### Abstract

In this paper, we study a generalization of the classical Voronoi diagram, called clustering induced Voronoi diagram (CIVD). Different from the traditional model, CIVD takes as its sites the power set U of an input set P of objects. For each subset C of P, CIVD uses an influence function F(C, q) to measure the total (or joint) influence of all objects in C on an arbitrary point q in the space ℝd, and determines the influence-based Voronoi cell in ℝd for C. This generalized model offers a number of new features (e.g., simultaneous clustering and space partition) to Voronoi diagram which are useful in various new applications. We investigate the general conditions for the influence function which ensure the existence of a small-size (e.g., nearly linear) approximate CIVD for a set P of n points in ℝd for some fixed d. To construct CIVD, we first present a standalone new technique, called approximate influence (AI) decomposition, for the general CIVD problem. With only O(n log n) time, the AI decomposition partitions the space ℝd into a nearly linear number of cells so that all points in each cell receive their approximate maximum influence from the same (possibly unknown) site (i.e., a subset of P). Based on this technique, we develop assignment algorithms to determine a proper site for each cell in the decomposition and form various (1-ε)-approximate CIVDs for some small fixed ε > 0. Particularly, we consider two representative CIVD problems, vector CIVD and density-based CIVD, and show that both of them admit fast assignment algorithms; consequently, their (1 - ε)-approximate CIVDs can be built in O(n log^{d+1} n) and O(n log2 n) time, respectively.

Original language | English (US) |
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Title of host publication | Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 |

Pages | 390-399 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 1 2013 |

Event | 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 - Berkeley, CA, United States Duration: Oct 27 2013 → Oct 29 2013 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 |
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Country | United States |

City | Berkeley, CA |

Period | 10/27/13 → 10/29/13 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

### Cite this

*Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013*(pp. 390-399). [6686175] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2013.49

}

*Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013.*, 6686175, Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 390-399, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013, Berkeley, CA, United States, 10/27/13. https://doi.org/10.1109/FOCS.2013.49

**On clustering induced voronoi diagrams.** / Chen, Danny Z.; Huang, Ziyun; Liu, Yangwei; Xu, Jinhui.

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

TY - GEN

T1 - On clustering induced voronoi diagrams

AU - Chen, Danny Z.

AU - Huang, Ziyun

AU - Liu, Yangwei

AU - Xu, Jinhui

PY - 2013/12/1

Y1 - 2013/12/1

N2 - In this paper, we study a generalization of the classical Voronoi diagram, called clustering induced Voronoi diagram (CIVD). Different from the traditional model, CIVD takes as its sites the power set U of an input set P of objects. For each subset C of P, CIVD uses an influence function F(C, q) to measure the total (or joint) influence of all objects in C on an arbitrary point q in the space ℝd, and determines the influence-based Voronoi cell in ℝd for C. This generalized model offers a number of new features (e.g., simultaneous clustering and space partition) to Voronoi diagram which are useful in various new applications. We investigate the general conditions for the influence function which ensure the existence of a small-size (e.g., nearly linear) approximate CIVD for a set P of n points in ℝd for some fixed d. To construct CIVD, we first present a standalone new technique, called approximate influence (AI) decomposition, for the general CIVD problem. With only O(n log n) time, the AI decomposition partitions the space ℝd into a nearly linear number of cells so that all points in each cell receive their approximate maximum influence from the same (possibly unknown) site (i.e., a subset of P). Based on this technique, we develop assignment algorithms to determine a proper site for each cell in the decomposition and form various (1-ε)-approximate CIVDs for some small fixed ε > 0. Particularly, we consider two representative CIVD problems, vector CIVD and density-based CIVD, and show that both of them admit fast assignment algorithms; consequently, their (1 - ε)-approximate CIVDs can be built in O(n logd+1 n) and O(n log2 n) time, respectively.

AB - In this paper, we study a generalization of the classical Voronoi diagram, called clustering induced Voronoi diagram (CIVD). Different from the traditional model, CIVD takes as its sites the power set U of an input set P of objects. For each subset C of P, CIVD uses an influence function F(C, q) to measure the total (or joint) influence of all objects in C on an arbitrary point q in the space ℝd, and determines the influence-based Voronoi cell in ℝd for C. This generalized model offers a number of new features (e.g., simultaneous clustering and space partition) to Voronoi diagram which are useful in various new applications. We investigate the general conditions for the influence function which ensure the existence of a small-size (e.g., nearly linear) approximate CIVD for a set P of n points in ℝd for some fixed d. To construct CIVD, we first present a standalone new technique, called approximate influence (AI) decomposition, for the general CIVD problem. With only O(n log n) time, the AI decomposition partitions the space ℝd into a nearly linear number of cells so that all points in each cell receive their approximate maximum influence from the same (possibly unknown) site (i.e., a subset of P). Based on this technique, we develop assignment algorithms to determine a proper site for each cell in the decomposition and form various (1-ε)-approximate CIVDs for some small fixed ε > 0. Particularly, we consider two representative CIVD problems, vector CIVD and density-based CIVD, and show that both of them admit fast assignment algorithms; consequently, their (1 - ε)-approximate CIVDs can be built in O(n logd+1 n) and O(n log2 n) time, respectively.

UR - http://www.scopus.com/inward/record.url?scp=84893457578&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893457578&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2013.49

DO - 10.1109/FOCS.2013.49

M3 - Conference contribution

AN - SCOPUS:84893457578

SN - 9780769551357

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 390

EP - 399

BT - Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013

ER -