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Agenda

The Quarter-State Sequence (Q-Sequence) to Represent the Floorplan and Applications to Layout Optimization. Sakanushi, K.; Kajitani, Y.; Circuits and Systems, 2000. IEEE APCCAS 2000. The 2000 IEEE Asia-Pacific Conference on , 4-6 Dec. 2000. Agenda. Introduction Preliminary

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Agenda

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  1. The Quarter-State Sequence(Q-Sequence) to Represent the Floorplan and Applications to Layout Optimization Sakanushi, K.; Kajitani, Y.;Circuits and Systems, 2000. IEEE APCCAS 2000. The 2000 IEEE Asia-Pacific Conference on , 4-6 Dec. 2000

  2. Agenda • Introduction • Preliminary • Coding of a Floorplan • Decoding of a Q-Sequence • Number of Floorplans • Observations • Conclusion

  3. a a b e e c b c d d f Introduction • Floorplan • Slicing floorplan • Non-slicing floorplan

  4. Agenda • Introduction • Preliminary • Coding of a Floorplan • Decoding of a Q-Sequence • Number of Floorplans • Observations • Conclusion

  5. a b e c d f Preliminary • Segs • Inside-segs • Wall-segs

  6. a b a b e e c d c d f f Preliminary • Equivalent with respect to the • Room-seg relation • Room-room relation a b

  7. Preliminary • T-junction • Prime Seg • Vertical seg • Horizontal seg a a

  8. Agenda • Introduction • Preliminary • Coding of a Floorplan • Decoding of a Q-Sequence • Number of Floorplans • Observations • Conclusion

  9. Coding of a Floorplan • Tail room • Associated room • Next room i a b a r j c r i j k k

  10. Coding of a Floorplan • Q-state of r i a b a r j c r i j k k rRRR rBBB

  11. Sub Q-Sequence • r ← left-top room • while ( r ≠ tail room ) • Attach the Q-state of r • r ← next room of r Sub Q-Sequence: a b e aRbBBcRdRReBf a R b BB c R d RR e B f c d f

  12. a b e c d f Pre-Sequence X Pre-Sequence X: R R B B B

  13. a b e c d f Q-Sequence • Q-sequence • pre-sequence X: RRBBB • sub Q-sequence: aRbBBcRdRReBf Q-Sequence: RRBBBaRbBBcRdRReBf

  14. Agenda • Introduction • Preliminary • Coding of a Floorplan • Decoding of a Q-Sequence • Number of Floorplans • Observations • Conclusion

  15. Decoding of a Q-Sequence • Procedure • Parent (Q to RQ) • Delete all the B’s. (The resultant is called the RQ-sequence.) • Replace every R with an open parenthesis “(”. • Replace every room name with a close parenthesis “)”. • Parent (Q to BQ)

  16. Parent (Q to RQ/Q to BQ) ( ( ) ( ) ) ( ) ( ( ) ) Parent (Q to RQ) RRBBBaRbBBcRdRReBf ( ( ( ) ) ( ( ) ) ) ( ) Parent (Q to BQ)

  17. Theorem 1 • A sequence consisting n R’s and n B’s and n room names is the Q-sequence if and only if following two conditions are satisfied: • Single: The subsequence between two rooms contains one or more single labels of R and B. • Parent: The parenthesis system by RQ-sequence or BQ-sequence is consistent.

  18. Agenda • Introduction • Preliminary • Coding of a Floorplan • Decoding of a Q-Sequence • Number of Floorplans • Observations • Conclusion

  19. Exact Number • Label the room names in reverse order of their appearance in the Q-sequence. • NR(x): the number of letters R in the right of x • NB(x): the number of letters B in the right of x • Consider a given Q-sequence: RRRBBB9B8RRR7B6R5BB4BB3R2R1 • NR(1) = 0, NR(2) = 1, …, NR(9) = 6 • NB(1) = 0, NB(2) = 0, …, NB(9) = 6

  20. Theorem 1’ • Single’: one of NR(x) - NR(x-1) and NB(x) - NB(x-1) is positive and the other is zero • Parent’: NR(x) < x, NB(x) < x

  21. Exact Number • C(n, r, b): the set of all Q-Sequence of satisfying NR(n) = r and NB(n) = b. • 0 ≤ b ≤ n-1, 0 ≤ r ≤ n-1 • |C(n, r, b)|: the number of distinct elements of the set. • |C(n, 0, b)| = |C(n, r, n-1)| = 1, for any n • |C(n, n-1, b)| = |C(n, r, 0)| = 1, for any n

  22. Exact Number • F(n): the number of distinct Q-sequences n-1 n-1 • F(n) = Σ Σ |C(n, r, b)| r=0 b=0 • C(n, r, b) is the union of • C(n-1, r’, b) over r’ < r or • C(n-1, r, b’) over b’ < b • |C(1, 0, 0)| = 1 (RB1)

  23. Exact Number 1 : n = 1, r = b = 0 0 : r, b ≥ n |C(n, r, b)| = r-1 b-1 Σ|C(n-1,r’,b)|+Σ|C(n-1,r,b’)| r’=0 b’=0 : otherwise

  24. Upper Bound • The variety of the parent system: • n R’s(or B’s) and n room names • Catalan(n) = 1/(n+1) * 2nCn ~ 22n / √(pi * n) • Reconstruct the Q-sequence with an RQ-sequence: • n-1Cn-m≤ n-1C(n-1)/2 • The variety of room names: • n!

  25. Upper Bound • The variety of the Q-sequence • F(n) ≤ Catalan(n) * n-1C(n-1)/2 * n! ≤ Catalan(n) * 2n-1 * n! ≤ 22n * 2n-1 * n! = 23n-1 * n! Upper Bound of Variety of Q-Sequence

  26. Agenda • Introduction • Preliminary • Coding of a Floorplan • Decoding of a Q-Sequence • Number of Floorplans • Observations • Conclusion

  27. Observations • BI(n): Upper bound of slicing floorplans • Catalan(n) * 2n-1 * n! • SL(n): Number of distinct slicing structure floorplans • SP(n): Number of distinct floorplans of sequence-pair • (n!)2 • SL(n) ≤ F(n) ≤ BI(n)

  28. Comparison of Sizes of Solution Spaces

  29. Agenda • Introduction • Preliminary • Coding of a Floorplan • Decoding of a Q-Sequence • Number of Floorplans • Observations • Conclusion

  30. Conclusion • The solution space is close to that of slicing structure than other researches. • Given a Q-sequence, the vertical (horizontal) constraint graph of the corresponding floorplan can be constructed directly. • Deletion or insertion of a room can be done in a constant time.

  31. Conclusion • Boundary check: listing the rooms that are adjacent to a specified seg is possible in a linear time.

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