Table Of ContentLecture Notes in Computer Science 2122
EditedbyG.Goos,J.Hartmanis,andJ.vanLeeuwen
3
Berlin
Heidelberg
NewYork
Barcelona
HongKong
London
Milan
Paris
Tokyo
Helmut Alt (Ed.)
Computational
Discrete Mathematics
Advanced Lectures
1 3
SeriesEditors
GerhardGoos,KarlsruheUniversity,Germany
JurisHartmanis,CornellUniversity,NY,USA
JanvanLeeuwen,UtrechtUniversity,TheNetherlands
VolumeEditor
HelmutAlt
FreieUniversita¤tBerlin,Institutfu¤rInformatik
Takustr.9,14195Berlin,Germany
E-mail:[email protected]
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ISBN3-540-42775-9
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Preface
In order to speed up doctoral education in Germany the “Deutsche Forschungs-
gemeinschaft” (DFG, German Research Association) in the late 1980s devel-
opedanewfundingconceptforgraduateprogramscalled“Graduiertenkollegs”.
Groups of university teachers could join together and submit proposals for doc-
toral studies within certain areas of research. If funded, the program would
supply scholarships for doctoral students including means for travel, literature,
scientificguests,meetings,andschools.Thescholarshipswouldallowdoctorands
to concentrate on their doctoral studies without the usual teaching duties and
to obtain their degree in a much shorter period of time.
It was the idea of Martin Aigner to join efforts of mathematicians and com-
puterscientistsinthenewlyreunifiedBerlinandtoapplyforagraduateprogram
withindiscretemathematicsandtheoreticalcomputerscience.Thefinalproposal
was submitted by researchers from the three universities of Berlin, Free Univer-
sity, Humboldt-University, and Technical University, and the then still existing
Academy of Sciences of the German Democratic Republic. The specific areas
covered were geometric pattern recognition, constructive approximation, com-
plexity theory, combinatorial optimization, graph theory, computational combi-
natorics, coding theory, and group theory. The proposal was accepted by the
DFG,theprogramwasnamed“AlgorithmischeDiskreteMathematik”(Compu-
tational Discrete Mathematics), and Emo Welzl was elected as its first speaker.
The program started with the first four doctorands on October 1st, 1991.
The program was extended twice by the DFG and ended in September 2000
after the maximal possible runtime of nine years. Twenty-five of the students
fundedobtainedtheirdoctoraldegree,mostofthemwithinatimeperiodsignif-
icantlybelowaverageandmanyofthemevenwithinthestandardtwoandahalf
years of funding by the program. All in all during the last years, discrete math-
ematics and algorithmics in general have been flourishing within Berlin mainly
due to the existence of the graduate program.
In order to enhance contact and cooperation between the various research
areas of the members of the program it had been decided at the beginning that
there should be a weekly meeting on Monday afternoons during semesters. Part
ofthismeetingconsistsofa60to90minutetutoriallecturebyfacultymembers
of the program or selected guests. These lectures should be understandable not
onlytospecialistsbuttoallmembersofresearchgroupsinvolvedintheprogram.
Still, they should be at a high scientific level including recent research results
within the different fields of the program. They have turned out to be very
popular and very fruitful for all groups involved including the faculty members
of the program who use the opportunity to learn about the research areas of
their colleagues. In order to make the material of these lectures available to a
larger public we created this booklet containing twelve selected lectures held
within the graduate program. It covers the whole range of areas represented
VI Preface
in the program including combinatorics, graph theory, coding theory, discrete
and computational geometry, optimization, and algorithmic aspects of algebra.
Particularly intriguing are the nonobvious connections between different areas
suchascombinatorics,linearalgebra,discretegeometry,andgraphtheoryinthe
contributionbyMartinAigner;algebraiccomputingandgraphtheoryintheone
by Gu¨nter Rote; or discrete geometry, graph theory, and coding theory in the
one by Gu¨nter M. Ziegler.
A preliminary version of this booklet was published internally on the day
of celebration of the end of the old graduate program and the start of a new
one called “Combinatorics, Geometry, and Computation” which is a European
graduate program of the same research groups together with partners at ETH
Zurich and other European institutions.
Berlin, June 1, 2001 Helmut Alt
Table of Contents
Lattice Paths and Determinants ..................................... 1
Martin Aigner (Freie Universita¨t Berlin)
The Nearest Neighbor .............................................. 13
Helmut Alt (Freie Universita¨t Berlin)
Explicit and Implicit Enforcing – Randomized Optimization............. 25
Bernd Ga¨rtner and Emo Welzl (ETH Zu¨rich)
Codes over Z4 ..................................................... 47
Tor Helleseth (University of Bergen)
Degree Bounds for Long Paths and Cycles in k-connected Graphs ........ 56
Heinz Adolf Jung (Technische Universita¨t Berlin)
Data Structures for Boolean Functions................................ 61
Christoph Meinel and Christian Stangier (Universita¨t Trier)
Scheduling under Uncertainty: Bounding the Makespan Distribution...... 79
Rolf H. Mo¨hring (Technische Universita¨t Berlin)
Random Graphs, Random Triangle-Free Graphs,
and Random Partial Orders ......................................... 98
Hans Ju¨rgen Pro¨mel and Anusch Taraz
(Humboldt–Universita¨t zu Berlin)
Division-Free Algorithms for the Determinant and the Pfaffian:
Algebraic and Combinatorial Approaches ............................. 119
Gu¨nter Rote (Freie Universita¨t Berlin)
Check Character Systems and Anti–symmetric Mappings ............... 136
Ralph-Hardo Schulz (Freie Universita¨t Berlin)
Algorithms in Pure Mathematics..................................... 148
Gernot Stroth (Martin-Luther-Universita¨t Halle Wittenberg)
Coloring Hamming Graphs, Optimal Binary Codes,
and the 0/1-Borsuk Problem in Low Dimensions ....................... 159
Gu¨nter M. Ziegler (Technische Universita¨t Berlin)
Author Index ................................................. 173
Author Index
Aigner, Martin 1 Rote, Gu¨nter 119
Alt, Helmut 13
Schulz, Ralph-Hardo 136
G¨artner, Bernd 25
Stangier, Christian 61
Stroth, Gernot 148
Helleseth, Tor 47
Jung, Heinz Adolf 56 Taraz, Anusch 98
Meinel, Christoph 61
Welzl, Emo 25
Mo¨hring, Rolf H. 79
Pr¨omel, Hans Ju¨rgen 98 Ziegler, Gu¨nter M. 159
Lattice Paths and Determinants
Martin Aigner
Department of Mathematics and Computer Science, Free University of Berlin
Arnimallee 3, 14195 Berlin, Germany
[email protected]
The essence of mathematics is proving theorems – and so, that is what mathe-
maticians do: they prove theorems. But to tell the truth, what they really want
to prove once in their lifetime, is a Lemma, like the one by Fatou in analysis,
the Lemma of Gauss in number theory, or the Burnside-Frobenius Lemma in
combinatorics.
Now what makes a mathematical statement a true Lemma? First, it should
be applicable to a wide variety of instances, even seemingly unrelated problems.
Secondly, the statement should, once you have seen it, be completely obvious.
Thereactionofthereadermightwellbeoneoffaintenvy:Whyhaven’tInoticed
this before? And thirdly, on an esthetic level, the Lemma including its proof
should be beautiful!
In this article we look at one such marvellous piece of mathematical reason-
ing, the Lemma of Gessel and Viennot which has become an instant classic in
combinatorial enumeration since its appearance in 1985 [1]. A similar version
which was largely overlooked was previously obtained by Lindstr¨om [2].
1 Four Problems
Let us consider the following four problems which do not appear to have much
in common – and yet they can all be solved by the Gessel–Viennot Lemma.
A. The following curious problem was posed by Berlekamp and answered by
Carlitz, Roselle and Scoville, see [4], p. 231. Suppose we are given a Young
diagram D corresponding to the partition λ : n1n2...nt,n1 ≥ n2 ≥ ... ≥ nt.
Given a square s of D, let (cid:7) be the lowest square in the same column as s, and
r the rightmost square in the same row as s. Insert into s the number f(s) of
paths from (cid:7) to r which stay inside D and use steps to the north and east only.
Figure 1 shows the diagram of λ : 765532 and the numbers f(s). Consider
the largest square matrix M contained in D. In the figure, the 4×4–matrix is
drawn with bold lines.
Problem. Prove that always detM =1.
B. Everybody knows the Catalan numbers Cn, defined by the convolution
recursion
(cid:1)n
C0 =1, Cn+1 = CkCn−k (n≥0).
k=0
H.Alt(Ed.):ComputationalDiscreteMathematics,LNCS2122,pp.1–12,2001.
(cid:2)c Springer-VerlagBerlinHeidelberg2001
2 Martin Aigner
142 46 16 7 2 1 1
47 16 6 3 1 1
19 7 3 2 1
5 2 1 1 1
2 1 1
1 1
Fig.1.
Stanley lists in his admirable book [4] some 70 combinatorial instances which
are counted by the Catalan numbers. One of the classical examples are Dyck
paths of length 2n. These are lattice paths from (0,0) to (2n,0) which never go
below the line y =0 and use only diagonal steps north–east or south–east. The
number of these Dyck paths (of length 2n) is precisely Cn.
The next figure shows the C3 =5 Dyck paths of length 6:
Fig.2.
One of the most elegant characterizations of the sequence (Cn) goes as fol-
lows: Associate to (Cn) the Hankel matrices Hn and H(cid:2)n (n≥0), where
C C ... C C C ... C
0 1 n 1 2 n+1
C C ... C C C ... C
Hn = 1 2 ... n+1 , H(cid:2)n = 2 3 ... n+2 .
C C ... C C C ... C
n n+1 2n n+1 n+2 2n+1
Of course, we may associate such a pair of Hankel matrices to any sequence
of real numbers.
Problem. Prove that (Cn) is the unique sequence of real numbers such that
detHn =detH(cid:2)n =1 for all n≥0.
C. The Stirling numbers of the second kind Sn,k are defined as the number of
partitions of an n-set into k blocks. A classical result states that the numbers
Sn,k satisfy the recursion
