Sweep Line Algorithm · CH08-320201: Algorithms and Data Structures 550 Visualization and Computer...

CH08-320201: Algorithms and Data Structures 550

Visualization and Computer Graphics LabJacobs University

Sweep Line Algorithm

• Event C: Intersection point– Report the point.– Swap the two intersecting line segments

in the sweep line status data structure.– For the new upper segment: test it

against its predecessor for an intersection.

– Insert intersection point (if found) into event queue.

– Similar for new lower segment (with successor).

• Complexity: – k such events– O(lg n) each – O(k lg n) total.



Example

s4

s2

s1s0

s3

e1

a1

b1

a0

b0

a2

b2

a4

a3

b3

b4 Sweep Line status:

s0, s1, s2, s3

Event Queue:

a4, b1, b2, b0, b3, b4



Example

s4

s2

s1s0

s3Actions:

Insert s4 to SLS

Test s4-s3 and s4-s2.

Add e1 to EQ

Sweep Line status:

s0, s1, s2, s4, s3

Event Queue:

b1, e1, b2, b0, b3, b4

e1

a1

b1

a0

b0

a2

b2

a4

a3

b3

b4



Example

s4

s2

s1s0

s3

e2

a1

b1

a0

b0

a2

b2

a4

a3

b3

b4

e1

Actions:

Delete s1 from SLS

Test s0-s2.

Add e2 to EQ

Sweep Line status:

s0, s2, s4, s3

Event Queue:

e1, e2, b2, b0, b3, b4



Example

s4

s2

s1s0

s3

a1

b1

a0

b0

a2

b2

a4

a3

b3

b4

e1

e2

Actions:

Swap s3 and s4 in SLS

Test s3-s2.

Sweep Line status:

s0, s2, s3, s4

Event Queue:

e2, b2, b0, b3, b4



Complexity Analysis

• Total time complexity: – O((n+k) lgn). – Best case: k = O(1), then this is much better than the naive

algorithm.– Even for k ≈ n, this is still the case.– Worst case: k ≈ n 2, then this is slightly worse than the naive

algorithm.• Total space complexity:

– Event queue: min-heap• O(n+k)

– Sweep line status: red-black tree• O(n).

– Total: O(n+k).



6.2 Convex Hull



Convex HullConvex Hull• The convex hull of a set of points is the smallest convex polygon

containing the set of points.• A convex polygon is a non-self-intersecting polygon whose internal

angles are all convex (i.e., less than 180 degrees)• In a convex polygon, a segment joining any two vertices lies entirely

inside the polygon.

convex nonconvex

Thursday March 12, 2015 557

segment not contained!



Convex HullConvex Hull• Think of a rubber band snapping around the points• Remember to think of special cases

– Collinearity: A point that lies on a segment of the convex is not included in the convex hull




ApplicationsApplications• Motion planning

– Find an optimal route that avoids obstacles for a robot• Bounding Box

– Obtain a closer bounding box in computer graphics• Pattern Matching

– Compare two objects using their convex hulls

obstacle

end




Finding a Convex HullFinding a Convex Hull

One algorithm for determining a convex polygon is one which, when following the points in counterclockwise order, always produces left-turns




Calculating OrientationCalculating Orientation• The orientation of three points (a, b, c) in the

plane is clockwise, counterclockwise, or collinear– Clockwise (CW): right turn– Counterclockwise (CCW): left turn– Collinear (COLL): no turn

• The orientation of three points is characterized by the sign of the determinant ∆(a, b, c)

a

b

c

a

c

b

c

ba

CW

CCW

COLL


111

),,(

cc

bb

aa

yxyxyx

cba =Δ

function isLeftTurn(a, b, c):return (b.x - a.x)*(c.y - a.y)–

(b.y - a.y)*(c.x - a.x) > 0



ExamplesExamples

a (0,0)

b (0.5,1)

c (1, 0.5)

a (0,0)

c (0.5,1)

b (1, 0.5)

c(2,2)b(1,1)

a(0,0)

CW

CCW

COLL


Using the isLeftTurn() method:

(0.5-0) * (0.5-0) - (1-0) * (1-0) = -0.75 (CW)

(1-0) * (1-0) - (0.5-0) * (0.5-0) = 0.75 (CCW)

(1-0) * (2-0) - (1-0) * (2-0) = 0 (COLL)



Orientations and convex hullOrientations and convex hull• Let a, b, c be three consecutive vertices of a polygon, in

counterclockwise order– b’ is not included in the hull if a’, b,’ c’ is non-convex,

i.e. orientation(a’, b’, c’) = CW or COLL– b is included in the hull if a, b, c is convex,

i.e. orientation(a, b, c) = CCW, and all other non-hull points have been removed

a’b’

c’

b

c

a




Graham Scan AlgorithmGraham Scan Algorithm

• Find the anchor point (the point with the smallest y value)• Sort points in CCW order around the anchor• We can sort by increasing angle of the line segments from the

anchor to the points with the x-axis


Anchor

2

3

4

56

7

8

1



Graham Scan AlgorithmGraham Scan Algorithm• The polygon is traversed in sorted order• A sequence H of vertices in the hull is maintained.• For each point a, add a to H.• While the last turn is a right turn, remove the second-to-last

point from H– In the image below, p,q,r forms a right turn.– Thus q is removed from H. – Similarly, o,p,r forms a right turn.– Thus p is removed from H.

p

q

r

H

op

r

H

n

or

H




Graham ScanGraham Scan

function graham_scan (pts):// Input: Set of points pts// Output: Hull of pointsfind anchor pointsort other points in CCW order around anchorhull = []for p in pts:

add p to hullwhile last turn is a “right turn”

remove 2nd to last pointadd anchor to hullreturn hull




AnalysisAnalysisfunction graham_scan(pts):

// Input: Set of points pts// Output: Hull of pointsfind anchor point // O(n)sort other points in CCW order around anchor // O(n lg n)hull = [] // O(1)for p in pts: // O(n)

add p to hull // O(1)while last turn is a “right turn” // O(1)

// Each point is looked at, at most, twice: // 1x left-turn, 1x right-turn.

remove 2nd to last point // O(1)add anchor to hull // O(1)return hull

Overall run time: O(n lg n)Space complexity: O(n) [often a stack is used]




Incremental AlgorithmIncremental Algorithm

• What if we already have a convex hull, and we just want to add one point q?

• Get the angle from the anchor to q and find points p and r, the hull points on either side of q

• If p,q,r forms a left turn, add q to the hull• Check if adding q creates a concave shape

– If you have right turns on either side of q, remove vertices until the shape becomes convex

– This is done in the same way as the static Graham Scan




Incremental AlgorithmIncremental Algorithm569Thursday March 12, 2015

H

Original Hull:q

H

Want to add point q:

p

r

q

H

q

H

Find p and r:

p

r

p, q, r forms a left turn, so add q:

r

q

H

on

o, p, q forms a right turn, so remove p

on

n, o, q forms a right turn, so remove o

r

q

H

n

s s s



Incremental AlgorithmIncremental AlgorithmNow we look at the other side:

r

q

Hs r

q

H

Since q, r, s is a left turn, we’re done!

s

q

H

n

Since m, n, q is a left turn, we’re done with that side.

• Remember that you can have right turns on either or both sides, so make sure to check in both directions and remove concave points!


m



AnalysisAnalysis

• Let n be the current size of the convex hull, stored in a binary search tree around the anchor for efficiency

• To check if a point q should be in the hull, we insert it into the tree and get its neighbors (p and r on the prior slides):O(lg n)

• We then traverse the ring, possibly deleting d points from the convex hull:O((1 + d) lg n)

• Therefore, incremental insertion is O(d log n) where d is the number of points removed by the insertion




A Divide-and-Conquer Approach

• First, sort all points by their x coordinate.• Then divide and conquer:

– Find the convex hull of the left half of points.– Find the convex hull of the right half of points.– Merge the two hulls into one.



A Divide-and-Conquer Approach



Merging Hulls

Idea:• Find the two lines that are upper and lower tangent to

the two hulls.• Then, remove points that are cut off.



Finding upper tangent line• Start with the right-most point of the left hull and

the left-most point of the right hull.• While the line is not an upper tangent to both hulls

– While the line is not an upper tangent to the left hull• Move to next point of left hull counter-clockwise.

– While the line is not an upper tangent to the right hull• Move to next point of right hull clockwise.



Finding lower tangent line• Start with the right-most point of the left hull and

the left-most point of the right hull.• While the line is not an lower tangent to both hulls

– While the line is not an lower tangent to the left hull• Move to next point of left hull clockwise.

– While the line is not an lower tangent to the right hull• Move to next point of right hull counter-clockwise.



Checking Tangentness• How can we check whether a line is an upper or lower

tangent?• Let‘s check upper tangentness for left hull:

– pr,pl,pl-ccw should be a left turn• Other cases are checked analogously



Lower Tangent Example• Initially, T=(4, 7) is only a lower

tangent for A. • The A loop does not execute,

but the B loop increments b to 11.• But now T=(4, 11) is no longer a

lower tangent for A, so the A loop decrements a to 0.

• Again, T=(0, 11) is not a lower tangent for B, so B is incremented to 12.

• T=(0, 12) is a lower tangent for both A and B, so T is returned.



Analysis • Preprocessing: Sort the points by x-coordinate.

– O(n lg n)• Let T(n) be the time complexity for n sorted points.• Divide: Split the set of points into two subsets of

(almost) same size.– O(1)

• Conquer: Recursive calls of algorithm for two subproblems of about half the size.– 2 T(n/2)

• Merge: Combine the two hulls by finding upper and lower tangent and removing points in between.– O(n)



Analysis

• Recurrence:T(n) = 2 T(n/2) + c n

• Solution (Case 2 of Master theorem)T(n) = O(n lg n)

• Altogether, pre-processing and divide-and-conquer approach lead to O(n lg n).



7. Outro



Algorithmic concepts

• Divide & Conquer• Dynamic Programming• Greedy Approaches



Efficient algorithms• Sorting• Searching• Power of a number• Fibonacci numbers• Matrix multiplication• VLSI layout• Longest Common Subsequence• Activity Selection problem• Knapsack problem• BFS and DFS• Minimum Spanning Trees• Shortest Paths• Maximum Flow• Line-segment Intersections• Convex Hull• Voronoi Diagrams



Runtime analysis

• Upper bounds• Lower bounds• Tight bounds• Solving recurrences



Data structures

• Array• Stack• Queue• Linked List• Trees• BST• Red-black Trees• Heap• Hash Tables• Graphs



Memory analysis

• Also asymptotic bounds• Additional memory• In-situ algorithms



The End

Sweep Line Algorithm · CH08-320201: Algorithms and Data Structures 550 Visualization and Computer...

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