Orthogonal line segment intersection

Transcription

1 Computtionl Geometry [csci 3250] Line segment intersection The prolem (wht) Computtionl Geometry [csci 3250] Orthogonl line segment intersection Applictions (why) Algorithms (how) A specil cse: Orthogonl line segments Generl cse nd Bentley-Otmn line sweep lgorithm Lur Tom Bowdoin College Lur Tom Bowdoin College Line segment intersection Line segment intersection Line segment intersection Prolem: Given set of line segments in 2D, find ll their pirwise intersections. Prolem: Given set of line segments in 2D, find ll their pirwise intersections. Prolem: Given set of line segments in 2D, find ll their pirwise intersections

2 Applictions Applictions Applictions Grphics: rendering => hidden surfces ==> intersections Motion plnning nd collision detection in utonomous systems/rootics Geogrphicl dt: River networks, rod networks, rilwys,.. R Applictions Applictions Applictions Geogrphicl dt: River networks, rod networks, rilwys,.. Mp overly in GIS Mp overly in GIS from: from: 12

3 Nive Nottion n: size of the input (numer of segments) k: size of output (numer of intersections) A specil cse: Prolem: Given set of n line segments in 2D, find ll their pirwise intersections. Algorithms Exercises: Give upper nd lower ounds for k, drw exmples tht chieve these ounds. Give strightforwrd lgorithm tht computes ll intersections nd nlyze its running time. Give scenrios when this lgorithm is efficient/inefficient. Wht is your intuition of n upper ound for this prolem? (how fst would you hope to e le to solve it?) Exercises Come up with strightforwrd lgorithm nd nlyze its time Improved lgorithm? Blnced Binry Serch Trees - review - Binry Serch Trees (BST) Opertions insert delete serch successor, predecessor trversls (in order,..) min, mx Blnced Binry Serch Trees (BBST) Binry serch trees + invrints tht constrin the tree to e lnced (nd thus hve logrithmic height) These invrints hve to e mintined when inserting nd deleting (so we cn think of the tree s self-lncing) BBST vrints red-lck trees AVL trees B-trees (,) trees

4 Exmple: Red-Blck trees Exmple: Red-Blck trees Exmple: Red-Blck trees Binry serch tree, nd Ech node is Red or Blck The children of Red node must e Blck The numer of Blck nodes on ny pth from the root to ny node tht does not hve two children must e the sme Theorem: A Red-Blck tree of n nodes hs height Thet( lg n). Theorem: After n insertion or deletion, the RB tree invrints cn e mintined in dditionl O(lg n) time. This is done y performing rottions nd recoloring nodes on the pth from the inserted/deleted node to the root. Note: esier to conceptulize the tree s contining explicit NULL leves, ll Blck the numer of Blck nodes on ny root-to-lef pth must e the sme Binry Serch Trees Opertions insert delete serch successor, predecessor trversls (in order,..) min, mx rnge serch (1D) 1D Rnge Serching Given set of vlues P = {x1, x2, x3, xn } Pre-process it in order to nswer rngeserch(,): return ll elements in P in intervl (,) 1D Rnge Serching Given set of vlues P = {x1, x2, x3, xn } Pre-process it in order to nswer rngeserch(,): return ll elements in P in intervl (,)

5 1D Rnge Serching Given set of vlues P = {x1, x2, x3, xn } Pre-process it in order to nswer rngeserch(,): return ll elements in P in intervl (,) 1D Rnge Serching Given set of vlues P = {x1, x2, x3, xn } Pre-process it in order to nswer rngeserch(,): return ll elements in P in intervl (,) 1D Rnge Serching Given set of vlues P = {x1, x2, x3, xn } Pre-process it in order to nswer rngeserch(,): return ll elements in P in intervl (,) If P is sttic If P is sttic If P is sttic Ides? Pre-precess: sort If P is dynmic: Rnge serch: inry serch, O( lg n + k) per query use BBST D rnge serching with Binry Serch Trees 1D rnge serching with Binry Serch Trees 1D rnge serching with Binry Serch Trees Exmple: rnge_serch(21, 53): return 21, 34, 35, 46, 51, 52 Exmple: rnge_serch(21, 53): return 21, 34, 35, 46, 51, 52 Exmple: rnge_serch(21, 53): return 21, 34, 35, 46, 51,

6 1D rnge serching with Binry Serch Trees 1D Rnge Serching with Red-Blck Trees 1D rnge serching with Binry Serch Trees Exmple: rnge_serch(21, 53): return 21, 34, 35, 46, 51, 52 Exmple: rnge_serch(10, 16): return 11, 13, 15 Rnge serch (,): return ll elements in this intervl D rnge serching with Binry Serch Trees Rnge serch (,): return ll elements in this intervl Cn e nswered in O( lg n+k), where k = O(n) is the size of output

7 solve the prolem ehind the line solve the prolem ehind the line //the events //our events //our events Sort X nd trverse the events in order Sort X nd trverse the events in order xstrt xend x solve the prolem ehind the line solve the prolem ehind the line solve the prolem ehind the line //our events //our events //our events Sort X nd trverse the events in order Sort X nd trverse the events in order Sort X nd trverse the events in order

8 solve the prolem ehind the line Events Line sweep technique Trverse events in order nd mintin n Active Structure (AS) AS contins ojects tht re ctive (strted ut not ended) in other words they re intersected y the present sweep line t certin events, insert in AS Events eginning of horizontl segment end of horizontl segment Events eginning of horizontl segment end of horizontl segment t certin events, delete from AS t other events, query AS verticl segment verticl segment

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10 Line sweep Pick n exmple nd simulte the lgorithm How do you implement the AS? Anlysis? Frequently used technique Line cn e horizontl or verticl or rdil or. Trverse events in order nd mintin n Active Structure (AS) AS mintins ojects tht re ctive (strted ut not ended) in other words they re intersected y the present sweep line t certin events, insert in AS t certin events, delete from AS t other events, query AS 58 59