Proximity Greedy triangulation, star of spokes

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ProximityGreedy triangulation, star of spokes
Star of spokes greedy triangulation The simple
greedy triangulation had two phases 1. Generation
and presort of possible edges. O(N2 log N)
time. 2. Determining if each possible edge
belongs in the triangulation. O(N3) time. The
star of spokes algorithm reduces the second
phase to O(N2 log N) overall by processing each
of the O(N2) possible edges in O(log N)
time. Here processing means to test a possible
edge for intersection with edges already in the
triangulation. We will see a way to perform that
test in O(log N) time. That time suggests a
search data structure of some kind.
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ProximityGreedy triangulation, star of spokes
Star of spokes Suppose the current possible edge
is edge p1pi. Consider a set of edges connecting
p1 to every other point in S. These edges, called
spokes, are not in the triangulation, there are
defined for the search structure. The point p1
and the spokes are together referred to as
STAR(p1), the star of spokes of p1. The spokes
are ordered, say counterclockwise, and they
subdivide the 2? angle around p1 into N - 1
angular intervals, called sectors.
pi
p1
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ProximityGreedy triangulation, star of spokes
Triangulation edges and sectors Suppose edge pjpk
is already part of the triangulation, i.e., a
triangulation edge. It spans a set of consecutive
sectors relative to p1, and the same is true for
any pl ? S - pj, pk. Observe that because
triangulation edges run from point to point, and
points define sectors, edges always entirely span
sectors, rather than partially spanning
them. Note also that triangulation edges do not
intersect each other (by definition), so the
triangulation edges present in each sector of
STAR(pl) pl ? S can be ordered from pl outwards.
pi
pj
p1
pk
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ProximityGreedy triangulation, star of spokes
What to check Recall that edge p1pi is the
possible triangulation edge that is being tested
for intersection with other triangulation
edges. Assume that in STAR(p1) point pi is
located in sector pkp1pj. To determine whether
edge p1pi can be added to the triangulation, it
must be determined if edge p1pi intersects any
other triangulation edge. It is sufficient to
check those triangulation edges that span sector
pkp1pj, and if there are gt 1 such edges, it is
sufficient to check the one closest to p1 in that
sector. Conceptually, if pi is closer to p1 than
the closest spanning edge, edge p1pi can be
added if not, it cannot. For each sector of
STAR(pl) for every pl ? S, the closest
triangulation edge must be maintained. Preparata
calls this a special case of planar point
location.
pi
pj
p1
pk
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ProximityGreedy triangulation, star of spokes
Example situation The figure shows STAR(p1) in
some typical situation. The heavy edges are
triangulation edges, and the lighter edges are
spokes. A possible triangulation edge p1pi can be
added iff pi is in the shaded area.
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ProximityGreedy triangulation, star of spokes
Search data structure The data structure to store
the spanning edge for each sector in STAR(pl) for
pl ? S must support two operations 1. Query to
determine if a possible triangulation
edge intersects a spanning triangulation
edge. 2. Insertion of new triangulation
edges. The second operation indicates that a
dynamic data structure is needed. We would like
the operations to execute in O(log N)
time. Because each insertion involves a sequence
of consecutive spanned sectors a segment tree is
suggested.
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ProximityGreedy triangulation, star of spokes
Sectors of STAR(pl) The sectors of STAR(pl) can
be numbered in counterclockwise order, from 1 to
N - 1.
3
1
2
2
1
4
3
4
pl
sector
5
9
5
Point in CCW sequence
9
6
8
7
7
6
8
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ProximityGreedy triangulation, star of spokes
Segment tree for STAR(pl) The sectors of STAR(pl)
are represented as the N - 1 leaves of a segment
tree, referred to as Tl, defined for interval
1,N.
Each node v of Tl has a field e(v) that stores an
edge. Field e(v) identifies the triangulation
edge closest to to pl in the sector(s) that
correspond to the node v.
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ProximityGreedy triangulation, star of spokes
Insertion When an edge pipj (arbitrary, not pipj
from earlier figure) is to be added to the
triangulation, it is inserted into the segment
tree Tl representing STAR(pl) for every pl ? S, l
? i, j. During a single insertion, at each
allocation node v in the segment tree Tl, edge
pipj is tested against the edge identified by
e(v). If edge pipj is closer, e(v) is updated to
identify edge pipj. No ambiguity about closest
is possible, because 1. Triangulation edges do
not intersect. 2. An edge is only allocated to a
node v if it spans all the sectors for that node.
1
2
not possible
Test and update of e(v) take constant time at
each node. Insertion into the segment tree will
visit O(log N) nodes, so the insertion time is
O(log N).
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ProximityGreedy triangulation, star of spokes
Query To test if edge pipj (arbitrary, not pipj
from earlier figure) intersects any edge of the
triangulation, search segment tree Tl
representing STAR(pl) with the number of the
sector contain pj. The search will trace a single
path of nodes in Ti from root to leaf. At each
node v on the path, test edge pipj for
intersection with e(v). If an intersection is
found, stop. If the search reaches a leaf without
finding any intersections, accept edge pipj for
the triangulation (add it to every segment
tree). A segment tree has depth O(log N), so
there are O(log N) nodes visited during the
query. The intersection test at each one require
O(1) time. ? The query requires O(log N) time.
pj
pj
pi
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ProximityGreedy triangulation, star of spokes
Star of spokes greedy triangulation
analysis Time O(N2 log N) O(N2 log N) for
presort of all possible edges, O(N2) edges ?
O(log N) processing for each Storage O(N2) O(N)
storage for the segment tree for each of N
points
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ProximityConstrained triangulation
Problem definition CONSTRAINED TRIANGULATION INSTA
NCE Set S p1, p2, ..., pN of N points in
the plane, and set E pi, pj, of
nonintersecting edges on S. QUESTION Join the
points in S with nonintersecting straight line
segments so that every region internal to the
convex hull of S is a triangle and every edge of
E is in the triangulation.
For this problem, optimization (e.g., edge length
minimization) is not required. A greedy method
will work (with the edges in E put into
the triangulation initially), but performance is
not optimum. Delaunay triangulation is not easily
adapted.
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ProximityConstrained triangulation
Assumptions Given S and E, assume 1. We will
triangulate not S, but instead the interior of
the smallest axis-parallel rectangle that
encloses S, considering S and E. The enclosing
rectangle is denoted rect(S). This adds ? 4
points beyond those of S to be triangulated. 2. No
two points of S have the same ordinate (y
coordinate) except the corners of rect(S). The
procedure can be modified to eliminate this
assumption, at the cost of introducing special
cases.
rect(S)
edge in E
point in S
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ProximityConstrained triangulation
S, E
(1) Inscribe S in minimum enclosing axis
parallel rectangle. O(N)
PSLG
rect(S), S, E
(2) Regularize rect(S), S, E. O(N log N)
Each region within regularized rect(S), S, E is a
monotone polygon.
Regularized rect(S), S, E
(3) Decompose regularized rect(S), S, E
into monotone polygons. O(N)
Monotone polygon
(4) Triangulate monotone polygons. O(N)
Triangulation of rect(S)
Comments Steps (1) and (3) are very
simple. Step(2) was previously described. Step(4)
will be explained.
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ProximityConstrained triangulation
Regularization of rect(S) The regularization
process operates on rect(S), S, and E. These
together form a PSLG. To regularize, apply the
plane-sweep algorithm to regularize a
PSLG described earlier for the chain method of
point location. Regularization requires O(N log
N) time. After regularization, these properties
hold (i) No two edges intersect, except at
vertices. (ii) Each vertex (except those with
largest y coordinate) is joined directly to at
least one vertex with a larger y
coordinate. (iii) Each vertex (except those with
smallest y coordinate) is joined directly to at
least one vertex with a smaller y coordinate.
rect(S)
edge in E
point in S
edgeadded during regularization
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ProximityConstrained triangulation
Monotone polygons A polygon P is monotone w. r.
t. a straight line l if P is simple and its
boundary is the union of two chains monotone
w.r.t. l. The regularization process partitions
the interior of rect(S) into a set of polygons,
each of which is monotone. For a proof, see
Preparata pp. 238-239. Furthermore, they are
monotone w.r.t. the y axis, i.e., the line of
monotonicity l is the y axis. The top and bottom
edges of rect(S) are exceptions to the y axis
monotonicity, and are handled as special cases.
P1
P2
P3
P4
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ProximityConstrained triangulation
Triangulating a monotone polygon,
introduction The algorithm to triangulate a
monotone polygon depends on its
monotonicity. Developed in 1978 by Garey,
Johnson, Preparata, and Tarjan, it is described
in both Preparata pp. 239-241 (1985) and Laszlo
pp. 128-135 (1996). The former uses y-monotone
polygons, the latter uses x-monotone. Initializat
ion Sort the N vertices of monotone polygon P in
order by decreasing y coordinate. (Here N is the
number of vertices of P, not S.) The sort can be
done in O(N) time, not O(N log N), by merging the
two monotone chains of P. Let u1, u2, , uN be
the sorted sequence of vertices, so y(u1) gt y(u2)
gt gt y(uN). Because of the regularization
process and the monotonicity of P, for every ui 1
? i lt N there exists uj 1 lt j ? N such that edge
uiuj is an edge of P.
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ProximityConstrained triangulation
Description of the processing The algorithm
processes one vertex at a time in order of
decreasing y coordinate, creating diagonals of
polygon P. Each diagonal bounds a triangle, and
leaves a polygon with one less side still to be
triangulated. Stack The algorithm uses a stack
to store vertices that have been visited but not
yet connected with a diagonal. The stack content
is v1, v2, , vi, where v1 is the bottom and vi
the top of the stack. At any time during the
execution, there are two invariants 1. The
vertices v1, v2, , vi on the stack from a
chain on the boundary of P, where y(v1) gt y(v2)
gt gt y(vi). 2. If i ? 3, angle vjvj1vj2 ? ?
for 1 ? j ? i - 2.
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ProximityConstrained triangulation
Algorithm By adjacent we mean connected by an
edge in P. Recall that v1 is the bottom of the
stack, vi is the top. 1. Push u1 and u2 on the
stack. 2. j 3 / j is index of current vertex
/ 3. u uj 4. Case (i) u is adjacent to v1
but not vi. add diagonals uv2, uv3, ,
uvi. pop vi, vi-1, , v1 from stack. push vi,
u on stack. Case (ii) u is adjacent to vi but
not v1. while i gt 1 and angle uvivi-1 lt
? add diagonal uvi-1 pop vi from
stack endwhile push u Case (iii) u
adjacent to both v1 and vi. add diagonals uv2,
uv3, , uvi-1. exit 5. j j 1 Go to step 3.
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ProximityConstrained triangulation
Algorithm cases Case (i) u is adjacent to v1
but not vi.
v1
v2
v3
v4 vi top of stack
u
Case (ii) u is adjacent to vi but not v1.
v1
v2
v3
v4
v5 vi top of stack
u
Case (iii) u adjacent to both v1 and vi.
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ProximityConstrained triangulation
Example, 1
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ProximityConstrained triangulation
Example, 2
v1
v2
v3
8, Case (ii)
u
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ProximityConstrained triangulation
Example, 3
9, Case (iii)
10, final
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ProximityConstrained triangulation
Proof of correctness The correctness of the
algorithm depends on the fact that all the added
diagonals lie inside the polygon P. For details,
see Preparata pp. 240-241, or Laszlo pp.
134-135. Analysis of triangulating a monotone
polygon The initial sort (merge) requires O(N)
time. Each of the N vertices is visited and
placed on the stack exactly once, except when the
while fails in case (ii). This happens at most
once per vertex, so that time can be charged to
the current vertex. ? The algorithm requires O(N)
time to triangulate a monotone polygon, where N
is the number of vertices of the polygon.
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ProximityConstrained triangulation
S, E
(1) Inscribe S in minimum enclosing axis
parallel rectangle. O(N)
PSLG
rect(S), S, E
(2) Regularize rect(S), S, E. O(N log N)
Each region within regularized rect(S), S, E is a
monotone polygon.
Regularized rect(S), S, E
(3) Decompose regularized rect(S), S, E
into monotone polygons. O(N)
Monotone polygon
(4) Triangulate monotone polygons. O(N)
Triangulation of rect(S)
Overall comments O(N log N) regularization
dominates time. O(N) for triangulating all
monotone polygons (here N is the number of
vertices in S and rect(S). We know TRIANGULATION
has lower bound in ?(N log N). ? This algorithm
is optimal, ?(N log N).
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ProximityTriangulating a simple polygon
Definitions In a simple polygon, edges intersect
only at vertices, and non-adjacent edges do not
intersect. Three consecutive vertices a, b, c of
a polygon form an ear of the polygon if segment
ac is a diagonal b is the ear tip. Meisters
Two Ears Theorem. Every polygon with N ? 4
vertices has at least two nonoverlapping ears.
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ProximityTriangulating a simple polygon
Example, simple polygon with ears
c
b
ear
a
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ProximityTriangulating a simple polygon
Triangulation by otectomy See ORourke, pp.
39-46. Let P be a simple polygon, with vertices
p1, p2, , pN. 1. if N gt 3 2. for each
potential ear diagonal pipi2 3. if pipi2 is
a diagonal 4. add diagonal pipi2 5. recurse
on P - pi1 6. endif 7. endfor 8. endif Ana
lysis Step 2 is O(N), search around P. Step 3 is
a test for diagonal by checking for
intersections, O(N). Step 5 can occur at most
O(N) times. Overall time required is O(N3).
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ProximityTriangulating a simple polygon
Example
p12
p5
7
p11
2
8
p10
10
9
p13
p7
p15
3
p6
11
p9
p14
p8
12
13
p16
4
p4
14
5
p17
1
6
15
p2
p18
p3
p1
Numbers indicate sequence in which diagonals were
added.