Convex Hull: Ordering the Points. - USI Informatics · Convex Hull: Ordering the Points. Elena...
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Convex Hull: Ordering the Points. Elena Khramtcova Algorithms and Data Structures. Faculty of Informatics, USI Spring 2013
Transcript of Convex Hull: Ordering the Points. - USI Informatics · Convex Hull: Ordering the Points. Elena...
Convex Hull: Ordering the Points.
Elena Khramtcova
Algorithms and Data Structures.Faculty of Informatics, USI
Spring 2013
Definition of convex hull (CH)
P – set of n points in the plane,
Problem(CH): Compute CH(P).
Definition of convex hull (CH)
P – set of n points in the plane,
Problem(CH): Compute CH(P).
CH(P): smallest convex set containing P.
Definition of convex hull (CH)
P – set of n points in the plane,
Problem(CH): Compute CH(P).
CH(P): smallest convex set containing P.
Definition of convex hull (CH)
P – set of n points in the plane,
Problem(CH): Compute CH(P).
h – # of hull points
p1
p2p3
p4
p5
p6
p7
Sorting can be reduced to CH
A – array of n numbers,
Problem: Compute B = sorted A.
(A[3],0)
(A[2],0)
(A[9],0)
(A[6],0)
(A[4],0)
(A[8],0)
(A[1],0)
(A[5],0)
(A[7],0)
Sorting can be reduced to CH
A – array of n numbers,
Problem: Compute B = sorted A.
Sorting can be reduced to CH
A – array of n numbers,
Problem: Compute B = sorted A.
Sorting can be reduced to CH
A – array of n numbers,
Problem: Compute B = sorted A.
p1
p2p3
p4 p5p6
p7
p8
p9
Sorting can be reduced to CH
A – array of n numbers,
Problem: Compute B = sorted A.
p1
p2p3
p4 p5p6
p7
p8
p9
(B[1],0) ... (B[9],0)
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
p2
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
p2
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
p2
p3
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
p2
p3
p4
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
p2
p3
p4p5
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
p2
p3
p4p5
p6
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh)
p1
p2
p3
p4p5
p6
p7
CH algorithm 1: Jarvis’s march/Jarvis’s wrap
1. Find the lowest point p1
2. Do
Find pnext : min.angle withsupporting line
While pnext ! = p1
Time complexity: O(nh) p1
p2
p3
p4p5
p6
p7
CH algorithm 2: ...
1. Divide points by avertical line in twoequal parts
2. Compute CH(Pleft) andCH(Pright) recursively
3. Find two bridges
4. Delete all edgesin-between the bridges
Time complexity: O(n log n)
CH algorithm 2: ...
1. Divide points by avertical line in twoequal parts
2. Compute CH(Pleft) andCH(Pright) recursively
3. Find two bridges
4. Delete all edgesin-between the bridges
Time complexity: O(n log n)
CH algorithm 2: ...
1. Divide points by avertical line in twoequal parts
2. Compute CH(Pleft) andCH(Pright) recursively
3. Find two bridges
4. Delete all edgesin-between the bridges
Time complexity: O(n log n)
CH algorithm 2: ...
1. Divide points by avertical line in twoequal parts
2. Compute CH(Pleft) andCH(Pright) recursively
3. Find two bridges
4. Delete all edgesin-between the bridges
Time complexity: O(n log n)
CH algorithm 2: ...
1. Divide points by avertical line in twoequal parts
2. Compute CH(Pleft) andCH(Pright) recursively
3. Find two bridges
4. Delete all edgesin-between the bridges
Time complexity: O(n log n)
CH algorithm 2: ...
1. Divide points by avertical line in twoequal parts
2. Compute CH(Pleft) andCH(Pright) recursively
3. Find two bridges
4. Delete all edgesin-between the bridges
Time complexity: O(n log n)
CH algorithm 2: Divide and Conquer
1. Divide points by avertical line in twoequal parts
2. Compute CH(Pleft) andCH(Pright) recursively
3. Find two bridges
4. Delete all edgesin-between the bridges
Time complexity: O(n log n)
CH algorithm 3: ...
1. Find points r and l
2. Divide the set by rl into Aand B
3. Return HULL(A, l , r)∪HULL(B, r , l)
HULL(A, l , r)if A = lr , return (l , r) else
1. Find z ∈ A: farthest fromlr
2. R – points to the right of lz
3. L – points to the left of zr
4. Return{HULL(L, l , z) ∪ {z}∪HULL(R, z , r)}
Time complexity: O(n2) w.c.,O(n log n) avg.
CH algorithm 3: ...
1. Find points r and l
2. Divide the set by rl into Aand B
3. Return HULL(A, l , r)∪HULL(B, r , l)
l
r
HULL(A, l , r)if A = lr , return (l , r) else
1. Find z ∈ A: farthest fromlr
2. R – points to the right of lz
3. L – points to the left of zr
4. Return{HULL(L, l , z) ∪ {z}∪HULL(R, z , r)}
Time complexity: O(n2) w.c.,O(n log n) avg.
CH algorithm 3: ...
1. Find points r and l
2. Divide the set by rl into Aand B
3. Return HULL(A, l , r)∪HULL(B, r , l)
l
r
HULL(A, l , r)if A = lr , return (l , r) else
1. Find z ∈ A: farthest fromlr
2. R – points to the right of lz
3. L – points to the left of zr
4. Return{HULL(L, l , z) ∪ {z}∪HULL(R, z , r)}
Time complexity: O(n2) w.c.,O(n log n) avg.
CH algorithm 3: ...
1. Find points r and l
2. Divide the set by rl into Aand B
3. Return HULL(A, l , r)∪HULL(B, r , l)
l
r
z
l
r
HULL(A, l , r)if A = lr , return (l , r) else
1. Find z ∈ A: farthest fromlr
2. R – points to the right of lz
3. L – points to the left of zr
4. Return{HULL(L, l , z) ∪ {z}∪HULL(R, z , r)}
Time complexity: O(n2) w.c.,O(n log n) avg.
CH algorithm 3: ...
1. Find points r and l
2. Divide the set by rl into Aand B
3. Return HULL(A, l , r)∪HULL(B, r , l)
z
l
r
HULL(A, l , r)if A = lr , return (l , r) else
1. Find z ∈ A: farthest fromlr
2. R – points to the right of lz
3. L – points to the left of zr
4. Return{HULL(L, l , z) ∪ {z}∪HULL(R, z , r)}
Time complexity: O(n2) w.c.,O(n log n) avg.
CH algorithm 3: ...
1. Find points r and l
2. Divide the set by rl into Aand B
3. Return HULL(A, l , r)∪HULL(B, r , l)
HULL(A, l , r)if A = lr , return (l , r) else
1. Find z ∈ A: farthest fromlr
2. R – points to the right of lz
3. L – points to the left of zr
4. Return{HULL(L, l , z) ∪ {z}∪HULL(R, z , r)}
Time complexity: O(n2) w.c.,O(n log n) avg.
CH algorithm 3: Quickhull
1. Find points r and l
2. Divide the set by rl into Aand B
3. Return HULL(A, l , r)∪HULL(B, r , l)
HULL(A, l , r)if A = lr , return (l , r) else
1. Find z ∈ A: farthest fromlr
2. R – points to the right of lz
3. L – points to the left of zr
4. Return{HULL(L, l , z) ∪ {z}∪HULL(R, z , r)}
Time complexity: O(n2) w.c.,O(n log n) avg.
CH algorithm 4: Heaphull. It does exist!
Uses a kinetic heap w.r.t. a certain “up” direction
Time complexity: O(n log2 n)
CH algorithm 4: Heaphull. It does exist!
⇒
up up
Uses a kinetic heap w.r.t. a certain “up” directionTime complexity: O(n log2 n)
Not covered here
I Graham scan
I Chan’s algorithm
I Randomized incremental construction
I 3- and d-dimension
