Conflict-Free Colorings of Rectangles Ranges
Given the range space (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P,\mathcal{R}$\end{document}), where P is a set...
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| creator | Elbassioni, Khaled Mustafa, Nabil H. |
| description | Given the range space (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}), where P is a set of n points in ℝ2 and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mathcal{R}$\end{document} is the family of subsets of P induced by all axis-parallel rectangles, the conflict-free coloring problem asks for a coloring of P with the minimum number of colors such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}) is conflict-free. We study the following question: Given P, is it possible to add a small set of points Q such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P \cup Q,\mathcal{R}$\end{document}) can be colored with fewer colors than (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document})? Our main result is the following: given P, and any ε ≥ 0, one can always add a set Q of O(n1 − ε) points such that P ∪ Q can be conflict-free colored using \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\tilde{O}(n^{\frac{3}{8}(1+\varepsilon)})$\end{document} colors. Moreover, the set Q and the conflict-free coloring can be computed in polynomial time, with high probability. Our result is obtained by introducing a general probabilistic re-coloring technique, which we call quasi-conflict-free coloring, and which may be of independent interest. A further application of this technique is also given. |
| doi_str_mv | 10.1007/11672142_20 |
| format | Conference Proceeding |
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\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}), where P is a set of n points in ℝ2 and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mathcal{R}$\end{document} is the family of subsets of P induced by all axis-parallel rectangles, the conflict-free coloring problem asks for a coloring of P with the minimum number of colors such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}) is conflict-free. We study the following question: Given P, is it possible to add a small set of points Q such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P \cup Q,\mathcal{R}$\end{document}) can be colored with fewer colors than (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document})? Our main result is the following: given P, and any ε ≥ 0, one can always add a set Q of O(n1 − ε) points such that P ∪ Q can be conflict-free colored using \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\tilde{O}(n^{\frac{3}{8}(1+\varepsilon)})$\end{document} colors. Moreover, the set Q and the conflict-free coloring can be computed in polynomial time, with high probability. Our result is obtained by introducing a general probabilistic re-coloring technique, which we call quasi-conflict-free coloring, and which may be of independent interest. A further application of this technique is also given.</description><identifier>ISSN: 0302-9743</identifier><identifier>ISBN: 9783540323013</identifier><identifier>ISBN: 3540323015</identifier><identifier>EISSN: 1611-3349</identifier><identifier>EISBN: 9783540322887</identifier><identifier>EISBN: 3540322884</identifier><identifier>DOI: 10.1007/11672142_20</identifier><language>eng</language><publisher>Berlin, Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algorithmics. Computability. Computer arithmetics ; Applied sciences ; Coloring Problem ; Computer science; control theory; systems ; Exact sciences and technology ; Free Coloring ; Grid Case ; Range Space ; Theoretical computing ; Unique Color</subject><ispartof>Lecture notes in computer science, 2006, p.254-263</ispartof><rights>Springer-Verlag Berlin Heidelberg 2006</rights><rights>2007 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/11672142_20$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/11672142_20$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>309,310,775,776,780,785,786,789,4036,4037,27902,38232,41418,42487</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=19162788$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><contributor>Durand, Bruno</contributor><contributor>Thomas, Wolfgang</contributor><creatorcontrib>Elbassioni, Khaled</creatorcontrib><creatorcontrib>Mustafa, Nabil H.</creatorcontrib><title>Conflict-Free Colorings of Rectangles Ranges</title><title>Lecture notes in computer science</title><description>Given the range space (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}), where P is a set of n points in ℝ2 and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mathcal{R}$\end{document} is the family of subsets of P induced by all axis-parallel rectangles, the conflict-free coloring problem asks for a coloring of P with the minimum number of colors such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}) is conflict-free. We study the following question: Given P, is it possible to add a small set of points Q such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P \cup Q,\mathcal{R}$\end{document}) can be colored with fewer colors than (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document})? Our main result is the following: given P, and any ε ≥ 0, one can always add a set Q of O(n1 − ε) points such that P ∪ Q can be conflict-free colored using \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\tilde{O}(n^{\frac{3}{8}(1+\varepsilon)})$\end{document} colors. Moreover, the set Q and the conflict-free coloring can be computed in polynomial time, with high probability. Our result is obtained by introducing a general probabilistic re-coloring technique, which we call quasi-conflict-free coloring, and which may be of independent interest. A further application of this technique is also given.</description><subject>Algorithmics. Computability. Computer arithmetics</subject><subject>Applied sciences</subject><subject>Coloring Problem</subject><subject>Computer science; control theory; systems</subject><subject>Exact sciences and technology</subject><subject>Free Coloring</subject><subject>Grid Case</subject><subject>Range Space</subject><subject>Theoretical computing</subject><subject>Unique Color</subject><issn>0302-9743</issn><issn>1611-3349</issn><isbn>9783540323013</isbn><isbn>3540323015</isbn><isbn>9783540322887</isbn><isbn>3540322884</isbn><fulltext>true</fulltext><rsrctype>conference_proceeding</rsrctype><creationdate>2006</creationdate><recordtype>conference_proceeding</recordtype><recordid>eNpNkE9Lw0AUxNd_YKg9-QVy8SAYfW_fNvv2KKFVoSAUPS-b7W6JxiRke_Hbm1IPncsc5scwjBC3CI8IoJ8QSy1RSSvhTMyNZlooICmZ9bnIsEQsiJS5OMkIkC5FBgSyMFrRtZin9AWTSAKyysRD1Xexbfy-WI0h5FXf9mPT7VLex3wT_N51uzakfDN5SDfiKro2hfm_z8TnavlRvRbr95e36nldDBLNvoiwlRyAgqJpEMstB-mRw4IwlioqjN7IaGpnKLga3ITV2rMGvWV2paaZuDv2Di5518bRdb5JdhibHzf-WjRYSs08cfdHLg2H0WG0dd9_J4tgD4_Zk8foD4P3VcA</recordid><startdate>2006</startdate><enddate>2006</enddate><creator>Elbassioni, Khaled</creator><creator>Mustafa, Nabil H.</creator><general>Springer Berlin Heidelberg</general><general>Springer</general><scope>IQODW</scope></search><sort><creationdate>2006</creationdate><title>Conflict-Free Colorings of Rectangles Ranges</title><author>Elbassioni, Khaled ; Mustafa, Nabil H.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p219t-f0d28e03e4333482d8e2c18e531f64f41fc92f9ba93eab0a433b7c8707d88a673</frbrgroupid><rsrctype>conference_proceedings</rsrctype><prefilter>conference_proceedings</prefilter><language>eng</language><creationdate>2006</creationdate><topic>Algorithmics. Computability. Computer arithmetics</topic><topic>Applied sciences</topic><topic>Coloring Problem</topic><topic>Computer science; control theory; systems</topic><topic>Exact sciences and technology</topic><topic>Free Coloring</topic><topic>Grid Case</topic><topic>Range Space</topic><topic>Theoretical computing</topic><topic>Unique Color</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Elbassioni, Khaled</creatorcontrib><creatorcontrib>Mustafa, Nabil H.</creatorcontrib><collection>Pascal-Francis</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Elbassioni, Khaled</au><au>Mustafa, Nabil H.</au><au>Durand, Bruno</au><au>Thomas, Wolfgang</au><format>book</format><genre>proceeding</genre><ristype>CONF</ristype><atitle>Conflict-Free Colorings of Rectangles Ranges</atitle><btitle>Lecture notes in computer science</btitle><date>2006</date><risdate>2006</risdate><spage>254</spage><epage>263</epage><pages>254-263</pages><issn>0302-9743</issn><eissn>1611-3349</eissn><isbn>9783540323013</isbn><isbn>3540323015</isbn><eisbn>9783540322887</eisbn><eisbn>3540322884</eisbn><abstract>Given the range space (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}), where P is a set of n points in ℝ2 and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\mathcal{R}$\end{document} is the family of subsets of P induced by all axis-parallel rectangles, the conflict-free coloring problem asks for a coloring of P with the minimum number of colors such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document}) is conflict-free. We study the following question: Given P, is it possible to add a small set of points Q such that (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P \cup Q,\mathcal{R}$\end{document}) can be colored with fewer colors than (\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$P,\mathcal{R}$\end{document})? Our main result is the following: given P, and any ε ≥ 0, one can always add a set Q of O(n1 − ε) points such that P ∪ Q can be conflict-free colored using \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$\tilde{O}(n^{\frac{3}{8}(1+\varepsilon)})$\end{document} colors. Moreover, the set Q and the conflict-free coloring can be computed in polynomial time, with high probability. Our result is obtained by introducing a general probabilistic re-coloring technique, which we call quasi-conflict-free coloring, and which may be of independent interest. A further application of this technique is also given.</abstract><cop>Berlin, Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/11672142_20</doi><tpages>10</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0302-9743 |
| ispartof | Lecture notes in computer science, 2006, p.254-263 |
| issn | 0302-9743 1611-3349 |
| language | eng |
| recordid | cdi_pascalfrancis_primary_19162788 |
| source | Springer Books |
| subjects | Algorithmics. Computability. Computer arithmetics Applied sciences Coloring Problem Computer science control theory systems Exact sciences and technology Free Coloring Grid Case Range Space Theoretical computing Unique Color |
| title | Conflict-Free Colorings of Rectangles Ranges |
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