{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22868"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22868","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Topics in combinatorial and computational geometry","abstract":"This thesis consists of two parts dealing with combinatorial and computational problems in geometry, respectively. In the first part three independent problems are considered: (1) We determine an upper bound $\\lfloor 11n/6\\rfloor$ + 1 for the number of extreme triples of n points in the plane, almost matching a known lower bound $\\lfloor 11n/6\\rfloor$; (2) we determine some bounds for the smallest dimension $d = \\Delta(j,k)$ such that for any j mass distributions in $\\IR\\sp{d}$, there are k hyperplanes so that each orthant contains a fraction 1/2$\\sp{k}$ of each of the masses; it is easily shown that $j(2\\sp{k}-1)/k \\le \\Delta(j,k)\\le j2\\sp{k-1}$; we believe the lower bound is tight, but can only prove it in a few cases (as a tool we prove a Borsuk-Ulam theorem on a product of balls, which is of independent interest); (3) for a collection B of pseudo-disks in the plane, we show the existence of a two-dimensional abstract simplicial complex, $\\chi \\subseteq 2\\sp{B}$, which has some nice topological properties, such that the inclusion-exclusion relation $\\mu(\\cup B) = \\Sigma \\sb{\\sigma\\in 2\\sp{B} - \\{\\phi\\}}(-1)\\sp{\\rm card\\ \\sigma -1}\\mu(\\cap\\sigma)$ holds when $\\chi$ is substituted for 2$\\sp{B}$. In the second part, using geometric sampling techniques, we give algorithms for three similar problems: (4) Computing the intersection of halfspaces in $\\IR\\sp3$; (5) computing the intersection of balls of equal radius in $\\IR\\sp3$; and (6) computing the Voronoi diagram of line segments in $\\IR\\sp2$; in each case we obtain a deterministic parallel algorithm for the EREW PRAM model that runs in time $O({\\rm log}\\sp2\\ n)$ and uses work $O(n\\ {\\rm log}\\ n)$ for a problem of size n (for ball intersection this is also the first optimal deterministic and sequential algorithm, using the Dobkin-Kirkpatrick decomposition, we can only achieve time $O(n\\ {\\rm log}\\sp2\\ n$)). Using the parallel algorithm for ball intersection, one obtains (7) a sequential deterministic algorithm for computing the diameter of a point set in $\\IR\\sp3$ that runs in time $O(n\\ {\\rm log}\\sp3\\ n)$. Using also geometric sampling techniques, (8) we describe an algorithm for computing the arrangement of n segments in the plane in time $O(\\log\\sp2 n)$ and using work $O(n\\ \\log\\ n + k)$ where k is the number of pairwise intersections, also in the EREW PRAM model (sequentially this results in an algorithm that outputs all the intersections in optimal time using O(n) space); and (9) assuming that certain sampling result can be derandomized in polynomial time, we describe a sequential algorithm for computing one face in an arrangement of segments that runs in time $O(n\\alpha\\sp2(n)\\ \\log\\ n)$ where $\\alpha(n)$ is a very slowly growing function.","abstract_html":"This thesis consists of two parts dealing with combinatorial and computational problems in geometry, respectively. In the first part three independent problems are considered: (1) We determine an upper bound $\\lfloor 11n/6\\rfloor$ + 1 for the number of extreme triples of n points in the plane, almost matching a known lower bound $\\lfloor 11n/6\\rfloor$; (2) we determine some bounds for the smallest dimension $d = \\Delta(j,k)$ such that for any j mass distributions in $\\IR\\sp{d}$, there are k hyperplanes so that each orthant contains a fraction 1/2$\\sp{k}$ of each of the masses; it is easily shown that $j(2\\sp{k}-1)/k \\le \\Delta(j,k)\\le j2\\sp{k-1}$; we believe the lower bound is tight, but can only prove it in a few cases (as a tool we prove a Borsuk-Ulam theorem on a product of balls, which is of independent interest); (3) for a collection B of pseudo-disks in the plane, we show the existence of a two-dimensional abstract simplicial complex, $\\chi \\subseteq 2\\sp{B}$, which has some nice topological properties, such that the inclusion-exclusion relation <span class=\"etd-inline-math\">&mu;(\\cup B) = \\Sigma \\sb{&sigma;\\in 2\\sp{B} - \\{\\phi\\}}(-1)\\sp{\\rm card &sigma; -1}&mu;(\\cap&sigma;)</span> holds when $\\chi$ is substituted for 2$\\sp{B}$. In the second part, using geometric sampling techniques, we give algorithms for three similar problems: (4) Computing the intersection of halfspaces in $\\IR\\sp3$; (5) computing the intersection of balls of equal radius in $\\IR\\sp3$; and (6) computing the Voronoi diagram of line segments in $\\IR\\sp2$; in each case we obtain a deterministic parallel algorithm for the EREW PRAM model that runs in time <span class=\"etd-inline-math\">O({\\rm log}\\sp2 n)</span> and uses work <span class=\"etd-inline-math\">O(n {\\rm log} n)</span> for a problem of size n (for ball intersection this is also the first optimal deterministic and sequential algorithm, using the Dobkin-Kirkpatrick decomposition, we can only achieve time <span class=\"etd-inline-math\">O(n {\\rm log}\\sp2 n</span>)). Using the parallel algorithm for ball intersection, one obtains (7) a sequential deterministic algorithm for computing the diameter of a point set in $\\IR\\sp3$ that runs in time <span class=\"etd-inline-math\">O(n {\\rm log}\\sp3 n)</span>. Using also geometric sampling techniques, (8) we describe an algorithm for computing the arrangement of n segments in the plane in time $O(\\log\\sp2 n)$ and using work <span class=\"etd-inline-math\">O(n \\log n + k)</span> where k is the number of pairwise intersections, also in the EREW PRAM model (sequentially this results in an algorithm that outputs all the intersections in optimal time using O(n) space); and (9) assuming that certain sampling result can be derandomized in polynomial time, we describe a sequential algorithm for computing one face in an arrangement of segments that runs in time <span class=\"etd-inline-math\">O(n&alpha;\\sp2(n) \\log n)</span> where <span class=\"etd-inline-math\">&alpha;(n)</span> is a very slowly growing function.","abstract_has_math":true,"creators":["Ramos, Edgar Arturo"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":["Edelsbrunner, Herbert"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:54:10Z","date_published":"2011-05-07T13:54:10Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Computer Science"],"languages":["eng"],"rights":["Copyright 1995 Ramos, Edgar Arturo"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624467","(UMI)AAI9624467"],"render_values":[{"text":"AAI9624467","href":null,"code":true},{"text":"(UMI)AAI9624467","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22868","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Edelsbrunner, Herbert"]},{"key":"dc:creator","label":"Author","values":["Ramos, Edgar Arturo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:54:10Z","10000-01-01","1995"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Computer Science"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1995 Ramos, Edgar Arturo"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624467","(UMI)AAI9624467","http://hdl.handle.net/2142/22868"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis consists of two parts dealing with combinatorial and computational problems in geometry, respectively. In the first part three independent problems are considered: (1) We determine an upper bound $\\lfloor 11n/6\\rfloor$ + 1 for the number of extreme triples of n points in the plane, almost matching a known lower bound $\\lfloor 11n/6\\rfloor$; (2) we determine some bounds for the smallest dimension $d = \\Delta(j,k)$ such that for any j mass distributions in $\\IR\\sp{d}$, there are k hyperplanes so that each orthant contains a fraction 1/2$\\sp{k}$ of each of the masses; it is easily shown that $j(2\\sp{k}-1)/k \\le \\Delta(j,k)\\le j2\\sp{k-1}$; we believe the lower bound is tight, but can only prove it in a few cases (as a tool we prove a Borsuk-Ulam theorem on a product of balls, which is of independent interest); (3) for a collection B of pseudo-disks in the plane, we show the existence of a two-dimensional abstract simplicial complex, $\\chi \\subseteq 2\\sp{B}$, which has some nice topological properties, such that the inclusion-exclusion relation $\\mu(\\cup B) = \\Sigma \\sb{\\sigma\\in 2\\sp{B} - \\{\\phi\\}}(-1)\\sp{\\rm card\\ \\sigma -1}\\mu(\\cap\\sigma)$ holds when $\\chi$ is substituted for 2$\\sp{B}$. In the second part, using geometric sampling techniques, we give algorithms for three similar problems: (4) Computing the intersection of halfspaces in $\\IR\\sp3$; (5) computing the intersection of balls of equal radius in $\\IR\\sp3$; and (6) computing the Voronoi diagram of line segments in $\\IR\\sp2$; in each case we obtain a deterministic parallel algorithm for the EREW PRAM model that runs in time $O({\\rm log}\\sp2\\ n)$ and uses work $O(n\\ {\\rm log}\\ n)$ for a problem of size n (for ball intersection this is also the first optimal deterministic and sequential algorithm, using the Dobkin-Kirkpatrick decomposition, we can only achieve time $O(n\\ {\\rm log}\\sp2\\ n$)). Using the parallel algorithm for ball intersection, one obtains (7) a sequential deterministic algorithm for computing the diameter of a point set in $\\IR\\sp3$ that runs in time $O(n\\ {\\rm log}\\sp3\\ n)$. Using also geometric sampling techniques, (8) we describe an algorithm for computing the arrangement of n segments in the plane in time $O(\\log\\sp2 n)$ and using work $O(n\\ \\log\\ n + k)$ where k is the number of pairwise intersections, also in the EREW PRAM model (sequentially this results in an algorithm that outputs all the intersections in optimal time using O(n) space); and (9) assuming that certain sampling result can be derandomized in polynomial time, we describe a sequential algorithm for computing one face in an arrangement of segments that runs in time $O(n\\alpha\\sp2(n)\\ \\log\\ n)$ where $\\alpha(n)$ is a very slowly growing function.","Made available in DSpace on 2011-05-07T13:54:10Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624467.pdf: 5930366 bytes, checksum: 19645fa8e1d97d1bac114b12d132e564 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:00:34Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:39-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Topics in combinatorial and computational geometry"]}]}],"canonical_facts":{"dc:contributor":["Edelsbrunner, Herbert"],"dc:creator":["Ramos, Edgar Arturo"],"dc:date":["2011-05-07T13:54:10Z","10000-01-01","1995"],"dc:description":["This thesis consists of two parts dealing with combinatorial and computational problems in geometry, respectively. In the first part three independent problems are considered: (1) We determine an upper bound $\\lfloor 11n/6\\rfloor$ + 1 for the number of extreme triples of n points in the plane, almost matching a known lower bound $\\lfloor 11n/6\\rfloor$; (2) we determine some bounds for the smallest dimension $d = \\Delta(j,k)$ such that for any j mass distributions in $\\IR\\sp{d}$, there are k hyperplanes so that each orthant contains a fraction 1/2$\\sp{k}$ of each of the masses; it is easily shown that $j(2\\sp{k}-1)/k \\le \\Delta(j,k)\\le j2\\sp{k-1}$; we believe the lower bound is tight, but can only prove it in a few cases (as a tool we prove a Borsuk-Ulam theorem on a product of balls, which is of independent interest); (3) for a collection B of pseudo-disks in the plane, we show the existence of a two-dimensional abstract simplicial complex, $\\chi \\subseteq 2\\sp{B}$, which has some nice topological properties, such that the inclusion-exclusion relation $\\mu(\\cup B) = \\Sigma \\sb{\\sigma\\in 2\\sp{B} - \\{\\phi\\}}(-1)\\sp{\\rm card\\ \\sigma -1}\\mu(\\cap\\sigma)$ holds when $\\chi$ is substituted for 2$\\sp{B}$. In the second part, using geometric sampling techniques, we give algorithms for three similar problems: (4) Computing the intersection of halfspaces in $\\IR\\sp3$; (5) computing the intersection of balls of equal radius in $\\IR\\sp3$; and (6) computing the Voronoi diagram of line segments in $\\IR\\sp2$; in each case we obtain a deterministic parallel algorithm for the EREW PRAM model that runs in time $O({\\rm log}\\sp2\\ n)$ and uses work $O(n\\ {\\rm log}\\ n)$ for a problem of size n (for ball intersection this is also the first optimal deterministic and sequential algorithm, using the Dobkin-Kirkpatrick decomposition, we can only achieve time $O(n\\ {\\rm log}\\sp2\\ n$)). Using the parallel algorithm for ball intersection, one obtains (7) a sequential deterministic algorithm for computing the diameter of a point set in $\\IR\\sp3$ that runs in time $O(n\\ {\\rm log}\\sp3\\ n)$. Using also geometric sampling techniques, (8) we describe an algorithm for computing the arrangement of n segments in the plane in time $O(\\log\\sp2 n)$ and using work $O(n\\ \\log\\ n + k)$ where k is the number of pairwise intersections, also in the EREW PRAM model (sequentially this results in an algorithm that outputs all the intersections in optimal time using O(n) space); and (9) assuming that certain sampling result can be derandomized in polynomial time, we describe a sequential algorithm for computing one face in an arrangement of segments that runs in time $O(n\\alpha\\sp2(n)\\ \\log\\ n)$ where $\\alpha(n)$ is a very slowly growing function.","Made available in DSpace on 2011-05-07T13:54:10Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624467.pdf: 5930366 bytes, checksum: 19645fa8e1d97d1bac114b12d132e564 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:00:34Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:39-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9624467","(UMI)AAI9624467","http://hdl.handle.net/2142/22868"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Ramos, Edgar Arturo"],"dc:subject":["Computer Science"],"dc:title":["Topics in combinatorial and computational geometry"],"dc:type":["text"],"thesis:degree_discipline":["Computer Science"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:20Z"}