Permutation codes, source coding and a generalisation of Bollob\'as-Lubell-Yamamoto-Meshalkin and Kraft inequalities

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by  Kristo Visk

Permutation codes, source coding and a
generalisation of Bolloba´s-Lubell-Yamamoto-
Meshalkin and Kraft inequalities
Kristo Visk and Ago-Erik Riet
Institute of Mathematics and Statistics
Faculty of Science and Technology, University of Tartu, J. Liivi 2, Tartu 50409, Estonia
6
1
0 Abstract—We develop a general framework to prove Kraft- Inthiswork,westudyaquestionofuniquedecodingofper-
2 type inequalities for prefix-free permutation codes for source mutation codes. This question is important for understanding
n coding with various notions of permutation code and prefix. theoretical limits for optimal (source) coding of information
a We also show that the McMillan-type converse theorem in stored in flash memories.
J most of these cases does not hold, and give a general form of
a counterexample. Our approach is more general and works To this end, we prove a generalisation of the Bolloba´s-
6
for other structures besides permutation codes. The classical Lubell-Yamamoto-Meshalkin (LYM) [4], [13], [17], [20] and
2
Kraft inequality for prefix-free codes as well as results about Kraft[11]inequalities,whichworksforcertaingradedposets.
permutationcodesfollowascorollariesofourmaintheoremand
] AscorollariesofourmaintheoremweobtainthataKraft-type
T main counterexample.
inequality holds for prefix-free permutation codes in different
Index Terms—Permutation codes, source coding, Kraft
I
. inequality, LYM inequality, prefix-free, Ulam distance, contexts, where we give several definitions of permutation
s
c subsequence-free, permutation pattern. codesandseveraldefinitionsofwhatitmeanstobea‘prefix’.
[ Ascorollariestoourgeneralcounterexample,weobtainthata
I. INTRODUCTION McMillan-type converse theorem fails in most of these cases,
1
v Non-volatile memory is a type of computer memory that but not for the case of the classical notion of prefix.
9 can store information after the device has been turned off.
5 Flash memory is a type of non-volatile storage device. Data II. NOTATION
0
7 isstoredontoaflash memorydevicebyinjectingchargesinto Denote [k]:={1,2,...,k}, let N:={1,2,...} and N0 :=
0 memorycells.Itispossibletoincreasethelevelofchargeofa {0}∪N.ForasetS,itscardinalityisdenoted#S,sometimes
particularcell,buttodecreasethelevelofchargeitisrequired
1. also written |S|.
to erase and overwrite a large block of cells.
0 Over time the drift of electrical charges in memory cells Analphabetisa finiteset S :={si |i∈[n]}of cardinality
6 n 6= 0. A symbol is an element s ∈ S, a finite sequence of
1 might occur. Drift may occur at different rates for different
symbols s ...s is a string. The length of a string is the
: cells, which makes sustaining required charge levels difficult, 1 k
v number of symbols it consists of. Let Sl := {s ...s | s ∈
becausechargelevelsineverycellwouldhavetobemonitored 1 l j
i S for all j ∈[l]} denote the set of strings of length l. Define
X separately.Moreover,whileincreasingthecharges,somecells
S0 := {ε} where ε is the unique string of length 0 called
r might get overcharged, resulting in an overshoot error. Since
a reducingthechargelevelsisacomplexprocess,thereliability btheetheempsteyt ostfrianlgl.fiLneitteSs∗trin:=gs.SFoj∈rNs0trSinjgs=t,Su0∈∪SS∗1w∪h.e.r.e
of flash memory devices decreases over time.
t = s s ...s ∈ Sk and u = s s ...s ∈ Sr their
To manage charge levels in memory cells more efficiently, i1 i2 ik j1 j2 jr
concatenation is tu = s ...s s ...s ∈ Sk+r. A string
multi-level memory cells are used. Single-level memory cells i1 ik j1 jr
t is a prefix to the string u if u = tw for some string w. If
atanygiventimeareeitherchargedorempty,whileinamulti-
w 6=ǫ, then t is called a proper prefix of u.
levelcellsystemthechargeofanindividualcellcanhavemore
A permutation of the set [k] is a bijection σ : [k] → [k].
thantwodifferentlevels.Ifthechargelevelsofcellsinablock
To denote permutations σ : i 7→ b (i ∈ [k]) we use vectors
of cells are different, i.e. arranged as a permutation, the drift i
(b ,...,b ) as wellas stringsb ...b . We also denoteS :=
and overshooterrors become easier to detect. By coding each 1 k 1 k k
{σ |σ :[k]→[k] is a bijection}fortheset(infact,group)of
block of cells into permutations and managing injections of
permutationson[k].Forexampleσ =2314=(2,3,1,4)∈S
chargesitispossibletoreducedriftandovershooterrors,using 4
is the permutation σ(1)=2, σ(2)=3, σ(3)=1, σ(4)=4.
the rank modulation scheme proposed in [9]. Permutation
We use notation
codes for error correction for flash memories which provide
further robustness have been studied for example in [3], [7], Tl :={(n ,...,n )|∀j :n ∈[k]; i6=j ⇒n 6=n }
k 1 l j i j
[2], [15], [6], [8]. Permutation codes were also proposed for
use in powerline communications, see for example [18]. for the set of l-element partial permutations on the set [k],
i.e. the injective mappings τ :[l]→[k]. Let We shall now state the Kraft inequality [11], [16], see
also [5] (Ch. 5) from classical source coding, which holds
T :=T1∪T2∪...∪Tk
k k k k forall uniquelydecodableclassical codes,in particularfor all
and write prefix-free classical codes.
Tk :={(n1,...,nl)|∀j :nj ∈[l],l≤k; i6=j ⇒ni 6=nj} Proposition III.1. Let c : S1 → S2∗ be a uniquely decod-
able classical code with parameter sequence (a ,a ,...) and
0 1
=S1∪S2∪...∪Sk. #S2 =r. Then
∞
Wesayσ ∈Sl isthepatternof τ ∈Tkl iftherelativeordering Kc := arii ≤1. (1)
of symbols is the same, i.e. if, for all i,j ∈ [l], we have Xi=0
σ(i)<σ(j) if and only if τ(i)<τ(j). A permutationσ ∈S
l ThenumberK istheKraftnumber,alsoknownastheKraft
is a pattern in a partial permutation τ ∈ Tm if there are c
k sumortheKraft-McMillannumber.NotethatPropositionIII.1
indices 1 ≤ b < b < ... < b ≤ m such that, for all
1 2 l states that the sum of densities of used codewords of a fixed
i,j ∈ [l], σ(i) < σ(j) if and only if τ(b ) < τ(b ). We call
i j length, over all lengths, is at most 1.
σ ∈Tma(notnecessarilyconsecutive)subsequenceofτ ∈Tl
k k McMillan [16] proved the following strong converse of
ifm≤l andtherearemindices1≤b <b <...<b ≤l
1 2 m Proposition III.1, known as (the converse part of) McMillan
with σ(i) = τ(b ) for all i ∈ [m]. Note that σ is a pattern in
i Theorem.Weremarkthatitsproofistheconstructionofacode
τ ∈T if and only if it is the pattern of a subsequence of τ.
k bypickingverticesinther-arycodetreegreedily,startingwith
For example 253 ∈ T3 is a subsequence in 2513 ∈ T3,
6 6 verticesclosertotherootandgoinginthelexicographicorder.
the pattern of 253 ∈ T3 is 132 ∈ S and thus 132 ∈ S is a
6 3 3
pattern in 2513∈T3. Proposition III.2. Let r ∈N. If non-negativeintegers a ∈
6 i
We call σ ∈ Tm a (consecutive) substring of τ ∈ Tl if N ∪{0}, i ∈ N , are such that inequality (1) holds, then
k k 0 0
m≤l andthereis0≤n≤l−mwithσ(i)=τ(i+n)forall there exists a prefix-free code c:S →S∗ with #S =r and
1 2 2
i∈[m]. If σ is a substring of τ then it is also a subsequence. parameter sequence (a ,a ,...).
0 1
For example 51 ∈ T2 is a substring in 2513 ∈ T4 but the
6 6
subsequence 253∈T3 is a not a substring in 2513∈T4. IV. PERMUTATIONCODES
6 6
We say σ ∈Sm is a substring pattern of τ ∈Tk if it is the We shalldefine permutationcodesin two ways, by restrict-
pattern of a substring of τ. So 21∈S2 is a substring pattern ing the output space of classical codes.
in 2513∈T4 since it is the pattern of the substring 51∈T2.
6 6 DefinitionIV.1. LetS beanalphabetandk ∈N.Wedefine
III. CODES, THEIRPREFIX-FREENESSAND UNIQUE a permutation code as an injection c:S →Tk. Note that
DECODABILITY
k
k
LetS1 andS2 bealphabets.A(non-singular,classical)code #Tk = l!.
is an injective map c : S1 → S2∗. An image c(s) is called a Xl=1(cid:18)l(cid:19)
codeword corresponding to symbol s ∈ S . The definition of
1 We have#S ≤ k k l!becauseoftheinjectivityofc.The
a code extendsto all stringsas c(s1...sn):=c(s1)...c(sn). l=1 l
It is easily checked the new map c : S∗ → S∗, called the parameter sequePnce of(cid:0)th(cid:1)e code is (a0,a1,...), where aj :=
extension of the code c, is well-defined.1A cod2e c : S1 → #{s∈S |c(s)∈Tkj}, as for classical codes.
S2∗ is uniquely decodable if its extension is injective, i.e. if Definition IV.2. A more restrictive definitionof a permuta-
ca(lls1j.∈..[snn]). A=cco(dse′1.c.:.sS′m)→imSp∗lieisspmrefi=x-fnreaenidf tshjer=e dso′j nfoort tion code is an injection c:S →Tk. Note that
1 2
exist s ,s ∈S with c(s ) a proper prefix of c(s ). k
i j 1 i j #T = l!.
Itiseasytoseethataprefix-freecodeisuniquelydecodable: k
we can read symbols in an output string c(s ...s ) = Xl=1
1 n
c(s1)...c(sn) from left to right, and prefix-freeness guaran- We have #S ≤ k l! because of the injectivity of c. The
l=1
tees that no proper prefix of c(s1) is a codeword and also parameter sequenPce of the code is then (a0,a1,...) where
that c(s1) is not a proper prefix to any codeword. Hence, aj :=#{s∈S |c(s)∈Sj}.
encounteringthesubstringc(s )atthebeginningoftheoutput
1
c(s ...s ), we are guaranteed that it arose by encoding A. Definitions of ‘prefix-freeness’ for permutation codes
1 n
the symbol s . We then continue by decoding c(s ...s )
1 2 n Let us define the permutation constant of a permutation
similarly. Because of this property, prefix-free codes are also
code c in these cases respectively as
called instantaneous, see for example [5] (Ch. 5).
pwahLreaermetecate:jrS:1s=e→qu#eSn{2∗csebe∈ofatScho1edec|.odTc(ehse)cn. ∈tThheeSs2jep}qa,uraemnjceet∈e(rasN0c,0oa,u1in,st..tthh.)ee, Pc :=Xl=k1 klall!, or, Pc :=Xl=k1 al!l.
(cid:0) (cid:1)
codewords of each length. ThisistheanalogueoftheKraftnumberfromclassicalcodes.
The extension,prefix-freenessand uniquedecodabilityof a A. Level-regular graded posets
permutation code c:S →Tk or c:S →Tk is understood as We shall consider a special kind of graded posets. A
that notion for the same code viewed as a code c:S →[k]∗. partially ordered set or a poset is a set P together with a
Now we give some notions analogous to prefix-freeness. binaryrelation≤thatisreflexive,transitiveandantisymmetric,
A permutation code c : S → Tk or c : S → Tk is i.e. for all a ∈ P, a ≤ a, for all a,b,c ∈ P, if a ≤ b and
{subsequence-, substring-, pattern- or substring-pattern-}free b≤c then a≤c andforall a,b∈P, if a≤b andb≤a then
if there are no two different codewords c(s1) 6= s(s2) such a=b. Write a<b for a≤b and a6=b. We say b covers a if
thatc(s1)isrespectivelya {subsequence,substring,patternor a<bandthereisnoc∈P witha<candc<b.Anelement
substring pattern} in c(s2). These notionscan also be defined a ∈P is minimal if there is no b ∈ P with b < a. A graded
for classical codes c : S1 → S2∗, with the subsequence- and posetis a posetP witha rankfunctionρ:P →N0 satisfying
substring-freeness being perhaps the more natural notions. ρ(c) = 0 for all minimal c ∈ P, and, ρ(b) = ρ(a)+1 if b
We shall see that often for prefix-free, subsequence-free, covers a, and, if a<b then ρ(a)<ρ(b).
substring-free, pattern-free or substring-pattern-free permu- A directed (multi)graph G = (V,E) is a set V, called its
tation codes, Pc ≤ 1, or, Pc ≤ 1, i.e. the analogue of vertexset,togetherwithamultisetE ⊆V×V oforderedpairs
Proposition III.1 holds. That is, the sum of densities of used of vertices, called its edge set (there may be multiple edges
codewords of fixed length, over all lengths, is at most 1. (u,v) for fixed u,v ∈V). An element v ∈V is a vertex and
However, we shall also see that P ≤ 1, or, P ≤ 1 an element e ∈ E is an edge. An edge e = (u,v) is directed
c c
for given code parameters does not in general imply that from u to v or goes from u to v. A vertex v is a neighbour
a subsequence-free, substring-free, pattern-free or substring- of a vertex u if there is an edge (u,v) or (v,u). A directed
pattern-freepermutationcode with these parametersexists. In graph is weakly connected, if its underlying undirected graph
some of these cases there is no analogue of Proposition III.2. is connected, i.e., without regard to directions of edges, one
canwalkfromanyvertexto anyothervertexalongedges(we
can walk from a vertex to any of its neighbours). The up-
V. A GENERALISATIONOF THELYM AND KRAFT
degree of a vertex u is #{e ∈ E | ∃v ∈ V : e = (u,v)},
INEQUALITIES
i.e.thenumberofedgesdirectedfromu,andthedown-degree
We shall state the Bolloba´s-Lubell-Yamamoto-Meshalkin of a vertex v is #{e ∈ E | ∃u ∈ V : e = (u,v)}, i.e. the
inequality, also known as the LYM inequality [4], [13], [17], number of edges directed to v. Sometimes the up- or down-
[20],see also [10] (Ch. 8).Considerthe setof allsubsetsofa degree is just called degree. For a graph G = (V,E) and a
finiteset[n],denotedbyP([n]),orderedbythesubsetrelation subset V′ ⊆V, the graph G′ =(V′,E∩(V′×V′)) is called
⊆. This is a poset. Denote [n](k) :={A⊆[n]:#A=k}. an induced subgraph of G, i.e. we keep all edges with both
endpoints in V′ and only them; then G′ is induced by V′.
Proposition V.1. Suppose A ⊆ P([n]) is an antichain,
TheHasse diagramofa gradedposetP isa directedgraph
i.e. A,B ∈A and A⊆B implies A=B. Then
with vertex set P, and an edge from a to b if and only if b
n #(A∩[n](i)) covers a; it is drawn with elements of the same rank on the
≤1. same horizontal level and elements of higher ranks higher.
n
Xi=0 i Example V.1. See Figure 1 for the Hasse diagram of the
(cid:0) (cid:1)
That, is, the sum of densities of A in each level, summed poset of subsets of {1,2} with the subset relation ⊆.
over all levels, is at most 1.
{1,2}
Nowweshallprovea commongeneralisationofthePropo-
sition III.1 for prefix-free codes and Proposition V.1. As
consequences we obtain some analogues of Proposition III.1, {1} {2}
namely P ≤ 1, and, P ≤ 1, for some of prefix-free,
c c
subsequence-free, substring-free, pattern-free and substring-
∅
pattern-free permutation codes. We remark that our theorem
follows from the so-called AZ identity for general finite
posets[1]butourproofhereisself-containedandmoresuited Fig.1. TheHassediagram oftheposetofsubsetsof{1,2}.
fortheapplicationswe havein mind.We alsoremarkthatour
proof follows closely the known proof of the Proposition V.1 Let us say a graded poset P is level-regular if the bipartite
via the Local LYM inequality, and our framework of level- (multi)graph induced by any two consecutive levels of its
regulargradedposets,tobedefined,ischosensothatthisproof Hassediagramisbiregular—thatis,allelementsonthesame
still works, while being generalenoughfor the corollarieswe level,i.e.withthesamerank,arecoveredbyanequalnumber
have in mind. of elements, and also cover an equal (perhaps different)
We shall also investigate when a converse statement such number of elements (with multiplicity).
asPropositionIII.2canhold.For exampleforPropositionV.1 Let A ⊆ P be any set of elements of the same rank,
such a converse statement does not hold in general. i.e. ρ(a) = ρ(b) for all a,b ∈ A. Then its upper shadow
δ+(A) := {b | ∃a ∈ A : b covers a} is the set of all u·#δ−(A(k))ontheotherhand—avertexoftheoriginalhas
elements covering some element of A, and its lower shadow d neighbours in the lower shadow and a vertex of the lower
δ−(A):={b|∃a∈A:acoversb} is defined analogously. shadow has at most u neighbours in the original. Hence
An antichain is a subset A ⊆ P whose elements are
#P(k) u #A(k)
pairwise incomparable, i.e. a6<b for all a,b∈A. = ≥ ,
#P(k−1) d #δ−(A(k))
B. AcommongeneralisationoftheLYMandKraftinequalities
and rearranging proves the Lemma.
OurmainTheoremisthefollowinggeneralisationofPropo- ThisprovestheTheoremifP isfinite.To provetheinfinite
sition V.1 and Proposition III.1. case, restrict the poset to levels up to N, for every N ∈ N,
and then by the finite case we have
TheoremV.2. LetP bealevel-regulargradedposetandlet
A ⊆ P be an antichain. Denote P(i) := {p ∈ P | ρ(p) = i} ∞ #A(i) N #A(i)
and A(i) := {a∈ A| ρ(a) = i}. Assume all levels are finite, LA = #P(i) =Nl→im∞ #P(i) ≤1.
i.e. #P(i) <∞ for each i∈N . Define the LYM number of Xi=1 Xi=1
0
the antichain as
∞ #A(i)
LA := #P(i). VI. FAILURE OFTHE CONVERSE MCMILLANTHEOREM
Xi=0 We may ask about the analogue of Proposition III.2 in this
Then general setting.
L ≤1.
A QuestionVI.1. LetP bean(infiniteorfinite)level-regular
Proof:First letusassumethattheposetP isfinite. Then gradedposet.Assumethatalllevelsarefinite,i.e.#P(i) <∞
theLYMnumberisafinitesum.Weshallproceedbyinduction for each i∈N0. Let ai ∈N0 for each i∈N0. Assume
on k :=max{ρ(a)|a∈A}. If k =0 then #A(i) =0 for all ∞
a
i
i>0 and ≤1.
∞ #P(i)
#A(i) #A(0) Xi=1
= ≤1,
#P(i) #P(0) Is it true that then there exists an antichain A ⊆ P with
Xi=0
#A(i) =a for each i?
asneeded.LetusdefineA′ :={a∈A|ρ(a)<k}∪{p∈P | i
ρ(p)=k−1, p∈δ−(A)}. We shall prove that, on replacing Theansweris in general“No” but“Yes”forexampleifthe
A by A′, its LYM number does not decrease, i.e. LA ≤LA′, Hasse diagram is a tree. The next Theorem gives the general
and that A′ is still an antichain. The claim now follows by formofourcounterexampletoQuestionVI.1.Wecanhopefor
induction, since LA′ ≤1 by the induction hypothesis. acounterexampleonlywhenLemmaV.3isnottight,i.e.when
Let a,b ∈ A′. We shall prove that a 6< b. If a,b ∈ A then #δ−(A(k))/#P(k−1) >#A(k)/#P(k).
a6<b as A is an antichain. If a,b∈P with ρ(a)=ρ(b) then Theorem VI.2. Consider two consecutive levels P(i) and
also a6<b by the definition of a graded poset. Suppose for a P(i+1) of a level-regular graded poset P. Suppose that the
contradictionthata<b.Theonlywayitmighthappeniswith
equal up-degrees from level i to level i+1 are u > 1 and
a∈A and b∈δ−(A) with ρ(b)=k−1. Butsince b is in the
the equal down-degrees from level i+1 to level i are d>1,
lower shadow of {a∈A|ρ(a)=k}, there exists c∈A with i.e. each element of P(i) is covered by u elements of P(i+1)
b<c. By transitivity, a<c with a,c∈A — a contradiction and that each element of P(i+1) covers d elements of P(i).
with A being an antichain. Hence A′ is an antichain.
Assume that the graph, induced by levels i and i+1 of the
Note that δ−(A(k)) ∩A(k−1) = ∅, as A is an antichain.
Hasse diagram, is weakly connected.Further assume thatthe
Hence, to prove that LA′ ≥LA, it is enough to show that greatest common divisor gcd(#P(i),#P(i+1))=:g >1.
Lemma V.3. Define ai := g−g1#P(i); ai+1 := g1#P(i+1); aj := 0 for
j 6=i,i+1, and note that they are integers. Then
#δ−(A(k)) #A(k)
≥ . ∞
#P(k−1) #P(k) ai
≤1
#P(i)
This is the analogue of what is known as the Local LYM Xi=1
inequality, which reads that shadows have greater density. but there is no antichain A with #A(j) =a for all j ∈N .
j 0
Proof: We shall prove the Lemma by degree consider-
ations of the Hasse diagram of levels k and k −1. Let the Proof: Fix any subset A(i+1) ⊆P(i+1) with #A(i+1) =
down-degree, i.e. the number of elements it covers, of each ai+1 = g1#P(i+1). Consider the graphGi =(Vi,Ei) induced
v ∈P(k) bed, andthe up-degree,i.e. thenumberof elements by V := A(i+1) ∪ δ−(A(i+1)). It is a bipartite graph with
i
covering it, of each w ∈ P(k−1) be u. The number of edges parts A(i+1) and δ−(A(i+1)), i.e. all edges e ∈ E are of
i
betweenthesetsP(k) andP(k−1) isd·#P(k) =u·#P(k−1). the form (v,w) with v ∈ δ−(A(i+1)) and w ∈ A(i+1). The
The number of edges in the subgraph induced by A(k) and average of down-degrees of vertices in A(i+1) is equal to d
δ−(A(k)) is equal to d·#A(k) on the one hand and at most since all neighbours of vertices of A(i+1) lying in the i-level
oftheHasse diagramofP arecontainedinthegraphG .The ThesubsequencerelationhasrelevancetoUlamcodes[19],
i
average of up-degrees of vertices in δ−(A(i+1)), however, is [6], [8]. A permutation code c : S → T has minimum
k
strictly less that u. To see this, note that all degrees are at Ulam distance d if and only if every string s ∈ [k]k−d+1
mostu.Butthereisavertexofdegreelessthanusincenotall is a subsequence in at most one codeword. For fixed-length
neighboursof all vertices of δ−(A(i+1)) in the graphinduced codes this was explored for example in [12], [14], [6], [8].
by P(i)∪P(i+1) (where all these degreesare u) lie in the set
VII. OPEN PROBLEMS
A(i+1). Indeed,otherwise the graphinduced by P(i)∪P(i+1)
would not be weakly connected, with δ−(A(i+1)) ∪ A(i+1) Does the analogue of Kraft inequality (1) hold with defini-
tions of ‘unique decodability’, expanding the freeness defini-
being disconnected from the rest of the graph. Hence,
tions, for classical or permutation codes? Does there exist an
#δ−(A(i+1)) #A(i+1) analogue of Huffman coding for these types of codes?
> .
#P(i) #P(i+1)
VIII. ACKNOWLEDGEMENTS
Thus a > #(P(i)\δ−(A(i+1))) and we cannot pick a
i i This work was supported in part by the Estonian Research
elements to our antichain A on level P(i). Since the choice
Council through the research grants PUT405, PUT620 and
of a elements comprisingA(i+1) was arbitrary,there is no
i+1 IUT20-57. The authors wish to thank Vitaly Skachek for
antichain A with the required properties.
helpful discussions.
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