Download Computational Geometry in Spatial Databases: Sweep-Line Method and Polygon Partitioning - and more Papers Computer Science in PDF only on Docsity!
ECS 266 – Spatial Databases Michael Gertz © 2008 UC Davis
3. Computational Geometry
ECS 266 – Spatial Databases
Overview
Learning Objectives
- Review the basic concepts of algorithms
- Understand some useful algorithmic strategies (e.g., the sweep-line method)
- Learn some algorithms for spatial databases (e.g., point in polygon, polygon intersection, ...)
Literature
- [RSV02] Chapter 5 and/or [BKO00] Chapters 1 and 2
- For many of the basic principles and algorithms presented in the following, you can find very useful resources on the Web that often go beyond what is presented in the following.
ECS 266 – Spatial Databases Michael Gertz © 2008 UC Davis 3. Computational Geometry^3
Background
Design of algorithms involves
1. its description at an abstract level by means of a pseudo-code , and
2. the proof that the algorithm is correct.
Analysis deals with the performance of an algorithm, such as its cost
in time or storage space, known as complexity analysis.
The data to be processed by an algorithm is organized in data structures.
Some notions:
Random Access Machine
Algorithm evaluation is a function f(n,c 1 , c 2 , ..., cm)
Upper bound: A function g(n) of the input size is said to be O(f(n)) if
there exists a constant c and integer n 0 such that g(n) ≤ c ∗ f(n) ∀ n > n 0
Lower bound: A function g(n) is said to be Ω (f(n)) if
there exists a c and n 0 such that g(n) ≥ c ∗ f(n) ∀ n > n 0
ECS 266 – Spatial Databases
Background (2)
The complexity of an algorithm is a function g(n) that gives the upper
bound of the number of operations performed by algorithm when the input
size is n.
- Worst-case complexity: the running time for any input of a given size
will be lower than the upper bound.
- Average-case complexity: g(n) is the average number of operations
over all problem instances for a given size.
Often, the complexity g(n) is approximated by its family O(f(n)) where
f(n) is n, log n, na^ a ≥ 2, an
An algorithm for a given problem is optimal if its complexity reaches the
lower bound over all the algorithms that solve the problem. Estimating
lower bound is an essential yet often very difficult task.
ECS 266 – Spatial Databases Michael Gertz © 2008 UC Davis 3. Computational Geometry^7
Sweep-Line Method
Basic idea:
- decompose input into vertical strips such that the information relevant to
the problem is located on the vertical lines that separate two strips.
- By sweeping a vertical line from left to right, stopping at the strip
boundaries, one can maintain the information needed for solving the
problem.
Algorithm may vary widely, based on what the geometric input is (sweep-
line of planar curves, line segments, rectangles etc)
Commonality among algorithms is the use of two data structures:
- Sweep-line active list L: this structure is updated as the line goes
through a finite number of positions, called events.
- Event list E: known beforehand (managed as a chained list) or
discovered step-by-step as sweep goes on (managed by a more
flexible structure, e.g., priority queue).
ECS 266 – Spatial Databases
Sweep-Line Method (2)
Example : Given a set of rectangles S. Find all pairwise intersections. Given the worst case complexity, an output sensitive algorithm is worthwhile.
Key observations:
- given a vertical line l and the set Sl of all rectangles cut by l , reporting pairwise intersections in Sl is reduced to testing the overlap of the projection of rectangles on the y axis. E.g., Sl = {A,C,E} in Fig. (a)
- When sweeping l from left to right, content of Sl changes at a finite number of locations of l , called event points, namely when l reaches the left or right vertical edges of each rectangle in S.
ECS 266 – Spatial Databases Michael Gertz © 2008 UC Davis 3. Computational Geometry^9
Sweep-Line Method (3)
Event points are the 2 × | S | left and right x coordinates, ordered in E. A data structure L is maintained for the list of active rectangles that intersect with l.
Operations at each event point:
- When left edge of rectangle r ∈ E is reached by l , insert r into L. r’s x interval intersects with those currently in L. Hence, the problem is reduced to searching for those rectangles in L that overlap in the y dimension.
- When right edge of rectangle r ∈ E is reached by l , r does not overlap with any of the remaining rectangles in E. Both cases are illustrated in Fig. (b) and (c).
ECS 266 – Spatial Databases
Sweep-Line Method (4)
Algorithm RectangleIntersection (S: set of rectangles)
begin sort the 2 n xmin and xmax values of S and place them in E L = {} while ( E <> {} ) do begin
p ← min(E) // extract the next event from E
if ( p is the left edge of a rectangle r ) then begin insert ( r, L ) search and report the rectangles in L intersecting r end if if ( p is the right edge of a rectangle r ) then delete (r, L) end while end Complexity: O(n log n+k), with k being the number of intersections
ECS 266 – Spatial Databases Michael Gertz © 2008 UC Davis 3. Computational Geometry^13
Polygon Partitioning (3)
Triangulation of a simple polygon consists of findings diagonals within
the polygon.
- Triangulation is non-deterministic
- Every triangulation of polygon P with n vertices has n-3 diagonals and results in n-2 triangles.
- Monotone polygons can be triangulated linearly, thus partitioning a simple polygon into monotone polygons is key to efficient triangulation. Such a partitioning (similar to the trapezoidalization) can be done in O(n log n).
- Subsequent triangulation can be done in linear time.
- There exists a linear algorithm for triangulating simple polygons.
ECS 266 – Spatial Databases
Some basics of 2-D Lines
Given two points p 1 (x 1 ,y 1 ) and p 2 (x 2 ,y 2 ) in a 2-dimensional space,
derive the explicit/implicit equation for the line.
Given a point p 1 (x 1 ,y 1 ) and a line, what is the distance between the point
and the line?
