5.4 Closest Pair of Points

Transcription

1 5.4 Closest Pair of Points

2 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric primitive. Graphics, computer vision, geographic information systems, molecular modeling, air traffic control. Special case of nearest neighbor, Euclidean MST, Voronoi. fast closest pair inspired fast algorithms for these problems Brute force. Check all pairs of points p and q with Θ(n 2 ) comparisons. -D version. O(n log n) easy if points are on a line. Assumption. No two points have same x coordinate. to make presentation cleaner 23

3 Closest Pair of Points: First Attempt Divide. Sub-divide region into 4 quadrants. L 24

4 Closest Pair of Points: First Attempt Divide. Sub-divide region into 4 quadrants. Obstacle. Impossible to ensure n/4 points in each piece. L 25

5 Closest Pair of Points Algorithm. Divide: draw vertical line L so that roughly ½n points on each side. L 26

6 Closest Pair of Points Algorithm. Divide: draw vertical line L so that roughly ½n points on each side. Conquer: find closest pair in each side recursively. L

7 Closest Pair of Points Algorithm. Divide: draw vertical line L so that roughly ½n points on each side. Conquer: find closest pair in each side recursively. Combine: find closest pair with one point in each side. Return best of 3 solutions. seems like Θ(n 2 ) L

8 Closest Pair of Points Find closest pair with one point in each side, assuming that distance < δ. L 2 2 δ = min(2, 2) 29

9 Closest Pair of Points Find closest pair with one point in each side, assuming that distance < δ. Observation: only need to consider points within δ of line L. L 2 2 δ = min(2, 2) δ 3

10 Closest Pair of Points Find closest pair with one point in each side, assuming that distance < δ. Observation: only need to consider points within δ of line L. Sort points in 2δ-strip by their y coordinate. 7 L δ = min(2, 2) 2 δ 3

11 Closest Pair of Points Find closest pair with one point in each side, assuming that distance < δ. Observation: only need to consider points within δ of line L. Sort points in 2δ-strip by their y coordinate. Only check distances of those within positions in sorted list! 7 L δ = min(2, 2) 2 δ 32

12 Closest Pair of Points Def. Let s i be the point in the 2δ-strip, with the i th smallest y-coordinate. Claim. If i j 2, then the distance between s i and s j is at least δ. Pf j No two points lie in same ½δ-by-½δ box. Two points at least 2 rows apart have distance 2(½δ). 2 rows 29 3 ½δ ½δ Fact. Still true if we replace 2 with 7. i ½δ δ δ 33

13 Closest Pair Algorithm Closest-Pair(p,, p n ) { Compute separation line L such that half the points are on one side and half on the other side. δ = Closest-Pair(left half) δ 2 = Closest-Pair(right half) δ = min(δ, δ 2 ) Delete all points further than δ from separation line L Sort remaining points by y-coordinate. Scan points in y-order and compare distance between each point and next neighbors. If any of these distances is less than δ, update δ. O(n log n) 2T(n / 2) O(n) O(n log n) O(n) } return δ. 34

14 Closest Pair of Points: Analysis Running time. T(n) 2T( n/2) + O(n log n) T(n) = O(n log 2 n) Q. Can we achieve O(n log n)? A. Yes. Don't sort points in strip from scratch each time. Each recursive call returns two lists: all points sorted by y coordinate, and all points sorted by x coordinate. Sort by merging two pre-sorted lists. T(n) 2T( n/2) + O(n) T(n) = O(n log n) 35

15 5.5 Integer Multiplication

16 37 Integer Arithmetic Add. Given two n-digit integers a and b, compute a + b. O(n) bit operations. Multiply. Given two n-digit integers a and b, compute a b. Brute force solution: Θ(n 2 ) bit operations. * + Add Multiply

17 Divide-and-Conquer Multiplication: Warmup To multiply two n-digit integers: Multiply four pairs of ½n-digit integers. Add two ½n-digit integers, and shift to obtain result. x = 2 n /2 x + x y = 2 n /2 y + y ( ) ( 2 n /2 y + y ) = 2 n x y + 2 n /2 x y + x y xy = 2 n /2 x + x ( ) + x y ( ) T(n) = 4T n/ Θ(n) 23 T(n) = Θ(n2 ) recursive calls add, shift assumes n is a power of 2 38

18 Karatsuba Multiplication To multiply two n-digit integers: Add two ½n digit integers. Multiply three ½n-digit integers. Add, subtract, and shift ½n-digit integers to obtain result. Theorem. [Karatsuba-Ofman, 962] Can multiply two n-digit integers x = 2 n /2 x + x y = 2 n /2 y + y xy = 2 n x y + 2 n /2 x y + x y in O(n.585 ) bit operations. ( ) + x y ( ) + x y = 2 n x y + 2 n /2 (x + x )(y + y ) x y x y A B A C C ( ) + T n /2 ( ) + T( + n /2 ) T(n) T n / recursive calls T(n) = O(n log 2 3 ) = O(n.585 ) + Θ(n) add, subtract, shift 39

19 Karatsuba: Recursion Tree T(n) = if n = 3T(n/2) + n otherwise log 2 n T(n) = n 2 3 = k= ( ) k ( ) +log 2 n = 3n log T(n) n T(n/2) T(n/2) T(n/2) 3(n/2) T(n/4) T(n/4) T(n/4) T(n/4) T(n/4) T(n/4) T(n/4) T(n/4) T(n/4) 9(n/4) T(n / 2 k ) 3 k (n / 2 k ) T(2) T(2) T(2) T(2) T(2) T(2) T(2) T(2) 3 lg n (2) 4