Minimal Rectangular Partitions of Digitized Bolbs

Minimal Rectangular Partitions of Digitized Bolbs
• By L.FERRARI, P.V. SANKAR, AND J.SKLANSKY, 1984
Introduction
Contents
1. What is the minimum number? Theorem One: P = N – L + 1 2. How to produce the partition with minimum number? Theorem Two: Max Matching equal no augmenting path.
• A rectilinear polygon R has a minimum number of N – L + 1 rectangular partitions where, N = Total number of concave vertices L = Maximum number of nonintersecting chords that can be drawn between cogrid concave vertices.
Proof: P<= N – L + 1
Proof : P <= N – L + 1
Part 2 : P >= N – L + 1
• Given a rectilinear polygon B and one of its possible partitions. The set of nonintersecting cogrid points contains a maximum number of L rectangle. • Prove that at least N – L + 1 rectangles are needed to cover the whole polygon.
Only circles of even vertices
Finding a maximum matching
• We can always find a maximum matching by reversing the augmenting path. • Bipartite Matching(G;M)
Contents
1. What is the minimum number? Theorem One: P = N – L + 1 2. How to produce the partition with minimum number? Theorem Two: Max Matching equals no augmenting path.
Proof Schedule
P <= N – L + 1
Construction of a solution with N – L + 1 rectangles By Lemma One P >= N – L + 1 Every partition must have at least N - L + 1 rectangles
• C is the set of cogrid concave chords. • C’ is the maximum set of nonintersecting cogrid concave chords. C’ contains L chords. • C’’ is the set of C – C’. Lemma 2 In the (L + 1) subregions divided by C’, treating the cogrid concave points forming the chords in C’’ as noncogrid points will not increase the number of rectangles.
Lemma 1
• For a rectilinear polygon whose boundary contains N noncogrid concave vertices, the minimum number of rectangular partition is N+1.
Proof of Lemma One
Some observations
• A is an important point. • A point is concave if it has an inner angle of 270° • Chord: a chord is a line segment separating the polygon A
From Maximum Matching to Vertex Cover
• Partition the vertices of G into subsets Si as follows: (1) S0 consist of all vertices unmatched by M. (2) For integer j >= 0, let S2j+1 be the set of vertices that: a. are adjacent to vertices in S2j via some edge e ∈E|M. b. have not been included in any previously-defined set Sk, k < j. Remark : Each vertex v∈S2j+1much be adjacent to another vertex u via an edge e ∈ M。 (3) S2j+2 contain these adjacent vertices (4) If there are no vertices adjacent to S2j , arbitrarily pick an unused vertex and continue in S2j+1. Vertices in odd levels form a vertex cover.
Illustration
Matching
• A matching M of G(V,E) is a subset of the edges E, such that no vertex in V is incident to more than one edge in M.
• Maximum matching
Theorem 2. Part 2
• Part 2 : If M is not maximum then there is an augmenting path w.r.t. M. • M is the maximum Matching, N is non-Maximum Matching • The components of union of M (solid) and N (dashed) can be classified as (|M| < |N|)
1. For every vertex in L, start DFS. 2. DFS() If current vertex is in L follow an matched edge else follow a free edge. If at any point we find an augmenting path, reverse it and continue with unmatched vertices.
a,b are boundary line defining A. Two chords (c,d) can be drawn from A α,β are arbitrarily small rectangles defined by (a,c) and (b,d) 1. α and β cannot be in the same rectangle. 2. Either c or d must exist to separate α and β. 3. Each noncogrid vertex will increase the total number of rectangles by at least one.
Alternating Path
• A path is alternating w.r.t. a matching if its edges are alternating between free and matched edges.
• Three types of alternating paths • 1. Between two matched vertices • 2. Between two free vertices (Augmenting Path) • 3. Between a matched vertex and a free v is not a maximum matching if and only if there exists an augmenting path w.r.t M. • Part One : If there exists an augmenting path, the matching is not maximum. • Proof : A reverse of matched and free edges can produce a matching with M+1.
P >= N – L + 1
• (3) Q: the total number of noncogrid concave vertices in the S subregions Q = N – 2L’ – 2L1’ – L2 P = min {L’ + 1 + M1 + M2 + M3+(N – 2L’ – 2L1– L2 )} = min {N – L’ + 1 + (M1-2L1) + (M2 – L2) + M3 } =N–L+1