UMass Lowell Computer Science 91.504 Advanced Algorithms

Computational Geometry Prof. Karen Daniels

Spring, 2001

UMass Lowell Computer Science 91.504 Advanced Algorithms

Computational Geometry Prof. Karen Daniels

Spring, 2001

Lecture 2Lecture 2

Chapter 2: Polygon PartitioningChapter 2: Polygon Partitioning

Chapter 3: 2D Convex HullsChapter 3: 2D Convex HullsMonday, 2/12/01Monday, 2/12/01

Chapter 2Chapter 2

Polygon PartitioningPolygon Partitioninguseful for triangulations useful for triangulations

and many other uses!and many other uses!

Monotone PartitioningMonotone Partitioning

A A chainchain is is monotonemonotone with respect to a with respect to a line L if every line orthogonal to L line L if every line orthogonal to L intersects the chain in at most 1 pointintersects the chain in at most 1 point

PP is monotone with respect to a line L if is monotone with respect to a line L if boundary of P can be split into 2 boundary of P can be split into 2 polygonal chains A and B such that each polygonal chains A and B such that each chain is monotone with respect to Lchain is monotone with respect to L

Monotonicity implies sorted order with Monotonicity implies sorted order with respect to Lrespect to L

Monotone polygon can be (greedily) Monotone polygon can be (greedily) triangulated in O(n) time triangulated in O(n) time

TrapezoidalizationTrapezoidalization

Partition into trapezoids Partition into trapezoids Horizontal line through each vertexHorizontal line through each vertex Diagonal with each “supporting” vertex Diagonal with each “supporting” vertex

yields monotone partitionyields monotone partition To trapezoidalize, To trapezoidalize, vertically sweep vertically sweep a linea line LL

presort vertices by presort vertices by yy (O(nlogn) time) (O(nlogn) time) maintain sorted list of edges intersecting Lmaintain sorted list of edges intersecting L

lg n lookup/insert/delete time (e.g. ht-balanced lg n lookup/insert/delete time (e.g. ht-balanced tree)tree)

for each vertexfor each vertex find edge left and right along sweep linefind edge left and right along sweep line

AlgorithmAlgorithm: POLYGON TRIANGULATION: MONOTONE PARTITION: POLYGON TRIANGULATION: MONOTONE PARTITIONSort vertices by y coordinateSort vertices by y coordinatePerform plane sweep to construct trapezoidalizationPerform plane sweep to construct trapezoidalizationPartition into monotone polygons by connecting from interior cuspsPartition into monotone polygons by connecting from interior cuspsTriangulation each monotone polygon in O(n) timeTriangulation each monotone polygon in O(n) time O(n lg n)O(n lg n)

Partition into Monotone MountainsPartition into Monotone Mountains

One monotone chain is a single One monotone chain is a single segmentsegment

Every strictly convex vertex is Every strictly convex vertex is an ear tip an ear tip (except maybe base (except maybe base endpoints)endpoints)

AlgorithmAlgorithm: TRIANGULATION of MONOTONE MOUNTAIN: TRIANGULATION of MONOTONE MOUNTAINIdentify base edgeIdentify base edgeInitialize internal angles at each nonbase vertexInitialize internal angles at each nonbase vertexLink nonbase strictly convex vertices into a listLink nonbase strictly convex vertices into a listwhile list nonempty dowhile list nonempty do

For convex vertex b, remove triangle abcFor convex vertex b, remove triangle abcOutput diagonal acOutput diagonal acUpdate angles and listUpdate angles and list O(n)O(n)

Linear-Time TriangulationLinear-Time Triangulation

1911 O(n2) Lennes1978 O(n log n) Garey et al.1983 O(n log r), r reflex Hertel & Melhorn1984 O(n log s), s sinuosity Chazelle & Incerpi1986 O(n log log n) Tarjan & Van Wyk1988 O(n + nt0), t0 int. triangles Toussaint1989 O(n log* n), randomized Clarkson, Tarjan & Van Wyk1990 O(n log* n) bounded integer coordinates Kirkpatrick, Klawe, Tarjan1990 O(n) Chazelle1991 O(n log*n), randomized Seidel

YearYear ComplexityComplexity AuthorsAuthors

Linear-Time TriangulationLinear-Time Triangulation

Chazelle’s Algorithm (High-Level Sketch)Chazelle’s Algorithm (High-Level Sketch) Computes visibility mapComputes visibility map

horizontal chords left and right from each vertexhorizontal chords left and right from each vertex

Algorithm is like MergeSort (divide-and-conquer)Algorithm is like MergeSort (divide-and-conquer) Partition polygon of n vertices into n/2 vertex chainsPartition polygon of n vertices into n/2 vertex chains Merge visibility maps of subchains to get one for chainMerge visibility maps of subchains to get one for chain

Improve this by dividing process into 2 phases:Improve this by dividing process into 2 phases:1) Coarse approximations of visibility maps for linear-time 1) Coarse approximations of visibility maps for linear-time

mergemerge

2) Refine coarse map into detailed map in linear time2) Refine coarse map into detailed map in linear time

Seidel’s Randomized TriangulationSeidel’s Randomized Triangulation

Simple, practical algorithmSimple, practical algorithm Randomized: Coin-flip for some decisionsRandomized: Coin-flip for some decisions Build trapezoidalization quicklyBuild trapezoidalization quickly

O(log n) expected cost for locating point in query O(log n) expected cost for locating point in query structurestructure

Coin-flip to decide which segment to add nextCoin-flip to decide which segment to add next

Trapezoidalize -> Monotone Mountain -> Triangulate

Convex PartitioningConvex Partitioning

Competing Goals:Competing Goals: minimize number of convex piecesminimize number of convex pieces minimize partitioning timeminimize partitioning time

To add points or not add points?To add points or not add points?

Theorem (Chazelle): Let be the fewest number of convex pieces into which a polygon may be partitioned. For a polygon of r reflex vertices:

112

rr

Chapter 3Chapter 3

2D Convex Hulls2D Convex Hulls

Convexity & Convex HullsConvexity & Convex Hulls

A convex combination of points A convex combination of points xx11, ..., x, ..., xkk is a sum of the form is a sum of the form

11xx11+...+ +...+ kkxxkk where where

Convex hull of a set of points is Convex hull of a set of points is the set of all convex the set of all convex combinations of points in the combinations of points in the set.set.

nonconvex polygonnonconvex polygon

convex hull of a point setconvex hull of a point set

10 1 ki andi

Naive Algorithms for Extreme PointsNaive Algorithms for Extreme Points

AlgorithmAlgorithm: INTERIOR POINTS: INTERIOR POINTSfor each i dofor each i do for each j = i dofor each j = i do for each k = j = i dofor each k = j = i do

for each L = k = j = i do for each L = k = j = i do if pif pL L in triangle(pin triangle(pii, p, pjj, p, pkk))

then pthen pLL is nonextreme is nonextreme O(nO(n44))

AlgorithmAlgorithm: EXTREME EDGES: EXTREME EDGESfor each i dofor each i do for each j = i dofor each j = i do for each k = j = i dofor each k = j = i do

if pif pk k is not left or on (pis not left or on (pii, p, pjj))

then (pthen (pi i , p, pjj) is not extreme) is not extreme O(nO(n33))

Gift WrappingGift Wrapping

Use one extreme edge as an Use one extreme edge as an anchor for finding the nextanchor for finding the next

O(nO(n22))

AlgorithmAlgorithm: GIFT WRAPPING: GIFT WRAPPING

ii00 index of the lowest point index of the lowest point

i ii i00repeatrepeat for each j = ifor each j = i Compute counterclockwise angle Compute counterclockwise angle from previous hull edge from previous hull edge k index of point with smallest k index of point with smallest Output (pOutput (pi i , p, pkk) as a hull edge) as a hull edge i ki kuntil i = iuntil i = i00

QuickHullQuickHull

Concentrate on points close to Concentrate on points close to hull boundaryhull boundary

Named for similarity to Named for similarity to Quicksort Quicksort

aa

bb

O(nO(n22))

AlgorithmAlgorithm: QUICK HULL: QUICK HULL

function QuickHull(a,b,S)function QuickHull(a,b,S)

if S = 0 return()if S = 0 return() elseelse c index of point with max distance from abc index of point with max distance from ab A points strictly right of (a,c)A points strictly right of (a,c) B points strictly right of (c,b)B points strictly right of (c,b) return QuickHull(a,c,A) + (c) + QuickHull(c,b,B)return QuickHull(a,c,A) + (c) + QuickHull(c,b,B)

Graham’s AlgorithmGraham’s Algorithm

Points sorted angularly provide Points sorted angularly provide “star-shaped” starting point“star-shaped” starting point

Prevent “dents” as you go via Prevent “dents” as you go via convexity testingconvexity testing

O(nlgn)O(nlgn)

AlgorithmAlgorithm: GRAHAM SCAN, Version B: GRAHAM SCAN, Version B

Find rightmost lowest point; label it pFind rightmost lowest point; label it p00..

Sort all other points angularly about pSort all other points angularly about p00.. In case of tie, delete point(s) closer to pIn case of tie, delete point(s) closer to p00..Stack S (pStack S (p11, p, p00) = (p) = (ptt, p, pt-1t-1); t indexes top); t indexes topi 2i 2while i < n dowhile i < n do

if pif pii is strictly left of p is strictly left of pt-1t-1pptt

then Push(pthen Push(pii, S) and set i i +1, S) and set i i +1else Pop(S)else Pop(S)

Lower Bound of O(nlgn)Lower Bound of O(nlgn)

Worst-case time to find convex hull of n points Worst-case time to find convex hull of n points in algebraic decision tree model is in in algebraic decision tree model is in (nlgn)(nlgn)

Proof uses sorting reduction:Proof uses sorting reduction: Given unsorted list of n numbers: (xGiven unsorted list of n numbers: (x11,x,x2 2 ,…, x,…, xnn)) Form unsorted set of points: (xForm unsorted set of points: (xii, x, xii

22) for each x) for each xii

Convex hull of points produces Convex hull of points produces sorted listsorted list!! Parabola: every point is on convex hullParabola: every point is on convex hull

Reduction is O(n) (which is o(nlgn))Reduction is O(n) (which is o(nlgn)) Finding convex hull of n points is therefore at least Finding convex hull of n points is therefore at least

as hard as sorting n points, so worst-case time is in as hard as sorting n points, so worst-case time is in (nlgn)(nlgn)

Parabola for sorting 2,1,3Parabola for sorting 2,1,3

Incremental AlgorithmIncremental Algorithm

Add points, one at a timeAdd points, one at a time update hull for each new pointupdate hull for each new point

Key step becomes adding a single Key step becomes adding a single point to an existing hull.point to an existing hull.

Idea is extended to 3D in Chapter Idea is extended to 3D in Chapter 4.4.

O(nO(n22))

AlgorithmAlgorithm: INCREMENTAL ALGORITHM: INCREMENTAL ALGORITHM

Let HLet H22 ConvexHull{p ConvexHull{p0 0 , p, p1 1 , p, p2 2 }}

for k 3 to n - 1 dofor k 3 to n - 1 do HHkk ConvexHull{ H ConvexHull{ Hk-1 k-1 U pU pk k }}

can be improved to O(nlgn)can be improved to O(nlgn)

Divide-and-ConquerDivide-and-Conquer

Divide-and-Conquer in a geometric settingDivide-and-Conquer in a geometric setting O(n) merge step is the challengeO(n) merge step is the challenge

Find upper and lower tangentsFind upper and lower tangents Lower tangent: find rightmost pt of A & Lower tangent: find rightmost pt of A &

leftmost pt of B; then “walk it downwards”leftmost pt of B; then “walk it downwards” Idea is extended to 3D in Chapter 4.Idea is extended to 3D in Chapter 4.

AlgorithmAlgorithm: DIVIDE-and-CONQUER: DIVIDE-and-CONQUER

Sort points by x coordinateSort points by x coordinate

Divide points into 2 sets A and B:Divide points into 2 sets A and B:

A contains left n/2 pointsA contains left n/2 points

B contains right n/2 pointsB contains right n/2 points

Compute ConvexHull(A) and ConvexHull(B) recursivelyCompute ConvexHull(A) and ConvexHull(B) recursively

Merge ConvexHull(A) and ConvexHull(B) Merge ConvexHull(A) and ConvexHull(B) O(nlgn)O(nlgn)

AABB

Homework Homework

11 Fri, 2/9 Fri, 2/9 Wed, 2/14 Wed, 2/14 Chapter 1 (O’Rourke)Chapter 1 (O’Rourke) problems 1 & 2problems 1 & 2

Extra credit may be turned in any time during Extra credit may be turned in any time during week of 2/12week of 2/12

22 Mon, 2/12 Mon, 2/12 Wed, 2/21 Wed, 2/21 Chapters 2,3 (O’Rourke)Chapters 2,3 (O’Rourke)problems 1 & 2problems 1 & 2

Extra credit may be turned in any time during Extra credit may be turned in any time during week of 2/21week of 2/21

HW#HW# AssignedAssigned DueDue ContentContent

Machine AccountsMachine Accounts

Each student has an account on my machine: Each student has an account on my machine: minkowski.cs.uml.eduminkowski.cs.uml.edu

Username is the same as your username on CS Username is the same as your username on CS Password is your initials followed by the last 5 digits on Password is your initials followed by the last 5 digits on

the bottom right hand corner of the back of your student the bottom right hand corner of the back of your student id card (1 exception)id card (1 exception)

To remotely log in, use a secure shell (e.g. ssh)To remotely log in, use a secure shell (e.g. ssh) To transfer files, use a secure FTP (e.g. sftpc)To transfer files, use a secure FTP (e.g. sftpc)

Saturn has an sftpc clientSaturn has an sftpc client Code to use with assignments will be there.Code to use with assignments will be there.

Preparing for Homework 2Preparing for Homework 2

Graham Scan 2D Convex Hull CodeGraham Scan 2D Convex Hull Code Background on research problem that Background on research problem that

motivates problem 2 motivates problem 2