Exact uniform sampling over catalan structures
We present a new framework for creating elegant algorithms for exact uniform sampling of important Catalan structures, such as triangulations of convex polygons, Dyck words, monotonic lattice paths and mountain ranges. Along with sampling, we obtain optimal coding, and optimal number of random bits...
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| creator | Angelopoulos, Alexandros Bakali, Eleni |
| description | We present a new framework for creating elegant algorithms for exact uniform
sampling of important Catalan structures, such as triangulations of convex
polygons, Dyck words, monotonic lattice paths and mountain ranges. Along with
sampling, we obtain optimal coding, and optimal number of random bits required
for the algorithm. The framework is based on an original two-parameter
recursive relation, where Ballot and Catalan numbers appear and which may be
regarded as to demonstrate a generalized reduction argument. We then describe
(a) a unique $n\times n$ matrix to be used for any of the problems -the common
pre-processing step of our framework- and (b) a linear height tree, where
leaves correspond one by one to all distinct solutions of each problem;
sampling is essentially done by selecting a path from the root to a leaf - the
main algorithm. Our main algorithm is linear for a number of the problems
mentioned. |
| doi_str_mv | 10.48550/arxiv.1803.03945 |
| format | Article |
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sampling of important Catalan structures, such as triangulations of convex
polygons, Dyck words, monotonic lattice paths and mountain ranges. Along with
sampling, we obtain optimal coding, and optimal number of random bits required
for the algorithm. The framework is based on an original two-parameter
recursive relation, where Ballot and Catalan numbers appear and which may be
regarded as to demonstrate a generalized reduction argument. We then describe
(a) a unique $n\times n$ matrix to be used for any of the problems -the common
pre-processing step of our framework- and (b) a linear height tree, where
leaves correspond one by one to all distinct solutions of each problem;
sampling is essentially done by selecting a path from the root to a leaf - the
main algorithm. Our main algorithm is linear for a number of the problems
mentioned.</description><identifier>DOI: 10.48550/arxiv.1803.03945</identifier><language>eng</language><subject>Computer Science - Computational Geometry ; Computer Science - Data Structures and Algorithms ; Computer Science - Discrete Mathematics</subject><creationdate>2018-03</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1803.03945$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1803.03945$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Angelopoulos, Alexandros</creatorcontrib><creatorcontrib>Bakali, Eleni</creatorcontrib><title>Exact uniform sampling over catalan structures</title><description>We present a new framework for creating elegant algorithms for exact uniform
sampling of important Catalan structures, such as triangulations of convex
polygons, Dyck words, monotonic lattice paths and mountain ranges. Along with
sampling, we obtain optimal coding, and optimal number of random bits required
for the algorithm. The framework is based on an original two-parameter
recursive relation, where Ballot and Catalan numbers appear and which may be
regarded as to demonstrate a generalized reduction argument. We then describe
(a) a unique $n\times n$ matrix to be used for any of the problems -the common
pre-processing step of our framework- and (b) a linear height tree, where
leaves correspond one by one to all distinct solutions of each problem;
sampling is essentially done by selecting a path from the root to a leaf - the
main algorithm. Our main algorithm is linear for a number of the problems
mentioned.</description><subject>Computer Science - Computational Geometry</subject><subject>Computer Science - Data Structures and Algorithms</subject><subject>Computer Science - Discrete Mathematics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotzr0KwjAUhuEsDqJegJO5gdbEc9Iko4h_ILh0L6cxkUJbJW1F7178mb7hhY-HsbkUKRqlxJLis3qk0ghIBVhUY5Zun-R6PrRVuMWGd9Tc66q98tvDR-6op5pa3vVxcP0QfTdlo0B152f_nbB8t803h-R03h8361NCmVaJQaW9E6XBkmypghMgPWYO8KK8uQQwVioSiBbByk8VWq-CJw3WQeZhwha_2y-4uMeqofgqPvDiC4c3zLc9FQ</recordid><startdate>20180311</startdate><enddate>20180311</enddate><creator>Angelopoulos, Alexandros</creator><creator>Bakali, Eleni</creator><scope>AKY</scope><scope>GOX</scope></search><sort><creationdate>20180311</creationdate><title>Exact uniform sampling over catalan structures</title><author>Angelopoulos, Alexandros ; Bakali, Eleni</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a675-8457ec0b84ba9b5fc031e46c34d5e8df38915a04494391c0310772fea739c36e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Computer Science - Computational Geometry</topic><topic>Computer Science - Data Structures and Algorithms</topic><topic>Computer Science - Discrete Mathematics</topic><toplevel>online_resources</toplevel><creatorcontrib>Angelopoulos, Alexandros</creatorcontrib><creatorcontrib>Bakali, Eleni</creatorcontrib><collection>arXiv Computer Science</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Angelopoulos, Alexandros</au><au>Bakali, Eleni</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Exact uniform sampling over catalan structures</atitle><date>2018-03-11</date><risdate>2018</risdate><abstract>We present a new framework for creating elegant algorithms for exact uniform
sampling of important Catalan structures, such as triangulations of convex
polygons, Dyck words, monotonic lattice paths and mountain ranges. Along with
sampling, we obtain optimal coding, and optimal number of random bits required
for the algorithm. The framework is based on an original two-parameter
recursive relation, where Ballot and Catalan numbers appear and which may be
regarded as to demonstrate a generalized reduction argument. We then describe
(a) a unique $n\times n$ matrix to be used for any of the problems -the common
pre-processing step of our framework- and (b) a linear height tree, where
leaves correspond one by one to all distinct solutions of each problem;
sampling is essentially done by selecting a path from the root to a leaf - the
main algorithm. Our main algorithm is linear for a number of the problems
mentioned.</abstract><doi>10.48550/arxiv.1803.03945</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Computer Science - Computational Geometry Computer Science - Data Structures and Algorithms Computer Science - Discrete Mathematics |
| title | Exact uniform sampling over catalan structures |
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