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Paper Abstract and Keywords
Presentation 2011-04-22 14:30
Reconstructing sets from distances given by graphs
Meng Li, Yota Otachi, Takeshi Tokuyama (Tohoku Univ.) COMP2011-7
Abstract (in Japanese) (See Japanese page) 
(in English) Given $n$ points in some Euclidean space, $\binom{n}{2}$ pairwise distances among the points can be easily calculated.
On the other hand, it is not always possible to reconstruct a point set up to motion uniquely from a multiset of $\binom{n}{2}$ distances, and the computational complexity for the reconstruction is not known even if it uniquely exists. Therefore, it is a natural question to ask how much additional information is required for designing an efficient reconstruction algorithm. It becomes trivial if we know which distance corresponds to which pair of points; that is, we have a distance matrix instead of a multiset of distances. Our interest is that ``what if we have a \emph{partial} distance matrix?'' Thus, we consider the case where partial information about the distance matrix is given: Precisely, we are given an weighted graph $G = (V, E)$ and find an embedding of $V$ into an Euclidean space such that the weight $w(e)$ of each edge $e$ gives the Euclidean distance between embedded points. Unfortunately, this problem is known to be strongly NP-hard for any Euclidean space $\mathbb{R}^{d}$ in general.
In this paper, we discuss complexity of the problem for important families of graphs. We first present a polynomial-time algorithm to embed chordal graphs into $\mathbb{R}^{d}$ for any positive integer $d$. Then, we prove that although embedding cycles in a line is NP-hard, it becomes easy in higher-dimensions. We also give a polynomial-time algorithm to find a point set on a line from distance information given as a graph with a small connected dominating set.
Keyword (in Japanese) (See Japanese page) 
(in English) Point set reconstruction / Weighted graph embedding / Graph turnpike problem / Chordal graph / Connected dominating set / / /  
Reference Info. IEICE Tech. Rep., vol. 111, no. 20, COMP2011-7, pp. 49-54, April 2011.
Paper # COMP2011-7 
Date of Issue 2011-04-15 (COMP) 
ISSN Print edition: ISSN 0913-5685  Online edition: ISSN 2432-6380
All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034)
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Conference Information
Committee COMP  
Conference Date 2011-04-22 - 2011-04-22 
Place (in Japanese) (See Japanese page) 
Place (in English) Kyoto University 
Topics (in Japanese) (See Japanese page) 
Topics (in English)  
Paper Information
Registration To COMP 
Conference Code 2011-04-COMP 
Language English 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) Reconstructing sets from distances given by graphs 
Sub Title (in English)  
Keyword(1) Point set reconstruction  
Keyword(2) Weighted graph embedding  
Keyword(3) Graph turnpike problem  
Keyword(4) Chordal graph  
Keyword(5) Connected dominating set  
1st Author's Name Meng Li  
1st Author's Affiliation Tohoku University (Tohoku Univ.)
2nd Author's Name Yota Otachi  
2nd Author's Affiliation Tohoku University (Tohoku Univ.)
3rd Author's Name Takeshi Tokuyama  
3rd Author's Affiliation Tohoku University (Tohoku Univ.)
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Date Time 2011-04-22 14:30:00 
Presentation Time 35 
Registration for COMP 
Paper # IEICE-COMP2011-7 
Volume (vol) IEICE-111 
Number (no) no.20 
Page pp.49-54 
#Pages IEICE-6 
Date of Issue IEICE-COMP-2011-04-15 

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