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Journal ArticleDOI

On a distribution problem in finite and countable sets

01 Sep 1973-Journal of Combinatorial Theory, Series A (Academic Press)-Vol. 15, Iss: 2, pp 129-137
TL;DR: Hall's theorem on distinct representatives is proved that there exists an infinite sequence on X with discrepancy at most 1 and this result is very close to the best possible.
About: This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-09-01 and is currently open access. It has received 28 citations till now. The article focuses on the topics: σ-finite measure & Countable set.
Citations
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Journal ArticleDOI
TL;DR: A simple algorithm for a chairman assignment is given which guarantees a small discrepancy and the situation that not only states form unions, but also unions form federations, etc., with one overall organization is investigated.

101 citations

Journal ArticleDOI
Roy L. Adler1, Bruce Kitchens1, Marco Martens1, Charles Tresser1, Chai Wah Wu1 
TL;DR: Some mathematical aspects of halftoning in digital printing, including the method of dithering, are described, with main emphasis on error diffusion.
Abstract: This paper describes some mathematical aspects of halftoning in digital printing. Halftoning is the technique of rendering a continuous range of colors using only a few discrete ones. There are two major classes of methods: dithering and error diffusion. Some discussion is presented concerning the method of dithering, but the main emphasis is on error diffusion.

49 citations

Journal ArticleDOI
TL;DR: An algorithm for a chairman assignment is given which, depending on the weights, guarantees a small discrepancy.

33 citations

Journal ArticleDOI
01 Jan 1973

25 citations

Journal ArticleDOI
Don Coppersmith1, Tomasz Nowicki1, Giuseppe A. Paleologo1, Charles Tresser1, Chai Wah Wu1 
TL;DR: It is shown that the greedy algorithm is optimal among online algorithms for the chairman assignment problem and the generalized carpool problem, and the bounds adapt to these cases.
Abstract: We study several classes of related scheduling problems including the carpool problem, its generalization to arbitrary inputs and the chairman assignment problem. We derive both lower and upper bounds for online algorithms solving these problems. We show that the greedy algorithm is optimal among online algorithms for the chairman assignment problem and the generalized carpool problem. We also consider geometric versions of these problems and show how the bounds adapt to these cases.

25 citations


Additional excerpts

  • ...For of.ine algorithms, B = 1 - 1 [Meijer 1973; Tijdeman 1973, 2(n-1) 1980] (tight bound)....

    [...]

References
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Journal ArticleDOI
TL;DR: In this paper, a set S of mn things is divided into m classes of n things each in two distinct ways, (a) and (b); so that there are m (a)-classes and m (b)-classes.
Abstract: Let a set S of mn things be divided into m classes of n things each in two distinct ways, (a) and (b); so that there are m (a)-classes and m (b)-classes. Then it is always possible to find a set R of m things of S which is at one and the same time a C.S.R. (= complete system of rcpresentatives) for the (a)-classes, and also a C.S.R. for the (b)-classes.

1,976 citations

Book ChapterDOI
TL;DR: In a recent issue of this journal Weyl as discussed by the authors proved a combinatorial lemma which was apparently considered first by P. Hall and was later generalized by Everett and Whaples.
Abstract: In a recent issue of this journal Weyl1 proved a combinatorial lemma which was apparently considered first by P. Hall2 Subsequently Everett and Whaples 3 published another proof and a generalization of the same lemma. Their proof of the generalization appears to duplicate the usual proof of Tychonoff’s theorem.4 The purpose of this note is to simplify the presentation by employing the statement rather than the proof of that result. At the same time we present a somewhat simpler proof of the original Hall lemma.

188 citations

Book ChapterDOI
TL;DR: In this paper, a distribution problem in finite sets and a few relations between the distribution problem and other combinatorial problems are discussed, and a problem that is helpful in the construction of low discrepancy sequences is described.
Abstract: Publisher Summary This chapter presents a distribution problem in finite sets and discusses a few relations between the distribution problem and other combinatorial problems. It describes a problem that is helpful in the construction of low discrepancy sequences. The problem arose from a method for explicitly constructing uniformly distributed sequences in a compact space. The chapter presents an equation to define a counting function and the construction of an infinite sequence, which is obtained by reading the rows of the array consecutively from the left to the right. There is a remarkable connection between distribution problems and combinatorial analysis. Incidentally, such relations between the theory of uniform distribution and combinatorial analysis have recently been investigated.

14 citations

Journal ArticleDOI
TL;DR: In this article, the authors showed that the existence of a system of distinct representatives of a given family of sets or subfamily thereof with certain properties being required can be shown to be a transfinite and symmetrized form of a theorem of A J Hoffman and H W Kuhn.
Abstract: 1 Introduction Currently the theory of systems of distinct representatives (and the closely allied theory of transversals) is being carefully examined and reworked, often in a more general context which allows for the transfinite situation This theory can be said to have had its beginning in 1935 when P Hall proved his now celebrated theorem for the existence of a system of distinct representatives of a finite family of sets In a no less significant paper M Hall, Jr (in 1948) extended P Hall's theorem to infinite families of finite sets Around these two theorems a considerable literature has grown (for an excellent survey and thorough bibliography see [10]) The two theorems have been refined in various ways by requiring that the system of distinct representatives have additional properties It is however true that these refinements can be obtained by applying the original theorems to a modified family of sets For finite families this is implicit in the work of Ford and Fulkerson [3] who show how most of these refinements can be obtained from their maximum flow-minimum cut theorem for flows in networks For finite or infinite families Mirsky and Perfect [10], [11] have shown how these refinements can be obtained from the original theorems of the two Halls and a generalization of a mapping theorem of Banach [1] In a recent paper [2] we obtained a further generalization of Banach's mapping theorem This theorem along with M Hall's theorem enables us to prove a very general theorem on systems of distinct representatives, which is in fact a transfinite and symmetrized form of a theorem of A J Hoffman and H W Kuhn The theorem we prove contains as special cases (that is, without further refinement) all theorems that we know which assert the existence of a system of distinct representatives of a given family of sets or subfamily thereof with certain properties being required We then can prove a theorem giving necessary and sufficient conditions that a family of sets possess a family of subsets whose cardinalities lie within prescribed bounds and where the frequencies of occurrences in these subsets of the elements lie within prescribed bounds This will be made more precise later From this we also obtain an extension to locally finite graphs of Ore's solution [12] of the so-called " subgraph problem for directed graphs" and for that matter a generalization of Ore's solution due to Ford and

10 citations