Discrete Analysis and Operations Research

342 Pages·1996·11.292 MB·English

by  S. V. Avgustinovich (auth.), Alekseǐ D. Korshunov (eds.)

Discrete Analysis and Operations Research
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 355
and
Discrete Analysis
Operations Research
edited by
Alekser D. Korshunov
Sobolev Institute ofM athematics,
Siberian Branch of the Russian Academy of Sciences,
Novosibirsk, Siberia, Russia
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13: 978-94-010-7217-5 e-ISBN-13: 978-94-009-1606-7
DOl: 10.1007/978-94-009-1606-7
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Printed on acid-free p(lper
The contributions to this volume have all been translated
from the first volume of the Russian journal
DiscreteAnalysis and Operations Research,
the Sobolev Institute of Mathematics,
Siberian Branch of the Russian Academy of Sciences,
Novosibirsk, Russia, © 1994.
All Rights Reserved
© 1996 Kluwer Academic Publishers
Softcover reprint of the hardcover 1st 1996
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TABLE OF CONTENTS
PREFACE VI
S. V. Avgustinovich, The Number of Distinct Subwords
of Fixed Length in the Morse-Hedlund Sequence 1 3-7 (2)
A. A. Evdokimov, Locally Isometric Embeddings
of Graphs and the Metric Prolongation Property 7 5-12(1)
D. G. Fon-Der-Flaass, Local Complementations
of Simple and Directed Graphs ................. . 15 43-62(1)
E. Kh. Gimadi, N. I. Glebov, and A. I. Serdyukov,
An Approximation Algorithm for the Traveling
Salesman Problem and Its Probabilistic Analysis 35 8-17 (2)
N. I. Glebov and A. V. Kostochka, On Minimum
Independent Dominating Sets in Graphs ....... . 45 7-21(4)
A. A. Kolokolov, Regular Partitions and Cuts
in Integer Programming ....................... . 59 18-39(2)
A. D. Korshunov, Complexity of Coverings of Number
Sets by Arithmetical Progressions .............. . 81 40-60(2.)
E. Sh. Kospanov, Circuit Realization of the Sorting
Problem ....................................... . 101 13-19(1)
A. V. Kostochka, A Refinement of the
Frank-Seb/J-Tardos Theorem and Its Applications 109 3-19(3)
A. V. Kostochka and N. Tulai, On the Length
of the Chinese Postman Tour in Regular Graphs 125 20-37(3)
N. N. Kuzyurin, An Integer Linear Programming
Algorithm Polynomial in the Average Case 143 38-48 (3)
A. A. Levin, Projections of the Hypercube on the Line
and the Plane ................................. . 153 22-32 (4)
V. V. Lozin, Canonical Decomposition of Graphs 163 49-59(3)
V. N. N oskov, Fault Detection in Parts of the Circuits
of Functional Elements ........................ . 173 60-96(3)
Table of Contents
VI
V. Nyu, On the External Stability Number
of the Generalized De Bruijn Graphs ........... . 211 61-66(2)
K. L. Rychkov, On the Lower Bounds
for the Complexity of Serial-Parallel Contact
Circuits Realizing Linear Boolean Functions 217 33-52(4)
S. V. Sevast'yanov, Efficient Scheduling in Open
Shops ......................................... . 235 20-42 (1)
S. V. Sevast'yanov, Nonstrict Vector Summation
in Scheduling Problems ........................ . 257 67-99 (2)
Yu. V. Shamardin, Worst-Case Analysis of Some
Algorithms for Solving the Subset-Sum Problem 289 53-63 (4)
V. 1. Shevchenko, On the Depth of Conditional Tests
for Controlling "Negation" Type Faults in Circuits
of Functional Gates ........................... . 301 63-74 (1)
L. A. Sholomov, Synthesis of Transitive Order
Relations Compatible with the Power of Criteria 313 64-92(4)
INDEX 339
PREFACE
This book contains translations of papers from the first volume of the new
Russian-language journal published at the Sobolev Institute of Mathematics (Sibe
rian Branch of the Russian Academy of Sciences, Novosibirsk) since 1994.
In 1994 the journal was titled Sibirskil Zhurnal Issledovaniya Operatsil.
Since 1995 this journal has the title DiskretnYl Analiz i Issledovanie Operatsil
(Discrete Analysis and Operations Research)
The aim of this journal is to bring together research papers in different areas
of discrete mathematics and computer science. The journal DiskretnYl Analiz
i Issledovanie Operatsil covers the following fields:
• discrete optimization • synthesis and complexity
• discrete structures and • of control systems
extremal problems • automata
• combinatorics • graphs
• control and reliability • game theory and its
of discrete devices applications
• mathematical models and • coding theory
methods of decision making • scheduling theory
• design and analysis • functional systems theory
of algori thms
Contributions presented to the journal can be original research papers and
occasional survey articles of moderate length.
A. D. Korshunov
THE NUMBER OF DISTINCT SUBWORDS
OF FIXED LENGTH IN THE
MORSE-HEDLUND SEQUENCEt)
S. V. Avgustinovich
An exact formula is obtained for the number of distinct subwords of length n in
the Morse-Hedlund sequence [1), i.e., the sequence in which the initial member is 0
and subsequent members are produced by unlimited application of the operation of
substituting 01 for 0 and 10 for 1. Earlier, only the bounds on the number of subwords
of fixed length were known [2), [3). The Morse-Hedlund sequence provides a classic
example of a sequence in which no subword occurs three times in succession [4).
Our interest in repetition-free sequences is motivated by their relevance to
issues of completeness of the set of words and investigations of languages with
forbidden subwords (see [5]). Many equivalent ways of defining the Morse-Hedlund
sequence are known, the most simple being the following: the ith member of the
sequence is 0 if the number of ones in the binary representation of i is even and
is 1 otherwise. Another way is iterative: Xo = 0, X2i = Xi, X2i+1 = Xi + 1 (mod 2),
i = 0, 1, ....
In this paper, a third way is used which is the most convenient for our pur
poses. Let the mapping tp transform symbol 0 into word 01 and symbol 1 into
word 10. For an arbitrary word S = Sl ... Sn in the alphabet {O, I}, define tp(S) =
tp(sI) ... tp(sn). Consider the collection of words 0, 01, 0110, 01101001, ... in
which each word is obtained from the preceding one by applying mapping tp, and,
as is easily seen, is the beginning of the succeeding word. Thus an infinite word
W = 0110100110010110 ... is defined which is called the Morse-Hedlund sequence
(see 1[1]). Denote the set of distinct subwords of length n in W by 9J1(n) and their
number by R(n). A. T. Kolotov (see [4J and [5]) obtained the following bounds on
R(n):
2n::::; R(n) ::::; 6n.
In this paper, an exact formula is obtained for R( n) which, in particular, implies
the following bounds:
3(n - 1) ::::; R(n) ::::; 10(n - 1)/3. (1)
t) This research was partially supported by the Russian Foundation for Funda
mental Research (Grant 93-01-01484).
A. D. Korshunov (ed.), Discrete Analysis and Operations Research, 1-5.
© 1996 Kluwer Academic Publishers.
2 S. V. A vgustinovich
The lower bound is exact for n = 2k+l and the upper, for n = 3·2k+l, k = 0,1, ....
1. Definitions and Notation
We call a word S admissible if it occurs as a subword in W. We call an
admissible word S right (left) binary if SO and SI (OS and IS) are admissible
words. Otherwise, we call S right (left) unary.
We call an ordered pair (i,j), where i, j E {1,2}, the type of a word S if S
is left i-nary and right j-nary. For example, a word of type (1,2) is left unary and
right binary.
We call a word of type (2,2)
stable if adding to it an arbitrary symbol on the left yields a right binary word,
unstable if adding to it an arbitrary symbol on the left yields a right unary
word.
We call the words 01 and 10 blocks.
It follows from the definitions that W can be partitioned into blocks. Consequently,
each admissible word also can be partitioned into blocks with the possible exception
of the first and/or the last symbols which might remain isolated. We call such
a partition of a word regular.
Let B( n) denote the number of right binary (hereafter, binary) words of length
n, and U(n) and N(n), the numbers of stable and unstable words of length n
respectively.
2. An Exact Formula for R(n)
Proposition 1. For each n, n > 1,
R(n) = R(n - 1) + B(n - 1).
PROOF. The above equality reflects the fact that words in 9J1( n) are obtained
by adding admissible symbols on the right of the words in 9J1( n - 1), each binary
word giving an additional variant as compared to a unary word.
Since R(I) = 2, we have
Corollary 1. For each n 2: 2,
n-1
L
R(n) = 2 + B(i).
;=1
The Number of Subwords in the Morse-Hedlund Sequence 3
Proposition 2. A word of type (1,2) becomes binary after adding to it an
admissible symbol on the left of it.
The proof is an immediate consequence of the definitions.
Proposition 3. Among all admissible words of length 2, only 01 and 10 have
type (2,2) and, moreover, both of them are stable.
Proposition 4. Among all admissible words of length 3, only 010 and 101
have type (2,2) and, in addition, both of them are unstable.
The validity of Propositions 3 and 4 is established by direct checking.
Proposition 5. All words of type (2,2) are divided into stable and unstable
words, the former being produced by applying operation '-P to the words 01 and 10,
and the latter by applying the same operation to the words 010 and 10l.
PROOF. Consider an arbitrary word S of type (2,2) and of length 1 ~ 5. It is
easily checked that two identical symbols do occur in S consecutively. Therefore,
there is a unique regular partition of S. If in this partition of word S an isolated
symbol occurred at one of the ends of the word, this symbol would uniquely be
complemented to a whole block, which contradicts the fact that S is binary. Con
sequently, the word S is partitioned into blocks and hence there is a word S' such
that '-P(S') = S.
Consider an occurrence of the word S in the word W which is regularly par
titioned into blocks. By definition, W = '-P(W), which defines a one-to-one corre
spondence between the symbols of Wand the blocks into which it is partitioned.
It readily follows from the uniqueness of partitioning S into blocks that to each
occurrence of the word S in the word '-P(W), there corresponds an occurrence of
the word S' in the word W. This is also the case for each admissible extension of S
by blocks to which an extension of S' by symbols corresponds. Thus S', like S, is
admissible and has type (2,2), Sand S' being either both stable or both unstable.
The same argument can be applied to the word S' which is twice as short as S;
therefore, we will eventually arrive at a word of length 3 or 4. However, among
words of length 4, only the words 0110 and 1001 have type (2,2) and, in addition,
0110 = '-P(01) and 1001 = '-P(10). The proposition is proved.
Since each application of the operation doubles the length of the word, we
obtain
Corollary 2. The following relations hold:
ifn = 2k, k = 1,2, ... ,
~
U(n) = {
otherwise;
if n = 3 . 2k, k = 0,1, ... ,
~
N(n) = {
otherwise.