 ∈ P2(m) are enumerated at least
j=1 z=1
xP1,j,z ! once by
Q Q
2.32 Cycle Order Enumeration Problem 1. An exhaustive enumeration of Combinations of
Micro-partitions of β w.r.t. β for 1≤i≤r–1 .
i+1 i
Givenasetofr−1unorderedlabeledpartitionsofm,to
enumerate all possible distinct r−1 orderedlabeled par-
2. Anexhaustiveenumerationofsetsofr–1Unordered
titions,eachofwhichreferadistinctchoiceofcompatible
labeled partitions for each choice of Combinations
permutation. p ∈/ C(p ,p ,...,p )w.r.t. p ,p ,...,p
i+1 1 2 i 1 2 i ofMicro-partitionsofβ w.r.t. β for 1≤i≤r–1
for 1 ≤ i ≤ r–1 . Each of enumerated r−1 ordered la- i+1 i
enumerated in the previous step.
beled partitions at each stage have to satisfy the criteria
for compatible permutations as far as labels at various
3. Restricted Cycle Order Enumeration of Sets of r–1
depths are concerned in order to correspond to a labeled
Ordered labeled partitions that result in compati-
(m,r) BTU.
ble permutations corresponding to each Unordered
labeled partition enumerated in the previous step.
2.33 Restricted Cycle Order Enumera-
tion Problem
4. We construct p ,p ,...,p corresponding to each
2 3 r
setofr–1Orderedlabeledpartitionsenumeratedin
Given a set of r−1 unordered labeled partitions of m
the previous step. ( p =I )
, to enumerate all possible distinct r−1 ordered labeled 1 m
partitions, each of which refer a distinct choice of com-
patible permutation. p ∈/ C(p ,p ,...,p ) w.r.t. Proof Since any labeled (m,r) BTU with the first per-
i+1 1 2 i
p ,p ,...,p for 1≤i≤r–1 . Eachofenumerated r−1 mutationp =I canbe representedby a set of r–1 sets
1 2 i 1 m
ordered labeled partitions at each stage have to satisfy of micro-partitions of β w.r.t. β for 1 ≤ i ≤ r−1 ,
i+1 i
the criteria for compatible permutations as far as labels mapping from micro-partitions to labels, and defined cy-
at various depths are concerned in order to correspond cleordersforeachpartitioncomponent,itfollowsthatfor
to a labeled (m,r) BTU. For Restricted Cycle Or- eachnon-isomorphic(m,r)BTU,thereexistsatleastone
der Enumeration, we impose the following additional setofmicro-partitionsofβ w.r.t. β for1≤i≤r−1,
i+1 i
constraints one set of mapping from micro-partitions to labels, and
one set of defined cycle ordersfor each partition compo-
1. Without loss of generality, we restrict p1 =Im. nent.
Without loss of generality, any non-isomorphic (m,r)
2. p = Ψ(β ) . The first set of the r−1 ordered la-
2 1
BTU can be mapped to a
beled partitions on m corresponding to choices for
one set of micro-partitions of β w.r.t. β for 1 ≤
p gets fixed due to this. i+1 i
2
i≤r−1,toonesetof mappingfrommicro-partitionsto
3. While choosing the second set of of the r −1 or- labels using Restricted Cycle Order Enumeration,
deredlabeledpartitionsofmcorrespondingtop ∈/ and one set of defined cycle orders for each partition
3
C(p ,p ) w.r.t. p ,p , we choose the first element component.
1 2 1 2
2.35 Corollary Micro-partitions Isomor- 3 Direct Construction
phism for (m,3) BTUs
3.1 Generalized Cycle Traversal for a
All non-isomorphic(m,3)BTUs inΦ(β1,β2)canbe enu- permutationrepresentation ofa(m,r)
merated by BTU
If labels l and l occur at the same depth, labels l and
1. An Exhaustive enumeration of Micro-partitions of 1 2 2
l occur at the same depth, ... , and finally labels l and
β w.r.t. β . 3 x
2 1
l occur at the same depth, in the permutation represen-
1
tation of a labeled (m,r) BTU, then there exists a cycle
2. AnExhaustiveenumerationof2Unorderedlabeled
connecting the labels l ,l ...,l .
partitionsforeachchoiceofCombinationsofMicro- 1 2, x
partitions of β w.r.t. β enumerated in the previ-
2 1
3.2 Known Cycle Conjecture for a (m,2)
ous step.
BTU
3. RestrictedCycleOrderEnumerationofSetof2Or-
The cycle lengths of a (m,2) BTU that is isomorphic to
dered labeled partitions that result in compatible Ψ(β) for some β ∈ P (m) given by y q = m are
permutations corresponding to each set of 2 Un- 2 i=1 i
{2∗q };1≤i≤y .
i
orderedlabeled partitionsenumeratedin the previ- P
Proof A (m,2) BTU that is isomorphic to Ψ(β) has no
ous step.
other cycles other than that of β ∈ P (m) given by
2
y q = m . The cycle length for a partition compo-
4. We construct p ,p corresponding to the set of 2 i=1 i
2 3 nent q is 2∗q . Hence, it follows that the cycle lengths
i i
Ordered labeled partitions enumerated in the pre- P
are are {2∗q };1≤i≤y .
i
vious step. ( p =I ).
1 m
3.3 Maximum possible girth of a (m,2)
2.36 Search Problem for BTU with best BTU
girth in Φ(β1,β2,...,βr−1) The maximum possible girth of a (m,2) BTU is 2∗m .
Proof This directly follows when we consider that every
Given β1,β2,...,βr−1 ∈ P2(m) , we exhaustively enu- (m,2)BTUcanbemappedtoΨ(β)whereβ ∈P (m). It
merate 2
isclearthatgirthofa(m,2)BTUis2∗min(q );1≤i≤y
i
where y q =m represents β ∈P (m) . Hence, it fol-
i=1 i 2
1. Combinations of Micro-partitions of βi+1 w.r.t. βi lowsthat the maximumpossible girthofa (m,2) BTUis
for 1≤i≤r–1 . 2∗m .P
2. A set of r–1 Unordered labeled partitions for each 3.4 Upper Bounds Known partition com-
choice of Combinations of Micro-partitions of β
i+1 ponent upper bound Conjecture
w.r.t. β for1≤i≤r–1enumeratedintheprevious
i
step. If u = min(q ;1 ≤ j ≤ y ;1 ≤ i ≤ r–1) where each par-
i,j i
tition βi is given by yj=i1qi,j =m for β1,β2,...,βr−1 ∈
3. Setsofr–1Orderedlabeledpartitionsthatresultin P (m) where each u,q ∈N , then the maximum possi-
2 i,j
P
compatible permutations correspondingto each set ble girth of all (m,r) BTUs in Φ(β1,β2,...,βr−1) is less
of r–1 Unordered labeled partition enumerated in than or equal to 2∗u . This is an upper bound on the
the previous step. possible possible girth.
Proof This follows directly from the fact that there
4. Weconstructp ,p ,...,p correspondingtothe set can be smaller cycles caused due to interactions between
2 3 r
ofr–1Orderedlabeledpartitionsenumeratedinthe thepartitionsβ1,β2,...,βr−1 ∈P2(m)andthemaximum
previous step. ( p1 =Im ). girthofall(m,r)BTUsinΦ(β1,β2,...,βr−1)islessthan
or equal to 2∗u , with strict equality when r=2.
5. We evaluate the girth for each of the enumer-
ated/constructed (m,r) BTUs. 4 Micro-partition cycles
6. We choose the BTU with the best girth at the end For a (m,r) BTU Φ(β1,β2,...,βr−1) where
of this process. β1,β2,...,βr−1 ∈P2(m) given by {p1,p2,...,pr};pi+1 ∈/
C(p ,p ,...,p ) for 1 ≤ i ≤ r–1. If each β refers 4.4 All cycles caused
1 2 i i
to yi q = m for 1 ≤ i ≤ r − 1, the micro-
j=1 i,j Given a labeled (m,r) BTU with compatible permuta-
partition cycles are the cycles caused by interaction
betwPeen the partition components of β and β where tions {p1,p2,...,pr} in Φ(β1,β2,...,βr−1) where each
u v β ∈ P (m) refers to yi q = m for 1 ≤ i ≤ r–1
u6=v;1≤u≤r−1;1≤v ≤r−1 are the cycles caused i 2 j=1 i,j
we categorize its cycles in the following manner
due to interactions between the known cycles of βu and P
β namely {q ,q ,...,q }and{q ,q ,...,q } 1. Known cycles : These cycles refer to the cycles
v u,1 u,2 u,yu v,1 v,2 v,yv
and corresponding micro-partitions {x };1 ≤ c ≤ {q } for 1≤j ≤y and 1≤i≤r–1 corresponding
u,v,c,d i,j i
yu;1≤d≤yv and {xv,u,d,c};1≤c≤yu;1≤d≤yv. to β1,β2,...,βr−1 .We have r ∗(r–1)/2 partitions
in total arising from all possible combinations of 2
permutations from the set of r permutations, out
4.1 When do micro-partition cycles
of which r–1 partitions are considered for known
arise?
cycles.
When x 6= 1 and x 6= 0 for some {c,d} where 2. Cycles due to other partitions: We consider gener-
u,v,c,d u,v,c,d
1 ≤ c ≤ y ;1 ≤ d ≤ y , we have more than one point alized partitions α ∈P (m) where |(u−v)|6=0,
u v u,v 2
from partition component q is used for creating parti- |(u−v)| 6= 1 and 1 ≤ u ≤ r;1 ≤ v ≤ r. The
u,c
tion component q , then we have an additional cycle number of partitions considered here are
v,d
referred to as micro-partition cycle. r∗(r–1)/2–(r–1)=r2/2–r/2+1 .
3. Micro-partition cycles if they arise due to interac-
4.2 Conjecture for Length of micro-
tions between the combinations of r∗(r–1)/2 per-
partition cycle mutations.
For a (m,r) BTU in Φ(β1,β2,...,βr−1) where 4. Hidden cycles caused due to interaction of all the
β1,β2,...,βr−1 ∈P2(m) , the maximum possible length above cycles.
ofmicro-partitioncycleismin{2∗(q /x +t −1)};
u,j u,v,j,z u
x ≥ 2;1 ≤ j < y ;1 ≤ z ≤ y where β refers to 4.5 Strategy for girth maximization
u,v,j,z u v i
yi q =mfor1≤i≤r−1;i∈N,whereGeneralized
j=1 i,j
Micro-partition between β ,β ∈ P (m);u 6= v; 1 ≤ u ≤ Inorderto constructa (m,r) BTUwith maximumgirth,
u v 2
Pr–1;1 ≤ v ≤ r–1 : x where 1 ≤ j ≤ y ;1 ≤ z ≤ y we choose
u,v,j,z u v
. t which is the number of partition components of β
u u 1. Partitions β1,β2,...,βr−1 ∈ P2(m) such that the
connected by the micro-partitioncycle , 1≤t ≤y .
u u known cycles are maximized.
Proof If x 6=1 and x 6=0 for some {c,d}
u,v,c,d u,v,c,d
where 1 ≤ c ≤ y ;1 ≤ d ≤ y , we have more than 2. By maximizing the length of the micro-
u v
one point frompartitioncomponent q is used for cre- partition cycle, we obtain optimal parameters for
u,c
ating partition component qv,d , then we have a micro- β1,β2,...,βr−1 .
partition cycle. Since the number of partition compo-
3. Wesearchforpermutationssuchthatthecyclesdue
nents of β connected by the micro-partition cycle is
u tootherpartitionsandhiddencycles causeddueto
t , where 1 ≤ t ≤ y , we have a smaller cycle which
u u u interaction of all the above cycles is maximized.
cantakethe maximumvalue {2∗(q /x +t −1)}
u,j u,v,c,d u
, by choosing the x points appropriately. Hence ,
u,v,c,d 4.6 Self Evident Fact About the Girth of
the maximum possible length of micro-partition cycle is
min{2∗(q /x +t −1)}; x ≥ 2;1 ≤ j < a member of Φ(β1,β2,...,βr−1)
u,j u,v,j,z u u,v,j,z
y ;1≤z ≤y .
u v If2∗visthelengthoftheminimummicro-partitioncycle,
and u ∈ N is the smallest partition component among
4.3 Corollary for (m,3) BTU given β1,β2,...,βr−1 ∈P2(m) i.e., u=min(qi,j;1≤j ≤
y ;1≤i≤r–1) where each β is givenby yi q =m,
i i j=1 i,j
For a (m,3) BTU in Φ(β ,β ) where β ,β ∈ P (m) , if and2∗w∈Nisthethelengthofthesmallestcyclecaused
1 2 1 2 2
P
micro-partitioncyclesexist,theminimumpossiblelength duetointeractionsbetweenthemicro-partitioncyclesand
of micro-partition cycle is min{2∗(q1,j/x1,j,z +t1−1)} cycles due to β1,β2,...,βr−1, the girth of a member of
where the micro-partitionsx1,j,z ≥2;1≤j <y1;1≤z ≤ Φ(β1,β2,...,βr−1) is 2∗min(u,v,w) or 2∗min(u,w) if
y where β refers to yi q = m . t which is the there are no micro-partition cycles or 2∗u for r = 2, in
2 i j=1 i,j 1
number of partition components of β connected by the which case the only cycles that arise due to one element
1
P
micro-partition cycle , max(t )=y . of P (m).
1 1 2
4.7 (k,2) BTU puncturing Conjecture y ;y 6= y ;y 6= y . Let us define h(w ,w ,k) =
2 2 3 3 1 1 2
min{|(w –w )|,k−|(w –w )|}
1 2 1 2
If a 0 element in a (k,2) BTU with cycle length 2∗k is
Wecanverifythattraversalslengthsbetweenanytwo
changed to 1, then length of new minimum cycle within
of the three points through B are 2 ∗h(x ,x ,k)+1,
the sub-block l ∈ N satisfies 4 ≤ l ≤ k if k is an even 1 1 2
2∗h(x ,x ,k)+1 and 2∗h(x ,x ,k)+1.
2 3 3 1
positive integer and 4 ≤l ≤ k+1 if k is an odd positive
Wecanverifythattraversalslengthsbetweenanytwo
integer.
of the three points through B are 2∗h(y ,y ,k)+1 ,
2 1 2
Proof Let us map the (k,2) BTU to a labeled
2∗h(y ,y ,k)+1and2∗h(y ,y ,k)+1. Thecorrespond-
2 3 3 1
directed graph with k vertices such that each vertex
ing cycle lengths are 2∗h(x ,x ,k)+2∗h(y ,y ,k)+2 ,
1 2 1 2
i;1 ≤ i ≤ k is connected to vertex i+1 mod k. If the
2∗h(x ,x ,k)+2∗h(y ,y ,k)+2 and 2∗h(x ,x ,k)+
2 3 2 3 3 1
distance between two vertexes is defined as the lenght of
2∗h(y ,y ,k)+2 .
3 1
shortest traversals in the same direction of the directed
If possible let the length of the minimum cycle be
edges,itis clearthatthe maximumdistancemeasuredin
greater than or equal to 2∗k, which implies that
termsofnumberofdirectedtraversalsfromonevertexto
h(x ,x ,k) + h(y ,y ,k) ≥ k − 1 , h(x ,x ,k) +
the next, between two vertexes on this labeled directed 1 2 1 2 2 3
h(y ,y ,k)≥k−1 and h(x ,x ,k)+h(y ,y ,k)≥k−1
graphis k/2 for even positive integers k and (k+1)/2 2 3 3 1 3 1
. Thisgivesrisetoacontradictionsincetheupperbound
for odd positive integers k, and the minimum distance
on maximum attainable value of
measured in terms of number of directed traversals from
min(|(x −x )|,|(x −x )|,|(x −x )|) is k/3 and
one vertex to the next, between two vertexes on this la- 1 2 2 3 3 1
similarly upper bound on maximum attainable value of
beled directed graph is 1. If a directed edge is connected
min(|(y −y )|,|(y −y )|,|(y −y )|) is k/3 .
betweentwovertexesofminimumdistanceof1,thisleads 1 2 2 3 3 1
Hence. The maximum attainable girth when three 1
toaminimumcyclelengthof4onthematrixrepresenta-
s are placed in CB(1,2) is strictly less than 2∗k.
tion. Ifadirectededgeisconnectedbetweentwovertexes
Case 3: Three 1 s in CB(2,1)
ofmaximumdistancek/2forevenpositiveintegersk and
By repeating the argument for three 1 s in CB(1,2)
(k+1)/2foroddpositiveintegersk,wegetacyclelength
we can show that the maximum attainable girth when
of k for even positive integers k and (k+1) for odd pos-
three1s areplacedinCB(2,1)is strictlylessthan2∗k .
itive integers k in the equivalent matrix representation.
Case 4: Two 1 s placed in CB(1,2) and one 1 placed
Hence, the length of new minimum cycle within the
sub-block l ∈N satisfies 4≤l ≤k if k is an evenpositive in CB(2,1)
Maximum value of traversallength from one point to
integer and 4≤l ≤k+1 if k is an odd positive integer.
another through B is k/3. Maximum value of traver-
1
sal length from one point to another through B is k/3
2
4.8 Three -one Conjecture . Hence. The maximum attainable girth when two 1 s
placedinCB(1,2)andone1placedinCB(2,1)isstrictly
Letusconsidera(2∗k,2)BTUconstructedwithp1 =I2∗k less than 2∗k .
andp asperΨ((k,k)),andifwehavetoadditionallycon-
2 Case 5: Two 1 s placed in CB(2,1) and one 1 placed
vert three 0 s in this BTU to 1 s, the girth is strictly less
in CB(1,2)
than 2∗k .
By repeating the argument for two 1 s placed in
Proof Girth of a (2 ∗ k,2) BTU constructed with
CB(1,2) and one 1 placed in CB(2,1) we can show that
p1 = I2∗k and p2 as per Ψ((k,k)) is 2∗k. Let us de- the maximum attainable girth when two 1 s placed in
notethe (2∗k,2)BTUconsistingofsub-matricesB and
1 CB(2,1)andone1placedinCB(1,2)isstrictlylessthan
B eachofwhicharek×k matricesthatrepresenta con-
2 2∗k .
stituent (k,2) BTU, and two k×k matrices referred to
Hence, given a (2 ∗ k,2) BTU constructed with
as CB(1,2) that shares its rows with B and columns
with B and CB(2,1) that shares its row1s with B and p1 = I2∗k and p2 as per Ψ((k,k)) , and if we have to
2 2 additionally convert three 0 s in this BTU to 1 s, the
columns with B .
1 girth is strictly less than 2∗k .
Let us consider different cases for placement of the
three 1 s.
4.9 Upper bound on the maximum at-
Case1: Ifa1isplacedinsideeitheroftheconstituent
(k,2) BTUs, by the previous theorem, the girth reduces tainable girth for the case when no
to k+1 if k is odd, and k if k is even. micro-partition cycles arises for a
Case 2: Three 1 s in CB(1,2) (k2,3) BTU
Let the positions of the three 1 s in CB(1,2) be
(x1,y1),(x2,y2),(x3,y3) such that 1 ≤ xi ≤ k;1 ≤ yi ≤ k Nomicro-partitioncyclesarisewhenβ1,β2 ∈P2(m)both
for 1 ≤ i ≤ 3 and x 6= x ;x 6= x ;x 6= x ;y 6= correspond to the partition k k = k2. Let us con-
1 2 2 3 3 1 1 j=1
P
struct a labeled (k2,2) BTU with p = I and p ∈S Now,thecyclelengthdueto{CB(1,k–1),CB(k–1,1)}
1 k2 2 k2
as per Ψ(β ). Let us consider the labeled (k2,2) BTU andsub-blocks1 andk is now 2∗k since the function for
1
as consisting of k constituent (k,2) BTUs which we re- traversallength between two points (x y ) and (x y )
1, 1 2, 2
fer to as sub-blocks numbers from {1,2,...,k} and k2–k is now 2∗|(x –x )|+1 and2∗|(y –y )|+1 insteadof
1 2 1 2
cross-blocks CB(i,j) where 1 ≤ i ≤ k;1 ≤ j ≤ k;i 6= j. 2∗min{|(x –x )|,k−|(x –x )|}+1 and
1 2 1 2
Each cross-block CB(i,j) shares its rows with sub-block 2∗min{|(y –y )|,k−|(y –y )|}+1 .
1 2 1 2
i and shares its columns with sub-block j . The same is true for {CB(2,1),CB(1,2)} , ... ,
Let p3 ∈ Sk2 be the permutation that maximizes the {CB(k,k–2),CB(k–2,k)} .
girth among all possible (k2,3) BTUs in Ψ(β1,β2). Now, let us examine the situation that leads to the
Theremustexistatleastonecross-blockineachcross- constraintonmaximumattainableminimumcyclelength
block row that has two 1 s from p3. There must exist at when no micro-partitions cycles arise, and see that the
least one cross-blockin each cross-blockcolumn that has maximum attainable minimum cycle length is better for
two 1 s from p3 . when micro-partitions cycles arise.
Hence, there must exist u,v such that u 6= v and Let us consider CB(u,v) and CB(v,u) with
1 ≤ u ≤ k;1 ≤ v ≤ k such that CB(u,v) and CB(v,u) with three 1 s between both the cross-blocks in the
have two 1 s and one 1 respectively. pair.
If we consider a 2∗k ×2∗k matrix consisting of a Case 1: Three 1 in CB(u,v) zero 1 s in CB(v,u) .
(k,2) BTU B1 and CB(u,v) in the same rows,and (k,2) Let the positions of the three 1 s in CB(u,v) be
BTU B2 and CB(u,v) in the same columns and conse- (x1,y1),(x2,y2),(x3,y3) such that 1 ≤ xi ≤ k;1 ≤ yi ≤ k
quently,B1 andCB(v,u)sharethesamecolumnsandB2 for1≤i≤3andx1 6=x2;x2 6=x3;x3 6=x1;y1 6=y2;y2 6=
and CB(v,u) share the same rows. y ;y 6=y . Let us define h (w ,w )=|(w –w )|
3 3 1 2 1 2 1 2
Byusing the previouslyprovedresult,girthis strictly Wecanverifythattraversalslengthsbetweenanytwo
less than 2∗k . of the three points through B are 2 ∗h (x ,x ) + 1 ,
1 2 1 2
Hence,the maximumattainable girth of(k2,3)BTU 2∗h (x ,x )+1 and 2∗h (x ,x )+1 .
2 2 3 2 3 1
with no micro-partition cycles is strictly less than 2∗k . Wecanverifythattraversalslengthsbetweenanytwo
of the three points through B are 2∗h (y ,y )+1 ,
2 2 1 2
4.10 Conjecture 2∗h (y ,y )+1 and 2∗h (y ,y )+1 .The correspond-
2 2 3 2 3 1
ing cycle lengths are 2∗h (x ,x )+2∗h (y ,y )+2 ,
1 1 2 2 1 2
The maximum attainable girth for a (k2,3) BTU for the
2∗h (x ,x )+2∗h (y ,y )+2 and 2∗h (x ,x )+2∗
2 2 3 2 2 3 2 3 1
case whenmicro-partitioncycles ariseis greaterthanthe
h (y ,y )+2 .
2 3 1
maximum attainable girth for a (k2,3) BTU for the case
Thus, the length of the minimum cycle is increased
when no micro-partition cycles arise.
comparedtotheprevioustraversallength2∗h(y ,y ,k)+
3 1
Proof For the case where no micro-partition cycles 1 where h(y ,y ,k)=min{|(y –y )|,k−|(y –y )|}.
3 1 1 2 1 2
arise, let p ∈S maximize the girth among all possible
3 k2 Case 2: We can similarly show that the one Cross-
(k2,3) BTUs in Ψ(β,β) where β ∈P (m) corresponds to
2 block has two 1s and other in the pair has one 1s, the
the partition kj=1k = k2 with p1 = Ik2 and p2 ∈ Sk2 lengthoftheminimumcycleisincreasedforthecasewhen
as per Ψ(β) . Let us consider the labeled (k2,2) BTU as micro-partitioncycles arise.
P
consistingof k constituent(k,2)BTUswhichwereferto Similarly, we can also show that for the case when
as sub-blocks numbers from {1,2,...,k} and k2–k cross- CB(u,v) and CB(v,u) with
blocks CB(i,j) where 1 ≤ i ≤ k;1 ≤ j ≤ k;i 6= j . Each withtwo1sbetweenboththecross-blocksinthepair,
cross-block CB(i,j) shares its rows with sub-block i and has the length of the maximum cycle increased.
shares its columns with sub-block j . Given any points corresponding to p , we can show
3
Let us choose the following β1,β2 ∈ P2(m) thatthetraversallengthincreasesfortheconsideredcase
that correspond to k k = k2 and 1 k2 = when micro-partition cycles arise.
j=1 j=1
k2 respectively. Without loss generality, p is Hence, the maximum attainable girth for a (k2,3)
P P 3
such that CB(1,k − 1),CB(2,1),...,CB(k,k − 2) BTU for the case when micro-partition cycles arise is
and CB(k − 1,1),CB(1,2),...,CB(k − 2,k) such that greater than the maximum attainable girth for a (k2,3)
each pair of cross-blocks {CB(1,k–1),CB(k–1,1)}, BTU for the case when no micro-partition cycles arise.
{CB(2,1),CB(1,2)} , ...,
{CB(k,k–2),CB(k–2,k)} have exactly two 1 s be- 4.11 Girth maximum Conjecture for
tweenthe twoofthem suchthateach1withcoordinates (k2,3) BTU
(x,y);1≤x≤k2;1≤y ≤k2
satisfies the constraint |(x–y)|≥k. If k ∈ N;k > 3 there exists a (k2,3) BTU in Φ(β ,β )
1 2
We now replace p with q ∈S as per Ψ(β ) . with maximum girth among all (k2,3) BTUs where each
2 2 k2 2
