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| Presentation | 2008-05-13 15:55 An O(n^{1.75})-time Algorithm for L(2,1)-labeling of Trees Toru Hasunuma (Univ. Tokushima), Toshimasa Ishii (Otaru Univ. of Commerce), Hirotaka Ono (Kyushu Univ.), Yushi Uno (Osaka Prefecture Univ.)  COMP2008-14 |
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| Abstract | (in Japanese) | (See Japanese page) |
| (in English) | An $L(2,1)$-labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ and $y$ are adjacent and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are at distance 2 for all $x$ and $y$ in $V(G)$. A $k$-$L(2,1)$-labeling is an assignment $f:V(G)\rightarrow\{0,\ldots ,k\}$, and the $L(2,1)$-labeling problem asks the minimum $k$, which we denote by $\lambda(G)$, among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2. Tree is one of a few classes for which the problem is polynomially solvable, but still only an $\mbox{O}(\Delta^{4.5} n)$ time algorithm for a tree $T$ has been known so far, where $\Delta$ is the maximum degree of $T$ and $n=|V(T)|$. In this paper, we first show that an existent necessary condition for $\lambda(T)=\Delta+1$ is also sufficient for a tree $T$ with $\Delta=\Omega(\sqrt{n})$, which leads a linear time algorithm for computing $\lambda(T)$ under this condition. We then show that $\lambda(T)$ can be computed in $\mbox{O}(\Delta^{1.5}n)$ time for any tree $T$. Combining these, we finally obtain an $\mO(n^{1.75})$ time algorithm, which greatly improves the currently best known result. |
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| Keyword | (in Japanese) | (See Japanese page) |
| (in English) | frequency/channel assignment / graph algorithm / $L(2,1)$-labeling / vertex coloring / / / / | |
| Reference Info. | IEICE Tech. Rep., vol. 108, no. 29, COMP2008-14, pp. 43-50, May 2008. | |
| Paper # | COMP2008-14 | |
| Date of Issue | 2008-05-06 (COMP) | |
| ISSN | Print edition: ISSN 0913-5685 Online edition: ISSN 2432-6380 | |
| Copyright and reproduction |
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 | COMP2008-14 |
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| Conference Information | |
| Committee | COMP |
| Conference Date | 2008-05-13 - 2008-05-13 |
| Place (in Japanese) | (See Japanese page) |
| Place (in English) | Kyushu Sangyo University |
| Topics (in Japanese) | (See Japanese page) |
| Topics (in English) | |
| Paper Information | |
| Registration To | COMP |
| Conference Code | 2008-05-COMP |
| Language | English (Japanese title is available) |
| Title (in Japanese) | (See Japanese page) |
| Sub Title (in Japanese) | (See Japanese page) |
| Title (in English) | An O(n^{1.75})-time Algorithm for L(2,1)-labeling of Trees |
| Sub Title (in English) | |
| Keyword(1) | frequency/channel assignment |
| Keyword(2) | graph algorithm |
| Keyword(3) | $L(2,1)$-labeling |
| Keyword(4) | vertex coloring |
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| 1st Author's Name | Toru Hasunuma |
| 1st Author's Affiliation | The University of Tokushima (Univ. Tokushima) |
| 2nd Author's Name | Toshimasa Ishii |
| 2nd Author's Affiliation | Otaru University of Commerce (Otaru Univ. of Commerce) |
| 3rd Author's Name | Hirotaka Ono |
| 3rd Author's Affiliation | Kyushu University (Kyushu Univ.) |
| 4th Author's Name | Yushi Uno |
| 4th Author's Affiliation | Osaka Prefecture University (Osaka Prefecture Univ.) |
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| Speaker | Author-3 |
| Date Time | 2008-05-13 15:55:00 |
| Presentation Time | 35 minutes |
| Registration for | COMP |
| Paper # | COMP2008-14 |
| Volume (vol) | vol.108 |
| Number (no) | no.29 |
| Page | pp.43-50 |
| #Pages | 8 |
| Date of Issue | 2008-05-06 (COMP) |
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