HKOI 2012 (Junior) Q4 - Penicillin

1. HKOI 2012 (Junior)Q4 - Penicillin Gary Wong For any question, please ask via Email: garywong612@gmail.com MSN: gary_wong612@hotmail.com

2. For your interest (if any) • The history in the problem is really true! • Penicillin refers to a group of antibiotics • Examples: • Penicillin G • Amoxicillin • Ampicillin • … • Bacteria and viruses are 2 different things

3. Problem Statement • Each of N points (xi, yi) can “grow” into an “unclean” square with side 2t at time t, with (xi, yi) as the “centre” • A special point P(h, k) can “grow” into a square with side 2ts at time t • “Growth” is bounded by a rectangle with dimension P x Q • Area covered by square centred at P will be always clean

4. Problem Statement • Find the area of unclean area • N.B.: Overlapped area should be counted repeatedly • Constraints: • 50% • N <= 50 • P, Q, s, t <= 100 • 100% • 1 <= N <= 10,000 • 1 <= P, Q, s, t <= 100,000 • 0 <= xi, h <= P • 0 <= yi, k <= Q

6. Statistics • Max. score: 100 • No. of max: 2 • Std dev: 27.38 • Mean: 5.29 • Disappointed…

7. Statistics • Disappointed…

8. Solution – 50% • N <= 50 • P, Q, s, t <= 100 • How about declaring a 2D array to mimic the agar plate? • Simulate the growth step by step! • Count how many times each cell was covered, clear all sterile part • Complexity? • Very roughly, O(NPQt) • In fact much less than this

9. Solution – 100% • 1 <= N <= 10,000 • 1 <= P, Q, s, t <= 100,000 • O(NPQt) cannot work anymore… • Note that • Each point is independent from each other, because overlapped areas are counted repeatedly • If you know how to do for one point, same for other (N-1) points

10. Solution – 100% • The problem reduces to: • Given 2 “growing” squares, find the brown area without covered by green square

11. Solution – 100% • Consider the brown square • Top-left corner: (xi – t, yi – t) • Bottom-right corner: (xi + t, yi + t)

12. Solution – 100% • But what if it reaches/exceeds boundaries? • Actual top-left corner • (max(0, xi – t), max(0, yi – t)) • Actual bottom-right corner • (min(P, xi + t), min(Q, yi + t))

13. Solution – 100% • Similarly, for green square, • Top-left corner • (max(0, h– st), max(0, k – st)) • Bottom-right corner • (min(P, h + st), min(Q, k + st))

14. Solution – 100% • Next task: how to find intersection area between the 2 squares?

15. Solution – 100% • Consider the corners of the intersection area! (a1, b1) (a2, b2) (c2, d2) (c1, d1)

16. Solution – 100% • For the intersection area, • Top-left corner • (max(a1,a2), max(b1,b2)) • Bottom-right corner • (min(c1,c2), min(d1,d2)) (a1, b1) (a2, b2) (c1, d1) (c2, d2)

17. Solution – 100% • How to detect the case of “no intersection”?

18. Solution – 100% • Area of brown square minus area of intersection • Do this for N times • Complexity? • O(N)

19. Common mistakes • Areas were not counted repeatedly • Cannot even pass the sample input in the problem • Misunderstand that penicillin can kill only one layer of bacteria • Forgot to use 64-bit integer type to store the answer

20. Any question?