### Abstract

A difference triangle set (DΔS) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct DΔSs, we present a new type of combinatorial design, monotonic directed (v, k, λ)-designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (DΔSs). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) DΔSs if the existence of a single regular DΔS is known.

Original language | English (US) |
---|---|

Pages (from-to) | 741-748 |

Number of pages | 8 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 18 |

Issue number | 4 |

DOIs | |

State | Published - 2005 |

### Fingerprint

### Keywords

- Difference triangle set
- Directed design
- Directed packing
- Golomb ruler
- Spanning ruler set

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*SIAM Journal on Discrete Mathematics*,

*18*(4), 741-748. https://doi.org/10.1137/S0895480103436761

**A recursive construction for regular difference triangle sets.** / Chu, Wensong; Colbourn, Charles; Golomb, Solomon W.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 18, no. 4, pp. 741-748. https://doi.org/10.1137/S0895480103436761

}

TY - JOUR

T1 - A recursive construction for regular difference triangle sets

AU - Chu, Wensong

AU - Colbourn, Charles

AU - Golomb, Solomon W.

PY - 2005

Y1 - 2005

N2 - A difference triangle set (DΔS) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct DΔSs, we present a new type of combinatorial design, monotonic directed (v, k, λ)-designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (DΔSs). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) DΔSs if the existence of a single regular DΔS is known.

AB - A difference triangle set (DΔS) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct DΔSs, we present a new type of combinatorial design, monotonic directed (v, k, λ)-designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (DΔSs). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) DΔSs if the existence of a single regular DΔS is known.

KW - Difference triangle set

KW - Directed design

KW - Directed packing

KW - Golomb ruler

KW - Spanning ruler set

UR - http://www.scopus.com/inward/record.url?scp=27844504690&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27844504690&partnerID=8YFLogxK

U2 - 10.1137/S0895480103436761

DO - 10.1137/S0895480103436761

M3 - Article

AN - SCOPUS:27844504690

VL - 18

SP - 741

EP - 748

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -