Set of marks along a ruler such that no two pairs of marks are the same distance apart
"OGR" redirects here. For other uses, see OGR (disambiguation).
Golomb ruler of order 4 and length 6. This ruler is both optimal and perfect.The perfect circular Golomb rulers (also called difference sets) with the specified order. (This preview should show multiple concentric circles. If not, click to view a larger version.)
In mathematics, a Golomb ruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its order, and the largest distance between two of its marks is its length. Translation and reflection of a Golomb ruler are considered trivial, so the smallest mark is customarily put at 0 and the next mark at the smaller of its two possible values. Golomb rulers can be viewed as a one-dimensional special case of Costas arrays.
The Golomb ruler was named for Solomon W. Golomb and discovered independently by Sidon (1932)[1] and Babcock (1953). Sophie Piccard also published early research on these sets, in 1939, stating as a theorem the claim that two Golomb rulers with the same distance set must be congruent. This turned out to be false for six-point rulers, but true otherwise.[2]
There is no requirement that a Golomb ruler be able to measure all distances up to its length, but if it does, it is called a perfect Golomb ruler. It has been proved that no perfect Golomb ruler exists for five or more marks.[3] A Golomb ruler is optimal if no shorter Golomb ruler of the same order exists. Creating Golomb rulers is easy, but proving the optimal Golomb ruler (or rulers) for a specified order is computationally very challenging.
Distributed.net has completed distributed massively parallel searches for optimal order-24 through order-28 Golomb rulers, each time confirming the suspected candidate ruler.[4][5][6][7][8]
Currently, the complexity of finding optimal Golomb rulers (OGRs) of arbitrary order n (where n is given in unary) is unknown.[clarification needed] In the past there was some speculation that it is an NP-hard problem.[3] Problems related to the construction of Golomb rulers are provably shown to be NP-hard, where it is also noted that no known NP-complete problem has similar flavor to finding Golomb rulers.[9]
^Sidon, S. (1932). "Ein Satz über trigonometrische Polynome und seine Anwendungen in der Theorie der Fourier-Reihen". Mathematische Annalen. 106: 536–539. doi:10.1007/BF01455900. S2CID 120087718.
^Bekir, Ahmad; Golomb, Solomon W. (2007). "There are no further counterexamples to S. Piccard's theorem". IEEE Transactions on Information Theory. 53 (8): 2864–2867. doi:10.1109/TIT.2007.899468. MR 2400501. S2CID 16689687..
In mathematics, a Golombruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number...
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sets are Golombrulers, and vice versa. To see this, suppose for a contradiction that S {\displaystyle S} is a Sidon set and not a Golombruler. Since it...
engineer GolombrulerGolomb coding All pages with titles containing Golomb Gołąb (surname) This page lists people with the surname Golomb. If an internal...
arrays can be regarded as two-dimensional cousins of the one-dimensional Golombruler construction, and, as well as being of mathematical interest, have similar...
A Golombruler is a sparse ruler that requires all of the differences a j − a i {\displaystyle a_{j}-a_{i}} be distinct. In general, a Golombruler with...
-perfect number. Unrelated to 6's being a perfect number, a Golombruler of length 6 is a "perfect ruler". Six is a congruent number. 6 is the second primary...
Multiple Access – High bandwidth channel access method Golombruler – Set of marks along a ruler such that no two pairs of marks are the same distance...
Golombruler is a finite set of points on a line such that no two pairs of points have the same distance. Sophie Piccard claimed that no two Golomb rulers...
{\displaystyle 4=7-3} Golombruler Sparse ruler All-interval tetrachord This article incorporates material from perfect ruler on PlanetMath, which is...
appearance in Bowling for Columbine Solomon W. Golomb – mathematician, invented the Golomb coding and Golombruler Jane Goodall – distinguished adjunct professor...
periodic paths. distributed.net's OGR project searched for optimal Golombrulers. The PrimeGrid project is searching for the smallest Riesel and Sierpiński...
Endocrinology and Metabolism Solomon W. Golomb – mathematician, invented the Golomb coding and Golombruler George Gorse (B.A. 1971) – Viola Horton Professor...
Euclidean space might determine. This book included early research on Golombrulers, finite sets of integer points in a one-dimensional space with the property...
of points at a distance of 1 has the same color. Whereas the hexagonal Golomb graph and the regular hexagonal tiling generate chromatic numbers of 4 and...
Archived from the original on 23 April 2020. Retrieved 11 June 2017. Golomb & Wistrich 2002, p. 249. Tucker 2005, p. 1001. Tucker 2005, p. 884. Tucker...
called a graceful graph. The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander...
Lionel Goldsmid-Stern-Salomons (1851–1925), British author and baronet David Golomb (1933–2019), Israeli politician David Gomberg (born 1953), American businessman...
war dramas and the Russian version of "Dancing with the Stars". Eliyahu Golomb (1893–1945), leader of the Jewish defense effort in Mandate Palestine and...
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