Greedy Algorithm to Find Minimum Length Triangulation

1
Greedy Algorithm to Find Minimum Length
Triangulation

• CSC6520 Algorithm
• Graduate Student Project
• Instructor Dr. Alexander Zelikovsky Student
  Yan Liu
• Spring 2001

2
Introduction

• Definition of Triangulation
• -- a set T of chords of a convex polygon P that
  divide the polygon into disjoint triangles.
• Minimum length triangulation(MLT)
• -- Given a convex polygon P(v1, , vn)
• -- Form Triangles by sides and
• chords of P.
• -- Find a triangulation that minimizes the sum
  of the lengths of the triangles in the
  triangulations.

3
Example of triangulating a convex polygon
v7
v7
v1
v1
v6
v6
v2
v2
v5
v5
v3
v3
v4
v4
Figure 1. Two ways of triangulating a convex
polygon.
4
Data Structure

• Heap a complete binary tree to hold all of
  diagonals.
• -- Build-Heap(E)
• -- Heapify (E, i)
• HeapSort algorithm
• -- to get the minimum length triangulation.
• Remove diagonals from set E.
• -- Heap-delete(E,i)
• Add new diagonals
• -- Heap-Insert(E)

5
Algorithms for MLT

• Given a convex polygon P(v1v n)
• Find the minimum length triangulation for P(v1v
  n)
• Algorithms
• -- Greedy-1 Algorithm
• -- Greedy Algorithm

6
Greedy-1 Algorithm for MLT

• Step 1. Put all of vertices of convex
• polygon P into set V.
• Step 2. Put all of next-neighbor diagonals
• into set E. For example, (k, k2) is the
• next-neighbor diagonal between the
• vertices v k and v k2.
• Step 3. Find the minimum diagonal (k,
• k2) in E, and insert it into the result set M
• of the minimum length triangulation of the
• convex polygon P.

7
Greedy-1 Algorithm for MLT -- continue

• Step 4. Remove the vertex v k1 from set
• V.
• Step 5. Remove all diagonals which are
• related to the removed vertex in set V in
• the last step.
• Step 6. Update E to add unexisted next-
• neighbor diagonals for the vertices v k
• and v k2 such as (k, k3), (k2, k-1).
• Step 7. Check if E is NOT empty, go
• back step 3 to repeat otherwise return
• the result M.

8
Example for Greedy 1 Algorithm
v7
v1
v6
v2
v5
v4
v3
E (1,3), (2,4),(3,5),(4,6), (5,7), (6,1),
(7,2) Vv1, v2,v3,v4,v5,v6,v7 M Ø
9
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E(1,3), (2,4),(3,5),(4,6), (5,7), (6,1) Vv1,
v2,v3,v4,v5,v6,v7 M(7, 2)
10
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E (1,3),(1,6),(2,4),(3,5),(4,6),
(5,7) Vv2,v3,v4,v5,v6,v7 M(7, 2)
11
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E(2,4),(3,5),(4,6), (5,7) Vv2,v3,v4,v5,v6,v7
M(7, 2)
12
Example for Greedy 1 Algorithm --continue
v7
Add new next diagonals
v1
v6
v2
v5
v3
v4
E(2,4),(3,5),(4,6), (5,7), (6,2), (7,3)
Vv2,v3,v4,v5,v6,v7 M(7, 2)
13
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E(2,4),(3,5),(4,6), (6,2), (7,3) Vv2,v3,v4,v5
,v6,v7 M(7, 2), (5,7)
14
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E(2,4),(3,5), (4,6), (6,2), (7,3) Vv2,v3,v4,v
5,v6,v7 M(7, 2), (5,7)
15
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E(2,4),(3,5), (7,3) Vv2,v3,v4,v5,v7
M(7, 2), (5,7)
16
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
Add new next-diagonals
v3
v4
E(2,4),(3,5), (7,3), (5, 2),(7,4) V
v2,v3,v4,v5,v7 M(7, 2), (5,7)
17
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E(3,5), (5, 2),(7,3), (7,4) V
v2,v3,v4,v5,v7 M(7, 2), (5,7), (2,4)
18
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E(3,5), (7,3), (5, 2),(7,4) V
v2,v3,v4,v5,v7 M(7, 2), (5,7), (2,4)
19
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E (5, 2), (7,4) V v2,v4,v5,v7 M(7, 2),
(5,7),(2,4)
20
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E (v5, v2) V v2,v4,v7 M(7, 2),
(5,7),(2,4), (4,7)
21
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E (v5, v2) V v2,v4,v7 M(7, 2),
(5,7),(2,4), (4,7)
22
Example for Greedy 1 Algorithm --continue
v7
v1
v6
v2
v5
v3
v4
E Ø V v2,v4,v7 M(7, 2), (5,7),(2,4),
(4,7)
23
Greedy Algorithm for MLT

• Step 1. Insert all diagonals between any two
  vertices (exclude any two neighbor vertices on
  the polygon P) into set E.
• For example, (i, j) is the diagonal between
  the vertices v i and v j.

24
Greedy Algorithm for MLT -- continue

• Step 2. Find the minimum diagonal in E and
• insert it into the result set M of the
• minimum length triangulation for graph G.
• Step 3. Remove those diagonals from E,
• which cross the chosen minimum diagonal
• in the last step.
• Step 4. Check if E is NOT empty, go back
• step 2 to repeat otherwise return the result
• M.

25
Example for Greedy Algorithm
v7
v1
v6
v2
v5
v4
v3
E(1,3), (1,4), (1,5), (1,6), (2,4),(2,5),(2,6),
(2,7), (3,5),(3,6), (3,7),(4,6), (5,7), (4,6),
(4,7) M Ø
26
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(1,3), (1,4), (1,5), (1,6), (2,4),(2,5),(2,6),
(2,7), (3,5),(3,6), (3,7),(5,7), (4,7) M
(4,6)
27
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(1,3), (1,4), (1,5), (1,6), (2,4), (2,5),
(v2,v6), (2,7), (3,5), (3,6), (3,7), (5,7),
(4,7) M (4,6)
28
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(1,3), (1,4), (1,6), (2,4), (2,6), (2,7),
(3,6), (3,7),(4,7) M (4,6)
29
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(1,3), (1,4), (1,6), (2,4), (2,6), (3,6),
(3,7), (4,7) M (4,6), (2,7)
30
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(1,3), (1,4), (1,6), (2,4), (2,6), (3,6),
(3,7), (4,7) M (4,6), (2,7)
31
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(2,4), (2,6), (3,6), (3,7), (4,7) M (4,6),
(2,7)
32
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(2,6), (3,6), (3,7), (4,7) M (4,6),
(2,7),(2,4)
33
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(2,6), (3,6), (3,7), (4,7) M (4,6),
(2,7),(2,4)
34
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(2,6), (4,7) M(4,6), (2,7),(2,4)
35
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(4,7) M (4,6), (2,7),(2,4), (2,6)
36
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E(4,7) M (4,6), (2,7),(2,4), (2,6)
37
Example for Greedy Algorithm --continue
v7
v1
v6
v2
v5
v4
v3
E Ø M (4,6), (2,7),(2,4), (2,6)
38
Summary

• Performance
• -- Greedy algorithm is between O(n2)
• and O(n3)
• -- Greedy-1 algorithm is O(nlgn)
• Result of MLT
• -- For a convex polygon P(V),
• using different algorithms may
• get different results.