Segment Tree and Its VLSI Layout Applications

1. Segment Tree and Its VLSI Layout Applications Shmuel Wimer Bar Ilan Univ., School of Engineering Segment Tree

2. 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 Segment Tree

3. Segment Tree

4. Segment Tree

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

6. Insertion and Deletion Segment Tree

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8. 73,77 1,257 65,97 81,97 73,81 1,129 74,75 75,77 77,81 73,75 65,129 97,129 65,81 97,115 97,105 105,115 105,109 105,107 Segment Tree

9. Segment Tree

10. 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 of VLSI aplication 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). Segment Tree

11. VLSI Layout Applications • Useful for calculating area and perimeter of polygon ensemble for lateral and fringe capacitance extraction. • Mask Boolean operations like UNION, INTERSECTION, DIFFERENCE, and more, where contour is required. • Construction (contour) problems are typically more difficult than reporting (area, perimeter) problems. • More applications: • Design rule checking. • Cross coupling capacitance • Very robust and simple for implementation. There are other, more efficient and more complex data structures. Segment Tree

12. This data is used by lithography for mask generation, obtained by UNION operation This is what mask designer is drawing We may be interested in calculating the underling area for extracting lateral capacitance and the perimeter for the extraction of fringe capacitance Segment Tree

13. This is what mask designer is drawing This is the diffusion mask for manufacturing obtained by Boolean DIFFERENCE Segment Tree

14. The electric field between the metal wires implies line-to-line capacitance, which is the reason of noise and power dissipation This is what mask designer is drawing L D We’d like to calculate the common parallel run L and the distance D so the line-to-line capacitance is L/D Segment Tree

15. 1D Measure of Union of Intervals Segment Tree

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17. 2D Measure (Area) of Union of Rectangles Segment Tree

18. scan-line Segment Tree

19. Efficient Calculation of m(xi) Segment Tree

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22. Perimeter of Union of Rectangles Perimeter is the length sum of: vertical edges horizontal edges Segment Tree

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29. The Contour of Union of Rectangles Segment Tree

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39. Completing the Cycles of Contour Segment Tree

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41. e3 e6 e8 e1 e10 e5 e7 e2 e9 e4 Segment Tree

42. 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. Segment Tree

43. Run-time Complexity Segment Tree

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45. U’ U” U’ U” |P(U’)+ P(U”)|=7 |P(U’)+ P(U”)|=5 Segment Tree

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