Improved stretch factor of Delaunay triangulations of points in convex position
Let S be a set of n points in the plane, and let DT ( S ) be the planar graph of the Delaunay triangulation of S . For a pair of points a , b ∈ S , denote by | ab | the Euclidean distance between a and b . Denote by DT ( a , b ) the shortest path in DT ( S ) between a and b , and let | DT ( a , b...
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| Veröffentlicht in: | Journal of combinatorial optimization 2023, Vol.45 (1), Article 3 |
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| creator | Tan, Xuehou Sakthip, Charatsanyakul Jiang, Bo Liu, Shimao |
| description | Let
S
be a set of
n
points in the plane, and let
DT
(
S
) be the planar graph of the Delaunay triangulation of
S
. For a pair of points
a
,
b
∈
S
, denote by |
ab
| the Euclidean distance between
a
and
b
. Denote by
DT
(
a
,
b
) the shortest path in
DT
(
S
) between
a
and
b
, and let |
DT
(
a
,
b
)| be the total length of
DT
(
a
,
b
). Dobkin et al. were the first to show that
DT
(
S
) can be used to approximate the complete graph of
S
in the sense that the stretch factor
|
D
T
(
a
,
b
)
|
|
a
b
|
is upper bounded by
(
(
1
+
5
)
/
2
)
π
≈
5.08
. Recently, Xia improved this factor to 1.998. Amani et al. have also shown that if the points of
S
are in
convex position
(i.e., they form the vertices of a convex polygon), then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. In this paper, we prove that if the points of
S
are in convex position, then the stretch factor of
DT
(
S
) is less than 1.84, improving upon the previously known factors of Delaunay triangulations or planar graphs in the convex case. |
| doi_str_mv | 10.1007/s10878-022-00940-4 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2736742801</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2736742801</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-613fa52ef7ebad224c3d46ed3c4047ce174a275c20793b577dc944eab3fea0473</originalsourceid><addsrcrecordid>eNp9kE1LAzEQhoMoWKt_wFPAc3TysZvdo9SvQqEXPYc0m9QtbVKTbNF_b-oK3jzNMO_zzgwvQtcUbimAvEsUGtkQYIwAtAKIOEETWklOWNPUp6XnDSN1C9U5ukhpA4WCSkzQcr7bx3CwHU452mzesdMmh4iDww92qwevv3COvfbrYatzH3w6SvvQ-5xw77EJ_mA_yyD1R_USnTm9Tfbqt07R29Pj6-yFLJbP89n9ghgmIZOacqcrZp20K90xJgzvRG07bgQIaSyVQjNZGQay5atKys60Qli94s7qQvApuhn3lu8_Bpuy2oQh-nJSMclrKVgDtFBspEwMKUXr1D72Ox2_FAV1DE6NwakSnPoJToli4qMpFdivbfxb_Y_rG5QicZY</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2736742801</pqid></control><display><type>article</type><title>Improved stretch factor of Delaunay triangulations of points in convex position</title><source>SpringerLink Journals - AutoHoldings</source><creator>Tan, Xuehou ; Sakthip, Charatsanyakul ; Jiang, Bo ; Liu, Shimao</creator><creatorcontrib>Tan, Xuehou ; Sakthip, Charatsanyakul ; Jiang, Bo ; Liu, Shimao</creatorcontrib><description>Let
S
be a set of
n
points in the plane, and let
DT
(
S
) be the planar graph of the Delaunay triangulation of
S
. For a pair of points
a
,
b
∈
S
, denote by |
ab
| the Euclidean distance between
a
and
b
. Denote by
DT
(
a
,
b
) the shortest path in
DT
(
S
) between
a
and
b
, and let |
DT
(
a
,
b
)| be the total length of
DT
(
a
,
b
). Dobkin et al. were the first to show that
DT
(
S
) can be used to approximate the complete graph of
S
in the sense that the stretch factor
|
D
T
(
a
,
b
)
|
|
a
b
|
is upper bounded by
(
(
1
+
5
)
/
2
)
π
≈
5.08
. Recently, Xia improved this factor to 1.998. Amani et al. have also shown that if the points of
S
are in
convex position
(i.e., they form the vertices of a convex polygon), then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. In this paper, we prove that if the points of
S
are in convex position, then the stretch factor of
DT
(
S
) is less than 1.84, improving upon the previously known factors of Delaunay triangulations or planar graphs in the convex case.</description><identifier>ISSN: 1382-6905</identifier><identifier>EISSN: 1573-2886</identifier><identifier>DOI: 10.1007/s10878-022-00940-4</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Apexes ; Combinatorics ; Convex and Discrete Geometry ; Delaunay triangulation ; Euclidean geometry ; Graph theory ; Graphs ; Mathematical Modeling and Industrial Mathematics ; Mathematics ; Mathematics and Statistics ; Operations Research/Decision Theory ; Optimization ; Shortest-path problems ; Theory of Computation</subject><ispartof>Journal of combinatorial optimization, 2023, Vol.45 (1), Article 3</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-613fa52ef7ebad224c3d46ed3c4047ce174a275c20793b577dc944eab3fea0473</cites><orcidid>0000-0002-9828-9927</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10878-022-00940-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10878-022-00940-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Tan, Xuehou</creatorcontrib><creatorcontrib>Sakthip, Charatsanyakul</creatorcontrib><creatorcontrib>Jiang, Bo</creatorcontrib><creatorcontrib>Liu, Shimao</creatorcontrib><title>Improved stretch factor of Delaunay triangulations of points in convex position</title><title>Journal of combinatorial optimization</title><addtitle>J Comb Optim</addtitle><description>Let
S
be a set of
n
points in the plane, and let
DT
(
S
) be the planar graph of the Delaunay triangulation of
S
. For a pair of points
a
,
b
∈
S
, denote by |
ab
| the Euclidean distance between
a
and
b
. Denote by
DT
(
a
,
b
) the shortest path in
DT
(
S
) between
a
and
b
, and let |
DT
(
a
,
b
)| be the total length of
DT
(
a
,
b
). Dobkin et al. were the first to show that
DT
(
S
) can be used to approximate the complete graph of
S
in the sense that the stretch factor
|
D
T
(
a
,
b
)
|
|
a
b
|
is upper bounded by
(
(
1
+
5
)
/
2
)
π
≈
5.08
. Recently, Xia improved this factor to 1.998. Amani et al. have also shown that if the points of
S
are in
convex position
(i.e., they form the vertices of a convex polygon), then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. In this paper, we prove that if the points of
S
are in convex position, then the stretch factor of
DT
(
S
) is less than 1.84, improving upon the previously known factors of Delaunay triangulations or planar graphs in the convex case.</description><subject>Apexes</subject><subject>Combinatorics</subject><subject>Convex and Discrete Geometry</subject><subject>Delaunay triangulation</subject><subject>Euclidean geometry</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Mathematical Modeling and Industrial Mathematics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Shortest-path problems</subject><subject>Theory of Computation</subject><issn>1382-6905</issn><issn>1573-2886</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LAzEQhoMoWKt_wFPAc3TysZvdo9SvQqEXPYc0m9QtbVKTbNF_b-oK3jzNMO_zzgwvQtcUbimAvEsUGtkQYIwAtAKIOEETWklOWNPUp6XnDSN1C9U5ukhpA4WCSkzQcr7bx3CwHU452mzesdMmh4iDww92qwevv3COvfbrYatzH3w6SvvQ-5xw77EJ_mA_yyD1R_USnTm9Tfbqt07R29Pj6-yFLJbP89n9ghgmIZOacqcrZp20K90xJgzvRG07bgQIaSyVQjNZGQay5atKys60Qli94s7qQvApuhn3lu8_Bpuy2oQh-nJSMclrKVgDtFBspEwMKUXr1D72Ox2_FAV1DE6NwakSnPoJToli4qMpFdivbfxb_Y_rG5QicZY</recordid><startdate>2023</startdate><enddate>2023</enddate><creator>Tan, Xuehou</creator><creator>Sakthip, Charatsanyakul</creator><creator>Jiang, Bo</creator><creator>Liu, Shimao</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-9828-9927</orcidid></search><sort><creationdate>2023</creationdate><title>Improved stretch factor of Delaunay triangulations of points in convex position</title><author>Tan, Xuehou ; Sakthip, Charatsanyakul ; Jiang, Bo ; Liu, Shimao</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-613fa52ef7ebad224c3d46ed3c4047ce174a275c20793b577dc944eab3fea0473</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Apexes</topic><topic>Combinatorics</topic><topic>Convex and Discrete Geometry</topic><topic>Delaunay triangulation</topic><topic>Euclidean geometry</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Mathematical Modeling and Industrial Mathematics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Shortest-path problems</topic><topic>Theory of Computation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Tan, Xuehou</creatorcontrib><creatorcontrib>Sakthip, Charatsanyakul</creatorcontrib><creatorcontrib>Jiang, Bo</creatorcontrib><creatorcontrib>Liu, Shimao</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of combinatorial optimization</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Tan, Xuehou</au><au>Sakthip, Charatsanyakul</au><au>Jiang, Bo</au><au>Liu, Shimao</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Improved stretch factor of Delaunay triangulations of points in convex position</atitle><jtitle>Journal of combinatorial optimization</jtitle><stitle>J Comb Optim</stitle><date>2023</date><risdate>2023</risdate><volume>45</volume><issue>1</issue><artnum>3</artnum><issn>1382-6905</issn><eissn>1573-2886</eissn><abstract>Let
S
be a set of
n
points in the plane, and let
DT
(
S
) be the planar graph of the Delaunay triangulation of
S
. For a pair of points
a
,
b
∈
S
, denote by |
ab
| the Euclidean distance between
a
and
b
. Denote by
DT
(
a
,
b
) the shortest path in
DT
(
S
) between
a
and
b
, and let |
DT
(
a
,
b
)| be the total length of
DT
(
a
,
b
). Dobkin et al. were the first to show that
DT
(
S
) can be used to approximate the complete graph of
S
in the sense that the stretch factor
|
D
T
(
a
,
b
)
|
|
a
b
|
is upper bounded by
(
(
1
+
5
)
/
2
)
π
≈
5.08
. Recently, Xia improved this factor to 1.998. Amani et al. have also shown that if the points of
S
are in
convex position
(i.e., they form the vertices of a convex polygon), then a planar graph with these vertices can be constructed such that its stretch factor is 1.88. In this paper, we prove that if the points of
S
are in convex position, then the stretch factor of
DT
(
S
) is less than 1.84, improving upon the previously known factors of Delaunay triangulations or planar graphs in the convex case.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10878-022-00940-4</doi><orcidid>https://orcid.org/0000-0002-9828-9927</orcidid></addata></record> |
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| ispartof | Journal of combinatorial optimization, 2023, Vol.45 (1), Article 3 |
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| subjects | Apexes Combinatorics Convex and Discrete Geometry Delaunay triangulation Euclidean geometry Graph theory Graphs Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Shortest-path problems Theory of Computation |
| title | Improved stretch factor of Delaunay triangulations of points in convex position |
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