### Abstract

We introduce the notion of incremental codes. Unlike a regular code of a given rate, which is an unordered set of elements with a large minimum distance, an incremental code is an ordered vector of elements each of whose prefixes is a good regular code (of the corresponding rate). Additionally, while the quality of a regular code is measured by its minimum distance, we measure the quality of an incremental code C by its competitive ratio A: the minimum distance of each prefix of C has to be at most a factor of A smaller than the minimum distance of the best regular code of the same rate. We first consider incremental codes over an arbitrary compact metric space M, and construct a 2-competitive code for M. When M is finite, the construction takes time O(|M|^{2}), exhausts the entire space, and is NP-hard to improve in general. We then concentrate on 2 specific spaces: the real interval [0, 1] and, most importantly, the Hamming space Fn. For the interval [0, 1] we construct an optimal (infinite) code of competitive ratio ln 4 ≈ 1.386. For the Hamming space F^{n} (where the generic 2-competitive constructive is not efficient), we show the following. If |F| ≥ q, we construct optimal (and efficient) 1-competitive code that exhausts Fn (has rate 1). For small alphabets (|F| < q), we show that 1-competitive codes do not exist and provide several efficient constructions of codes achieving constant competitive ratios. In particular, our best construction has rate (1−o(1)) and competitive ratio (2+o(1)), essentially matching the bounds in the generic construction.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 4th International Workshop on Approximation, Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001, Proceedings |

Publisher | Springer Verlag |

Pages | 75-90 |

Number of pages | 16 |

Volume | 2129 |

ISBN (Print) | 3540424709 |

State | Published - 2015 |

Event | 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001 - Berkeley, United States Duration: Aug 18 2001 → Aug 20 2001 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2129 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001 |
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Country | United States |

City | Berkeley |

Period | 8/18/01 → 8/20/01 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 4th International Workshop on Approximation, Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001, Proceedings*(Vol. 2129, pp. 75-90). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2129). Springer Verlag.

**Incremental codes.** / Dodis, Yevgeniy; Halevi, Shai.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 4th International Workshop on Approximation, Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001, Proceedings.*vol. 2129, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2129, Springer Verlag, pp. 75-90, 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001, Berkeley, United States, 8/18/01.

}

TY - GEN

T1 - Incremental codes

AU - Dodis, Yevgeniy

AU - Halevi, Shai

PY - 2015

Y1 - 2015

N2 - We introduce the notion of incremental codes. Unlike a regular code of a given rate, which is an unordered set of elements with a large minimum distance, an incremental code is an ordered vector of elements each of whose prefixes is a good regular code (of the corresponding rate). Additionally, while the quality of a regular code is measured by its minimum distance, we measure the quality of an incremental code C by its competitive ratio A: the minimum distance of each prefix of C has to be at most a factor of A smaller than the minimum distance of the best regular code of the same rate. We first consider incremental codes over an arbitrary compact metric space M, and construct a 2-competitive code for M. When M is finite, the construction takes time O(|M|2), exhausts the entire space, and is NP-hard to improve in general. We then concentrate on 2 specific spaces: the real interval [0, 1] and, most importantly, the Hamming space Fn. For the interval [0, 1] we construct an optimal (infinite) code of competitive ratio ln 4 ≈ 1.386. For the Hamming space Fn (where the generic 2-competitive constructive is not efficient), we show the following. If |F| ≥ q, we construct optimal (and efficient) 1-competitive code that exhausts Fn (has rate 1). For small alphabets (|F| < q), we show that 1-competitive codes do not exist and provide several efficient constructions of codes achieving constant competitive ratios. In particular, our best construction has rate (1−o(1)) and competitive ratio (2+o(1)), essentially matching the bounds in the generic construction.

AB - We introduce the notion of incremental codes. Unlike a regular code of a given rate, which is an unordered set of elements with a large minimum distance, an incremental code is an ordered vector of elements each of whose prefixes is a good regular code (of the corresponding rate). Additionally, while the quality of a regular code is measured by its minimum distance, we measure the quality of an incremental code C by its competitive ratio A: the minimum distance of each prefix of C has to be at most a factor of A smaller than the minimum distance of the best regular code of the same rate. We first consider incremental codes over an arbitrary compact metric space M, and construct a 2-competitive code for M. When M is finite, the construction takes time O(|M|2), exhausts the entire space, and is NP-hard to improve in general. We then concentrate on 2 specific spaces: the real interval [0, 1] and, most importantly, the Hamming space Fn. For the interval [0, 1] we construct an optimal (infinite) code of competitive ratio ln 4 ≈ 1.386. For the Hamming space Fn (where the generic 2-competitive constructive is not efficient), we show the following. If |F| ≥ q, we construct optimal (and efficient) 1-competitive code that exhausts Fn (has rate 1). For small alphabets (|F| < q), we show that 1-competitive codes do not exist and provide several efficient constructions of codes achieving constant competitive ratios. In particular, our best construction has rate (1−o(1)) and competitive ratio (2+o(1)), essentially matching the bounds in the generic construction.

UR - http://www.scopus.com/inward/record.url?scp=84923111103&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923111103&partnerID=8YFLogxK

M3 - Conference contribution

SN - 3540424709

VL - 2129

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 75

EP - 90

BT - Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 4th International Workshop on Approximation, Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001, Proceedings

PB - Springer Verlag

ER -