{"id":15118,"date":"2022-06-18T19:35:13","date_gmt":"2022-06-18T23:35:13","guid":{"rendered":"https:\/\/carleton.ca\/scs\/?page_id=15118"},"modified":"2026-06-08T14:51:43","modified_gmt":"2026-06-08T18:51:43","slug":"tr-239-killing-two-birds-with-one-stone","status":"publish","type":"page","link":"https:\/\/carleton.ca\/scs\/research\/scs-technical-reports\/technical-reports-1994\/tr-239-killing-two-birds-with-one-stone\/","title":{"rendered":"TR-239: Killing Two Birds with One Stone"},"content":{"rendered":"\n<section class=\"w-screen px-6 cu-section cu-section--white ml-offset-center md:px-8 lg:px-14\">\n    <div class=\"space-y-6 cu-max-w-child-5xl  md:space-y-10 cu-prose-first-last\">\n\n            <div class=\"cu-textmedia flex flex-col lg:flex-row mx-auto gap-6 md:gap-10 my-6 md:my-12 first:mt-0 max-w-5xl\">\n        <div class=\"justify-start cu-textmedia-content cu-prose-first-last\" style=\"flex: 0 0 100%;\">\n            <header class=\"font-light prose-xl cu-pageheader md:prose-2xl cu-component-updated cu-prose-first-last\">\n                                    <h1 class=\"cu-prose-first-last font-semibold !mt-2 mb-4 md:mb-6 relative after:absolute after:h-px after:bottom-0 after:bg-cu-red after:left-px text-3xl md:text-4xl lg:text-5xl lg:leading-[3.5rem] pb-5 after:w-10 text-cu-black-700 not-prose\">\n                        TR-239: Killing Two Birds with One Stone\n                    <\/h1>\n                \n                                \n                            <\/header>\n\n                    <\/div>\n\n            <\/div>\n\n    <\/div>\n<\/section>\n\n\n\n<p>Carleton University<br><a href=\"https:\/\/carleton.ca\/scs\/research\/scs-technical-reports\/technical-reports-1994\/\">Technical Report<\/a>&nbsp;<strong>TR-239<\/strong><br>April 1994<\/p>\n\n\n\n<h2 id=\"killing-two-birds-with-one-stone\" class=\"wp-block-heading\">Killing Two Birds with One Stone<\/h2>\n\n\n\n<p>Evangelos Kranakis, Danny Krizanc, Anil Maheshwari, J.R. Sack, Jorge Urrutia<\/p>\n\n\n\n<h3 id=\"abstract\" class=\"wp-block-heading\">Abstract<\/h3>\n\n\n\n<p>We consider geometric and graph-theoretic applications of the well-known paradigm \u201ckilling two birds with one stone\u201d. In the plane, this gives rise to a stage graph as follows: vertices are the points, and { u, v} is an edge if and only if the (infinite, straight) line segment joining u to v intersects the stage. We give algorithms for \u201coptimal\u201d ray shootings of a collection of points in the plane from a given stage, including a characterization of stage graphs in time O(n2 ), and show how to recognize all dominances in O(log n) time using 0( n + k \/ log n) processors on the EREW PRAM, where k is the total number of dominances. Similar problems occur when we have a fixed number k of stages on the plane. In this case, { u, v} is an edge if and only if the (straight) line segment uv intersects one of the k stages. We consider the problem of constructing stage representations of arbitrary graphs and give upper and lower bounds on the number of stages thus required. We also study the 3-dimensional case. This gives rise to 3 ray shooting graphs; these are formed from a set of points in IR.3 and a \u201cstage\u201d which is a compact plane convex set. 3 ray shooting graphs are shown to be comparability graphs, but the converse is shown not to be true. We also provide a characterization of such graphs in terms of<\/p>\n\n\n\n<p><a href=\"https:\/\/carleton.ca\/scs\/wp-content\/uploads\/sites\/260\/TR-239.pdf\">TR-239.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Carleton UniversityTechnical Report&nbsp;TR-239April 1994 Killing Two Birds with One Stone Evangelos Kranakis, Danny Krizanc, Anil Maheshwari, J.R. Sack, Jorge Urrutia Abstract We consider geometric and graph-theoretic applications of the well-known paradigm \u201ckilling two birds with one stone\u201d. In the plane, this gives rise to a stage graph as follows: vertices are the points, and { [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":11914,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"_cu_dining_location_slug":"","footnotes":"","_links_to":"","_links_to_target":""},"cu_page_type":[],"class_list":["post-15118","page","type-page","status-publish","hentry"],"acf":{"cu_post_thumbnail":""},"_links":{"self":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/15118","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/comments?post=15118"}],"version-history":[{"count":3,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/15118\/revisions"}],"predecessor-version":[{"id":24440,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/15118\/revisions\/24440"}],"up":[{"embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/pages\/11914"}],"wp:attachment":[{"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/media?parent=15118"}],"wp:term":[{"taxonomy":"cu_page_type","embeddable":true,"href":"https:\/\/carleton.ca\/scs\/wp-json\/wp\/v2\/cu_page_type?post=15118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}