Segment Tree and Its Usage for geometric Computations

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Presentation transcript:

Segment Tree and Its Usage for geometric Computations Shmuel Wimer Bar Ilan Univ., Eng. Faculty Technion, EE Faculty April 2012 Segment Tree

Motivation We may be interested in calculating the underling area, perimeter or contour construction April 2012 Segment Tree

Segment Tree Definition Introduced by J. L. Bentley in 1977 Data structure designed to handle intervals on the real line Intervals end points belong to a fixed set of abscissas Abscissas can be normalized to range [1,N] without loss of generality by using a lookup table Given an interval [l,r], the segment tree T(l,r) is a rooted binary tree defined recursively April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

4,15 9,15 4,9 6,9 4,6 12,15 9,12 5,6 4,5 7,9 6,7 10,12 9,10 13,15 12,13 8,9 7,8 11,12 10,11 14,15 13,14 April 2012 Segment Tree

Insertion and Deletion April 2012 Segment Tree

Insertion and Deletion April 2012 Segment Tree

April 2012 Segment Tree

1,257 1,129 65,129 97,129 65,97 65,81 81,97 97,115 105,109 97,105 105,115 105,107 73,81 73,77 73,75 74,75 75,77 77,81 April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

Allocation and De-Allocation Depends on application. If we wish to know the cardinality of cover of [B[v],E[v]] then a counter C[v] is associated with node v: C[v] = C[V]+1 is allocation for INSERT C[v] = C[V]-1 is de-allocation for DELETE In many cases C[v] indicates the presence of material, so we’ll be interested in whether C[v] > 0 (material exists) or C[v]==0 (no material). April 2012 Segment Tree

1D Measure of Union of Intervals April 2012 Segment Tree

1D Measure of Union of Intervals April 2012 Segment Tree

April 2012 Segment Tree

2D Measure (Area) of Union of Rectangles April 2012 Segment Tree

2D Measure (Area) of Union of Rectangles April 2012 Segment Tree

scan-line April 2012 Segment Tree

Efficient Calculation of m(xi) April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

Perimeter of Union of Rectangles Perimeter is the length sum of: vertical edges horizontal edges April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

The Contour of Union of Rectangles April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

Completing the Cycles of Contour April 2012 Segment Tree

April 2012 Segment Tree

e1 e3 e2 e6 e5 e8 e7 e4 e9 e10 April 2012 Segment Tree

There’s no ambiguity in deciding whether to go to left or right triplet when an horizontal edge is decided. It follows that a pair of successive triplets defines horizontal edges. Consequently, once two successive triplets are traversed and define a new horizontal edge, the number of triplets on both the left and the right parts of the list must be even. Therefore, if the index of a triplet is even, its left adjacent triplet is paired, otherwise, the right triplet is paired. April 2012 Segment Tree

Run-time Complexity April 2012 Segment Tree

April 2012 Segment Tree

U’ U” |P(U’)+ P(U”)|=7 U’ U” |P(U’)+ P(U”)|=5 April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

April 2012 Segment Tree

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