Golomb Rulers
نویسندگان
چکیده
The Math Factor podcast posed the problem of finding the smallest number of inch marks on a 12 inch ruler so that one could still measure any integer length from 1 to 12. One needs only four additional marks besides 0 and 12; for example 1, 4, 7, 10 works. This entertaining problem lead to others during the next few minutes (you can listen at mathfactor.uark.edu/2005/10) and inspired us to look for generalizations. After several false starts and numerous literature searches we uncovered the fascinating theory of Golomb and minimal spanning rulers, a generalization to the natural numbers and relations to an unsolved conjecture of Erdös and Turan. We begin the discussion with our first problem—– which led us to Golomb rulers. A property of the ruler of length 6 with marks at 0, 1, 4, 6 is that each of the lengths 1, 2, 3, 4, 5, and 6 can be measured and it can be done in only one way. Can one choose marks on a ruler of length 12 so that each length from 1 to 12 can measured in only one way?
منابع مشابه
A review of the available construction methods for Golomb rulers
We collect the main construction methods for Golomb rulers available in the literature along with their proofs. In particular, we demonstrate that the Bose-Chowla method yields Golomb rulers that appear as the main diagonal of a special subfamily of Golomb Costas arrays. We also show that Golomb rulers can be composed to yield longer Golomb rulers.
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A Golomb ruler is a sequence of distinct integers (the markings of the ruler) whose pairwise differences are distinct. Golomb rulers, also known as Sidon sets and B2 sets, can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for optimal Golomb rulers (those of minimal length given a fixed number of mark...
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A Golomb ruler with M marks can be defined as a set {ai} of integers so that all differences δij = aj − ai, i 6= j, are distinct. An order 2 Golomb ruler is a Golomb ruler such that all differences δijk` = |δk` − δij |, {i, j} 6 = {k, `}, are distinct as much as possible. Contruction of optimum order 2 Golomb ruler, i.e., of rulers of minimum length, is a highly combinatorial problem which has ...
متن کاملGenetic Algorithm Approach to the Search for Golomb Rulers
GOLOMB RULERS Stephen W. Soliday soliday@gar eld.ncat.edu Abdollah Homaifar homaifar@gar eld.ncat.edu Department of Electrical Engineering North Carolina A&T State University Greensboro, North Carolina 27411 Gary L. Lebby lebby@gar eld.ncat.edu Abstract The success of genetic algorithm in nding relatively good solutions to NP-complete problems such as the traveling salesman problem and job-shop...
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We consider Golomb rulers and their construction. Common rulers feature marks at every unit measure, distances can often be measured with numerous pairs of marks. On Golomb rulers, for every distance there are at most two marks measuring it. The construction of optimal—with respect to shortest length for given number of marks or maximum number of marks for given length—is nontrivial, various pr...
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AGolombRuler is a rulerwith integermarkswhere the distances between every twomarks are distinct. Golomb Rulers find diverse applications in computer science and electrical engineering. According to our knowledge the computational complexity of problems related to the construction of Golomb Rulers is unknown. We provide natural definitions for problems related to the construction of such rulers....
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