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An arrangement of lines: A(H)
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Elements of the arrangement
Vertices – intersection of lines
Edges – portions of lines bounded by vertices, except when unbounded at one end
Faces – regions bounded by edges and vertices
Counts of the elements
Number of vertices (V) : n(n-1)/2
Number of edges (E): n^2 (interesting)
Number of faces (F): n^2 – n(n-1)/2 + 1 (follows from the modified Euler formula V-E+F = 1)
Computing an arrangement
We must output a data structure that encapsulates the mutual relationships of the elements (vertices, edges and faces), corresponding to the pictorial representation of an A(H) as in the first slide
EG paper only shows how to traverse A(H)
Sweepline paradigm
The standard sweepline paradigm requires sorting the O(n^2) intersection points
This would require O(n^2 log n) time
Topological sweep
Gets rid of the log n factor, by processing the intersection points without having to sort them !!
It does this at the expense of just O(n) additional space needed by lots of extra book-keeping.
A partial order
We can define the following partial order on elements of the arrangement
An element A is above element B if A is above B at every vertical line that intersects both A and B
The above relationship is acyclic The inverse of above is below
Consequences
There exists a unique element in A(H) that is not below any other and a unique element that is not above any other.
These are called respectively the top-most and the bottom-most element
Prove uniqueness from the acyclicity of the above partial order
Cuts
A cut is a sequence of edges (c1, c
2, ...,c
n),
one from each line of A(H), such that for each i (1.. n-1) there is a (unique) face f
i such that c
i
is above fi , and c
(i+1) is below f
i
c1 is below the top-most face and c
n is above
the bottom-most face
A cut – pictorially
c1c2
c3
c4
c5
Ordering the cuts
A cut C is is to the left of a cut C' if for each line l in A(H), c
i on l from C is to the left of or
identical with cj from C' on l
Thus there is a lefmost cut and a rightmost one
Important Fact
In a given cut there exists an i such that ci
and c(i+1)
have a common right end-point The so-called topological sweep exploits this
to move from the leftmost cut to the rightmost one in a series of elementary moves
In each such move, the topological line moves across such a common right end-point
We demonstrate this on the example arrangement in the next few slides
Sweeping the arrangement
The leftmost cut
Sweeping A(H) : 1st elementary move
2nd elementary move
3rd elementary move
4th elementary move
5th elementary move
6th elementary move
7th elementary move
8th elementary move
9th elementary move
10th elementary move
The rightmost cut
Representing a cut
A cut is represented by an array C[1..n], where each C[i] = ( λi, ρi , µi), representing respectively the index of the line that defines the left and right end-point of the edge c i and the index of the line on which it lies
From one cut to the next..
This is done in an elementary step An elementary step is one in which the
topological sweep moves across an intersection defined by c
i and c
(i+1) for some i in
the current cut Such an i always exists except when the cut is
the rightmost one
Implementing an elementary move
An elementary move is implemented with the help of two data structures derived from a cut C – namely the upper horizon tree T+(C) and the lower horizon tree T-(C)
Upper horizon tree
Lower horizon tree
Initializing the horizon trees
Data Structures for horizon trees
An array HTU[1..n] for the upper horizon tree, where HTL[i] = (λi, µi), where λi (µi ) is the index of the line that defines the left (right) end point of the segment from li that belongs to the upper horizon tree
λi = -1 if segment left-unbounded and µi = 0 if right-unbounded
A similar definition for HTL[1..n]
Initializing HTU (and HTL similarly)
Updating HTU (and similarly HTL)
Example HTU[1..5]
HTU[1]= (-1,2)HTU[2]= (-1,5)HTU[3]= (5,4)HTU[4]= (5,0)HTU[5]= (3,0)
Example HTL[1..5]
HTL[1]= (-1,0)HTL[2]= (-1,1)HTL[3]= (5,1)HTL[4]= (5,3)HTL[5]= (3,1)
Data Structures for horizon trees
Array M[1..n] stores the index of the line on which ci lies
Array N[1..n] stores the description of a cut; N[i] = (λi, µi), where λi (µi ) is the index of the line that defines the left-end (right-end) of ci
N[1..n] can be obtained from HTL[1..n], HTU[1..n] and M[1..n]
Example N[1..5]
M[1..5] = [1,2, 5, 3, 4] (slide 32) This gives: N[1..5] = [(-1,2), (-1,1), (3,1),
(5,4),(5,3)]
Data Structures for horizon trees
Finally, we have a stack I that stores the indices i such that ci and c(i+1) have a common right end-point.
This is obtained from N[1..n] by examining pairs of entries in N[1..n] and checking if µi =
µi+1 + 1, for i= 1, .., n-1, and stacking the i for which the above holds
Example I
I=[1,4 …….., for the example N[1..5]
Updating HTU (and similarly HTL)
Updating all the other data structures
It is easy to update M[1..n] N[i] = HTL[M[i]] ∩ HTU[M[i]] From N[i] we can update the stack I
Analysis
The analysis shows that the cost of traversing the bays associated with a fixed line l is O(n) and hence O(n2) for all n lines.
Amortized Analysis
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