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.
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CH08-320201: Algorithms and Data Structures 550 Example
a3 b4 Sweep Line status: s3 e1 s4 s0, s1, s2, s3 Event Queue: a2 a4 b3 a4, b1, b2, b0, b3, b4 s2 b1 b0 s1 b2 s0 a1 a0
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CH08-320201: Algorithms and Data Structures 551 Example
a3 b4 Actions: s3 Insert s4 to SLS e1 s4 Test s4-s3 and s4-s2. a2 a4 b3 Add e1 to EQ s2 b1 Sweep Line status: b0 s1 b2 s0, s1, s2, s4, s3 s0 a1 a0 Event Queue: b1, e1, b2, b0, b3, b4
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CH08-320201: Algorithms and Data Structures 552 Example
a3 b4 Actions: s3 e1 s4 Delete s1 from SLS Test s0-s2. a2 a4 b3 Add e2 to EQ s2 Sweep Line status: e2 b0 b1 s1 b2 s0, s2, s4, s3 s0 a1 a0 Event Queue: e1, e2, b2, b0, b3, b4
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CH08-320201: Algorithms and Data Structures 553 Example
a3 b4 Actions: s3 e1 s4 Swap s3 and s4 in SLS Test s3-s2. a2 a4 b3 Sweep Line status: s2 s0, s2, s3, s4 e2 b0 b1 s1 b2 Event Queue: s0 a1 a0 e2, b2, b0, b3, b4
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CH08-320201: Algorithms and Data Structures 554 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).
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CH08-320201: Algorithms and Data Structures 555 6.2 Convex Hull
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CH08-320201: Algorithms and Data Structures 556 Thursday March 12, 2015 557 ConvexConvex HullHull
• 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.
segment not contained!
convex nonconvex
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CH08-320201: Algorithms and Data Structures 557 Thursday March 12, 2015 558 ConvexConvex HullHull
• 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
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CH08-320201: Algorithms and Data Structures 558 Thursday March 12, 2015 559 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
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CH08-320201: Algorithms and Data Structures 559 Thursday March 12, 2015 560 FindingFinding aa ConvexConvex HullHull
One algorithm for determining a convex polygon is one which, when following the points in counterclockwise order, always produces left-turns
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CH08-320201: Algorithms and Data Structures 560 Thursday March 12, 2015 561 CalculatingCalculating OrientationOrientation
• The orientation of three points (a, b, c) in the b plane is clockwise, counterclockwise, or collinear c CW – Clockwise (CW): right turn – Counterclockwise (CCW): left turn a – Collinear (COLL): no turn c • The orientation of three points is characterized by the sign of the determinant ∆(a, b, c) b CCW xa ya 1 a
Δ(a,b,c) = xb yb 1 c b xc yc 1 COLL a function isLeftTurn(a, b, c): return (b.x - a.x)*(c.y - a.y)– (b.y - a.y)*(c.x - a.x) > 0 Jacobs University Visualization and Computer Graphics Lab
CH08-320201: Algorithms and Data Structures 561 Thursday March 12, 2015 562 ExamplesExamples
Using the isLeftTurn() method: b (0.5,1)
(0.5-0) * (0.5-0) - (1-0) * (1-0) c (1, 0.5) CW = -0.75 (CW) a (0,0)
c (0.5,1) (1-0) * (1-0) - (0.5-0) * (0.5-0) b (1, 0.5) CCW = 0.75 (CCW) a (0,0) c(2,2) (1-0) * (2-0) - (1-0) * (2-0) b(1,1) = 0 (COLL) a(0,0) COLL
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CH08-320201: Algorithms and Data Structures 562 Thursday March 12, 2015 563 OrientationsOrientations andand convexconvex hullhull
•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
c’ c
b’ a’ b
a
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CH08-320201: Algorithms and Data Structures 563 Thursday March 12, 2015 564 GrahamGraham ScanScan AlgorithmAlgorithm
• 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
7 4 5 6 3
8 2 Anchor 1
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CH08-320201: Algorithms and Data Structures 564 Thursday March 12, 2015 565 GrahamGraham ScanScan AlgorithmAlgorithm
• 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. r r o r o p p n
q H H H
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CH08-320201: Algorithms and Data Structures 565 Thursday March 12, 2015 566 GrahamGraham ScanScan
function graham_scan (pts): // Input: Set of points pts // Output: Hull of points find anchor point sort other points in CCW order around anchor hull = [] for p in pts: add p to hull while last turn is a “right turn” remove 2nd to last point add anchor to hull return hull
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CH08-320201: Algorithms and Data Structures 566 Thursday March 12, 2015 567 AnalysisAnalysis
function graham_scan(pts): // Input: Set of points pts // Output: Hull of points find 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] Jacobs University Visualization and Computer Graphics Lab
CH08-320201: Algorithms and Data Structures 567 Thursday March 12, 2015 568 IncrementalIncremental AlgorithmAlgorithm
• 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
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CH08-320201: Algorithms and Data Structures 568 Thursday March 12, 2015 569 IncrementalIncremental AlgorithmAlgorithm Original Hull: Want to add point q: Find p and r: q q p, q, r
p H
o, p, q r n H H q forms ao left q forms a right q forms a right turn, so add q: turn, so remove p turn, so remove o s n, o, q n o n
p s
H H H r r r s Jacobs University Visualization and Computer Graphics Lab
CH08-320201: Algorithms and Data Structures 569 Thursday March 12, 2015 570 IncrementalIncremental AlgorithmAlgorithm
Since m, n, q is a left Now we look at the Since q, r, s is a left turn, we’re done with thatq other side: q turn, we’re done! side. q
n m
H sH r r s H • Remember that you can have right turns on either or both sides, so make sure to check in both directions and remove concave points!
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CH08-320201: Algorithms and Data Structures 570 Thursday March 12, 2015 571 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
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CH08-320201: Algorithms and Data Structures 571 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.
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CH08-320201: Algorithms and Data Structures 572 A Divide-and-Conquer Approach
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CH08-320201: Algorithms and Data Structures 573 Merging Hulls
Idea: • Find the two lines that are upper and lower tangent to the two hulls. • Then, remove points that are cut off.
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CH08-320201: Algorithms and Data Structures 574 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.
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CH08-320201: Algorithms and Data Structures 575 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.
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CH08-320201: Algorithms and Data Structures 576 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
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CH08-320201: Algorithms and Data Structures 577 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.
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CH08-320201: Algorithms and Data Structures 578 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)
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CH08-320201: Algorithms and Data Structures 579 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).
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CH08-320201: Algorithms and Data Structures 580 7. Outro
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CH08-320201: Algorithms and Data Structures 653 Algorithmic concepts
• Divide & Conquer • Dynamic Programming • Greedy Approaches
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CH08-320201: Algorithms and Data Structures 654 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
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CH08-320201: Algorithms and Data Structures 655 Runtime analysis
• Upper bounds •Lower bounds •Tight bounds • Solving recurrences
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CH08-320201: Algorithms and Data Structures 656 Data structures
•Array • Stack •Queue •Linked List • Trees •BST • Red-black Trees •Heap •Hash Tables •Graphs
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CH08-320201: Algorithms and Data Structures 657 Memory analysis
•Also asymptotic bounds • Additional memory •In-situ algorithms
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CH08-320201: Algorithms and Data Structures 658 The End
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CH08-320201: Algorithms and Data Structures 659