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An arrangement of lines: A(H) 111 1 1111 22 1 3 3 4 4 5 1111111111 5155 3455 1 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, c2, ...,cn), one from each line of A(H), such that for each i (1.. n-1) there is a (unique) face fi such that ci is above fi , and c(i+1) is below fi c1 is below the top-most face and cn is above the bottom-most face A cut – pictorially c2 c 3 c4 c5 c1 Ordering the cuts A cut C is is to the left of a cut C' if for each line l in A(H), ci 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 ci 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 ci 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