Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation
A sliding k -transmitter inside an orthogonal polygon P , for a fixed k ≥ 0 , is a point guard that travels along an axis-parallel line segment s in P . The sliding k -transmitter can see a point p ∈ P if the perpendicular from p onto s intersects the boundary of P in at most k points. In the Minimu...
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| Veröffentlicht in: | Algorithmica 2019-01, Vol.81 (1), p.69-97 |
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| container_title | Algorithmica |
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| creator | Biedl, Therese Chan, Timothy M. Lee, Stephanie Mehrabi, Saeed Montecchiani, Fabrizio Vosoughpour, Hamideh Yu, Ziting |
| description | A sliding
k
-transmitter inside an orthogonal polygon
P
, for a fixed
k
≥
0
, is a point guard that travels along an axis-parallel line segment
s
in
P
. The sliding
k
-transmitter can see a point
p
∈
P
if the perpendicular from
p
onto
s
intersects the boundary of
P
in at most
k
points. In the Minimum Sliding
k
-Transmitters (
ST
k
) problem, the objective is to guard
P
with the minimum number of sliding
k
-transmitters. In this paper, we give a constant-factor approximation algorithm for the
ST
k
problem on
P
for any fixed
k
≥
0
. Moreover, we show that the
ST
0
problem is
NP
-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For
k
>
0
, the problem is
NP
-hard even in the extremely restricted case where
P
is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard
P
relative to the number of vertices of
P
. |
| doi_str_mv | 10.1007/s00453-018-0433-6 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2165690961</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2165690961</sourcerecordid><originalsourceid>FETCH-LOGICAL-c268t-85abb6a987fe17ea3ff4e6945c48dd953d108e8502fd6150a32c5f1200ee93e53</originalsourceid><addsrcrecordid>eNp1kLFOwzAQhi0EEqXwAGyWmA13duw4bFUFLVKlDpSByXITp01Jk2K7At4elyIxMd3yff_d_YRcI9wiQH4XADIpGKBmkAnB1AkZYCY4A5nhKRkA5pplCvNzchHCBgB5XqgBeZ3sra-abkXnPq77Vd_Zlo58pBPbts43LtCPJq7pc9v8UG9s4W0Xtk2Mzod7Ok1250KgtqvoaLfz_WeztbHpu0tyVts2uKvfOSQvjw-L8ZTN5pOn8WjGSq50ZFra5VLZQue1w9xZUdeZU0Umy0xXVSFFhaCdlsDrSqEEK3gpa-QAzhXCSTEkN8fctPt970I0m37v0xvBcFRSFVAoTBQeqdL3IXhXm51Ph_ovg2AODZpjgyY1aA4NGpUcfnRCYruV83_J_0vfy2Jzxg</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2165690961</pqid></control><display><type>article</type><title>Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation</title><source>SpringerLink Journals - AutoHoldings</source><creator>Biedl, Therese ; Chan, Timothy M. ; Lee, Stephanie ; Mehrabi, Saeed ; Montecchiani, Fabrizio ; Vosoughpour, Hamideh ; Yu, Ziting</creator><creatorcontrib>Biedl, Therese ; Chan, Timothy M. ; Lee, Stephanie ; Mehrabi, Saeed ; Montecchiani, Fabrizio ; Vosoughpour, Hamideh ; Yu, Ziting</creatorcontrib><description>A sliding
k
-transmitter inside an orthogonal polygon
P
, for a fixed
k
≥
0
, is a point guard that travels along an axis-parallel line segment
s
in
P
. The sliding
k
-transmitter can see a point
p
∈
P
if the perpendicular from
p
onto
s
intersects the boundary of
P
in at most
k
points. In the Minimum Sliding
k
-Transmitters (
ST
k
) problem, the objective is to guard
P
with the minimum number of sliding
k
-transmitters. In this paper, we give a constant-factor approximation algorithm for the
ST
k
problem on
P
for any fixed
k
≥
0
. Moreover, we show that the
ST
0
problem is
NP
-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For
k
>
0
, the problem is
NP
-hard even in the extremely restricted case where
P
is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard
P
relative to the number of vertices of
P
.</description><identifier>ISSN: 0178-4617</identifier><identifier>EISSN: 1432-0541</identifier><identifier>DOI: 10.1007/s00453-018-0433-6</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithm Analysis and Problem Complexity ; Algorithms ; Apexes ; Approximation ; Art galleries & museums ; Computer Science ; Computer Systems Organization and Communication Networks ; Data Structures and Information Theory ; Lower bounds ; Mathematical analysis ; Mathematics of Computing ; Sliding ; Theory of Computation ; Transmitters</subject><ispartof>Algorithmica, 2019-01, Vol.81 (1), p.69-97</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2018</rights><rights>Copyright Springer Science & Business Media 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c268t-85abb6a987fe17ea3ff4e6945c48dd953d108e8502fd6150a32c5f1200ee93e53</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00453-018-0433-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00453-018-0433-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Biedl, Therese</creatorcontrib><creatorcontrib>Chan, Timothy M.</creatorcontrib><creatorcontrib>Lee, Stephanie</creatorcontrib><creatorcontrib>Mehrabi, Saeed</creatorcontrib><creatorcontrib>Montecchiani, Fabrizio</creatorcontrib><creatorcontrib>Vosoughpour, Hamideh</creatorcontrib><creatorcontrib>Yu, Ziting</creatorcontrib><title>Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation</title><title>Algorithmica</title><addtitle>Algorithmica</addtitle><description>A sliding
k
-transmitter inside an orthogonal polygon
P
, for a fixed
k
≥
0
, is a point guard that travels along an axis-parallel line segment
s
in
P
. The sliding
k
-transmitter can see a point
p
∈
P
if the perpendicular from
p
onto
s
intersects the boundary of
P
in at most
k
points. In the Minimum Sliding
k
-Transmitters (
ST
k
) problem, the objective is to guard
P
with the minimum number of sliding
k
-transmitters. In this paper, we give a constant-factor approximation algorithm for the
ST
k
problem on
P
for any fixed
k
≥
0
. Moreover, we show that the
ST
0
problem is
NP
-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For
k
>
0
, the problem is
NP
-hard even in the extremely restricted case where
P
is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard
P
relative to the number of vertices of
P
.</description><subject>Algorithm Analysis and Problem Complexity</subject><subject>Algorithms</subject><subject>Apexes</subject><subject>Approximation</subject><subject>Art galleries & museums</subject><subject>Computer Science</subject><subject>Computer Systems Organization and Communication Networks</subject><subject>Data Structures and Information Theory</subject><subject>Lower bounds</subject><subject>Mathematical analysis</subject><subject>Mathematics of Computing</subject><subject>Sliding</subject><subject>Theory of Computation</subject><subject>Transmitters</subject><issn>0178-4617</issn><issn>1432-0541</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1kLFOwzAQhi0EEqXwAGyWmA13duw4bFUFLVKlDpSByXITp01Jk2K7At4elyIxMd3yff_d_YRcI9wiQH4XADIpGKBmkAnB1AkZYCY4A5nhKRkA5pplCvNzchHCBgB5XqgBeZ3sra-abkXnPq77Vd_Zlo58pBPbts43LtCPJq7pc9v8UG9s4W0Xtk2Mzod7Ok1250KgtqvoaLfz_WeztbHpu0tyVts2uKvfOSQvjw-L8ZTN5pOn8WjGSq50ZFra5VLZQue1w9xZUdeZU0Umy0xXVSFFhaCdlsDrSqEEK3gpa-QAzhXCSTEkN8fctPt970I0m37v0xvBcFRSFVAoTBQeqdL3IXhXm51Ph_ovg2AODZpjgyY1aA4NGpUcfnRCYruV83_J_0vfy2Jzxg</recordid><startdate>20190101</startdate><enddate>20190101</enddate><creator>Biedl, Therese</creator><creator>Chan, Timothy M.</creator><creator>Lee, Stephanie</creator><creator>Mehrabi, Saeed</creator><creator>Montecchiani, Fabrizio</creator><creator>Vosoughpour, Hamideh</creator><creator>Yu, Ziting</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20190101</creationdate><title>Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation</title><author>Biedl, Therese ; Chan, Timothy M. ; Lee, Stephanie ; Mehrabi, Saeed ; Montecchiani, Fabrizio ; Vosoughpour, Hamideh ; Yu, Ziting</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c268t-85abb6a987fe17ea3ff4e6945c48dd953d108e8502fd6150a32c5f1200ee93e53</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Algorithm Analysis and Problem Complexity</topic><topic>Algorithms</topic><topic>Apexes</topic><topic>Approximation</topic><topic>Art galleries & museums</topic><topic>Computer Science</topic><topic>Computer Systems Organization and Communication Networks</topic><topic>Data Structures and Information Theory</topic><topic>Lower bounds</topic><topic>Mathematical analysis</topic><topic>Mathematics of Computing</topic><topic>Sliding</topic><topic>Theory of Computation</topic><topic>Transmitters</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Biedl, Therese</creatorcontrib><creatorcontrib>Chan, Timothy M.</creatorcontrib><creatorcontrib>Lee, Stephanie</creatorcontrib><creatorcontrib>Mehrabi, Saeed</creatorcontrib><creatorcontrib>Montecchiani, Fabrizio</creatorcontrib><creatorcontrib>Vosoughpour, Hamideh</creatorcontrib><creatorcontrib>Yu, Ziting</creatorcontrib><collection>CrossRef</collection><jtitle>Algorithmica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Biedl, Therese</au><au>Chan, Timothy M.</au><au>Lee, Stephanie</au><au>Mehrabi, Saeed</au><au>Montecchiani, Fabrizio</au><au>Vosoughpour, Hamideh</au><au>Yu, Ziting</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation</atitle><jtitle>Algorithmica</jtitle><stitle>Algorithmica</stitle><date>2019-01-01</date><risdate>2019</risdate><volume>81</volume><issue>1</issue><spage>69</spage><epage>97</epage><pages>69-97</pages><issn>0178-4617</issn><eissn>1432-0541</eissn><abstract>A sliding
k
-transmitter inside an orthogonal polygon
P
, for a fixed
k
≥
0
, is a point guard that travels along an axis-parallel line segment
s
in
P
. The sliding
k
-transmitter can see a point
p
∈
P
if the perpendicular from
p
onto
s
intersects the boundary of
P
in at most
k
points. In the Minimum Sliding
k
-Transmitters (
ST
k
) problem, the objective is to guard
P
with the minimum number of sliding
k
-transmitters. In this paper, we give a constant-factor approximation algorithm for the
ST
k
problem on
P
for any fixed
k
≥
0
. Moreover, we show that the
ST
0
problem is
NP
-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For
k
>
0
, the problem is
NP
-hard even in the extremely restricted case where
P
is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard
P
relative to the number of vertices of
P
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00453-018-0433-6</doi><tpages>29</tpages></addata></record> |
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| identifier | ISSN: 0178-4617 |
| ispartof | Algorithmica, 2019-01, Vol.81 (1), p.69-97 |
| issn | 0178-4617 1432-0541 |
| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Algorithm Analysis and Problem Complexity Algorithms Apexes Approximation Art galleries & museums Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Lower bounds Mathematical analysis Mathematics of Computing Sliding Theory of Computation Transmitters |
| title | Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation |
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