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Toothpicks

Toothpicks. HKOI 2012 Senior Q2. Problem. N non-overlapping rectangles Using toothpicks to form the shape . Solution. Lets view it in a simple way. Break to 1x1 blocks. Compute the perimeters of the squares. Solution. Compute the perimeter. 4 x (number of squares) ?. Solution.

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Toothpicks

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  1. Toothpicks HKOI 2012 Senior Q2

  2. Problem • N non-overlapping rectangles • Using toothpicks to form the shape

  3. Solution • Lets view it in a simple way Break to 1x1 blocks Compute the perimeters of the squares

  4. Solution • Compute the perimeter 4 x (number of squares) ?

  5. Solution • Compute the perimeter Overlapped.. We counted twice 4 x (number of squares) ?

  6. Solution • Compute the perimeter Overlapped.. We counted twice 4 x (number of squares) – Overlapped lines

  7. Solution • Counting overlapped length xx.x xxxx Draw the squares on a ‘map’ A 2D array!

  8. Solution • Counting overlapped length xx.x xxxx Count how many adjacency ‘x’s! xx.x xxxx xx.x xxxx xx.x xxxx xx.x xxxx xx.x xxxx xx.x xxxx xx.x xxxx

  9. Solution • How about negative coordinates? Pascal: Easy. var a:array[-100..100][-100..100] of boolean; Shift all rectangles by adding 100 to them! C/C++?

  10. Algorithm 1 • Break the rectangles to 1x1 squares • Draw the squares on a map • Count the overlapped length Time complexity? Number of 1x1 squares Depend on size of rectangles In 50% test cases, absolute values of coordinates is at most 100 Work for 50% test cases

  11. Speed up • Do the easy part first - fill the inner part of the rectangle m Number of toothpicks needed for a n x m rectangle = 2nm + n + m n

  12. Speed up • How about the overlapped length? First check which lines are overlapping..

  13. Speed up • How about the overlapped length? First check which lines are overlapping..

  14. Speed up • How about the overlapped length? Compute the overlapped length

  15. Solution • Break each rectangle into 2 vertical line and 2 horizontal line For each pair of line, compute their overlapped length

  16. Algorithm 2 • Sum the toothpicks needed for inner part • Break N rectangles to 2N horizontallines and 2N vertical lines • For each pair of lines, compute their overlapped length (if any), subtract it from answer Time complexity: O(N2)

  17. Extra – We can do faster For each pair of line, compute their overlapped length Do we really need to go through all pair of lines?

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