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 Paper Abstract and Keywords Presentation 2011-04-22 14:30 Reconstructing sets from distances given by graphsMeng 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 Copyrightandreproduction 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) Download PDF COMP2011-7

 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 Keyword(6) Keyword(7) Keyword(8) 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.) 4th Author's Name 4th Author's Affiliation () 5th Author's Name 5th Author's Affiliation () 6th Author's Name 6th Author's Affiliation () 7th Author's Name 7th Author's Affiliation () 8th Author's Name 8th Author's Affiliation () 9th Author's Name 9th Author's Affiliation () 10th Author's Name 10th Author's Affiliation () 11th Author's Name 11th Author's Affiliation () 12th Author's Name 12th Author's Affiliation () 13th Author's Name 13th Author's Affiliation () 14th Author's Name 14th Author's Affiliation () 15th Author's Name 15th Author's Affiliation () 16th Author's Name 16th Author's Affiliation () 17th Author's Name 17th Author's Affiliation () 18th Author's Name 18th Author's Affiliation () Speaker 1 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