Divide and conquer algorithms. March 12, 2018 CSCI211 - Sprenkle. What is a recurrence rela&on? How can you compute D&C running &mes?

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1 Objec&ves Divide and conquer algorithms Ø Coun&ng inversions Ø Closest pairs of points March 1, 018 CSCI11 - Sprenkle 1 Review What is a recurrence rela&on? How can you compute D&C running &mes? March 1, 018 CSCI11 - Sprenkle 1

2 Coun&ng Inversions: Divide-and-Conquer Towards a solu+on Assume number represents where item should be in the list, i.e., where it is in someone else s list Should be at position 5 What was the approach so far? March 1, 018 CSCI11 - Sprenkle Coun&ng Inversions: Divide-and-Conquer Divide: separate list into two pieces Conquer: recursively count inversions in each half Combine: count inversions where a i and a j are in different halves, and return sum of three quan&&es Divide: O(1) blue-blue inversions 8 green-green inversions 9 blue-green inversions 5-, 4-, 8-, 8-, 8-7, 10-, 10-9, 10-, 10-7 Total = = Conquer: T(n / ) Combine: Θ(n ) What would make figuring out blue-green inversions easier? March 1, 018 CSCI11 - Sprenkle 4

3 Coun&ng Inversions: Combine Combine: count blue-green inversions Ø Assume each half is sorted Ø Count inversions Ø Merge into sorted whole to maintain sorted invariant What does sorting do for us? What is our algorithm for counting the inversions and merging? Note: these lists are only subsets of the original list March 1, 018 CSCI11 - Sprenkle 5 Coun&ng Inversions: Combine Combine: count blue-green inversions Ø Assume each half is sorted Ø Count inversions Ø Merge into sorted whole to maintain sorted invariant Count: O(n) 1 blue-green inversions: Merge: O(n) We ll run through an example in a bit March 1, 018 CSCI11 - Sprenkle

4 Merge-and-Count(A,B): i=0 j=0 inversions = 0 output = [] while i < A.size and j < B.size: output.append( min(a[i], B[j]) ) if B[j] < A[i]: inversions += A.size i update i or j Append the remainder of the non-exhausted list to the output return inversions and output March 1, 018 CSCI11 - Sprenkle 7 Precondition: A and B are sorted Merge-and-Count(A,B): i=0 (front of list A) j=0 (front of list B) inversions = 0 output = [] while A not empty and B not empty: output.append( min(a[i], B[j]) ) if B[j] < A[i]: inversions += A.size i (remaining elements in A) update i or j (whichever had smaller element) Append the remainder of the non-exhausted list to the output return inversions and output March 1, 018 CSCI11 - Sprenkle 8 4

5 Step Given, A B Total: March 1, 018 CSCI11 - Sprenkle 9 Given, Total: March 1, 018 CSCI11 - Sprenkle 10 5

6 Given, Total: March 1, 018 CSCI11 - Sprenkle 11 Given, Total: March 1, 018 CSCI11 - Sprenkle 1

7 Given, Total: March 1, 018 CSCI11 - Sprenkle 1 Given, Total: March 1, 018 CSCI11 - Sprenkle 14 7

8 Given, Total: March 1, 018 CSCI11 - Sprenkle 15 Given, Total: March 1, 018 CSCI11 - Sprenkle 1 8

9 Given, Total: March 1, 018 CSCI11 - Sprenkle 17 Given, Total: + March 1, 018 CSCI11 - Sprenkle 18 9

10 Given, Total: + March 1, 018 CSCI11 - Sprenkle 19 Given, Total: + March 1, 018 CSCI11 - Sprenkle 0 10

11 Given, Total: + March 1, 018 CSCI11 - Sprenkle 1 Given, Total: + + March 1, 018 CSCI11 - Sprenkle 11

12 Given, Total: + + March 1, 018 CSCI11 - Sprenkle Given, Total: March 1, 018 CSCI11 - Sprenkle 4 1

13 Given, Total: March 1, 018 CSCI11 - Sprenkle 5 Given, Total: March 1, 018 CSCI11 - Sprenkle 1

14 Given, Total: March 1, 018 CSCI11 - Sprenkle 7 Given, Total: March 1, 018 CSCI11 - Sprenkle 8 14

15 Given, first half exhausted Total: March 1, 018 CSCI11 - Sprenkle 9 Given, Total: March 1, 018 CSCI11 - Sprenkle 0 15

16 Given, Total: March 1, 018 CSCI11 - Sprenkle 1 Given, i = Total: March 1, 018 CSCI11 - Sprenkle 1

17 Given, Total: = 1 March 1, 018 CSCI11 - Sprenkle Coun&ng Inversions: Implementa&on Sort-and-Count(L) if list L has one element return 0 and the list L Divide the list into two halves A and B (i A, A) = Sort-and-Count(A) (i B, B) = Sort-and-Count(B) (i, L) = Merge-and-Count(A, B) total_inversions = i A + i B + i return total_inversions and the sorted list L March 1, 018 CSCI11 - Sprenkle 4 17

18 Coun&ng Inversions: Implementa&on Sort-and-Count(L) if list L has one element return 0 and the list L Divide the list into two halves A and B (i A, A) = Sort-and-Count(A) (i B, B) = Sort-and-Count(B) (i, L) = Merge-and-Count(A, B) total_inversions = i A + i B + i return total_inversions and the sorted list L Merge-and-Count Ø Pre-condi&on. A and B are sorted. Ø Post-condi&on. L is sorted. March 1, 018 CSCI11 - Sprenkle 5 Coun&ng Inversions: Implementa&on Sort-and-Count(L) if list L has one element return 0 and the list L Divide the list into two halves A and B (i A, A) = Sort-and-Count(A) (i B, B) = Sort-and-Count(B) (i, L) = Merge-and-Count(A, B) total_inversions = i A + i B + i return total_inversions and the sorted list L Merge-and-Count Ø Pre-condi&on. A and B are sorted. Ø Post-condi&on. L is sorted. Recurrence relation? Runtime of algorithm? March 1, 018 CSCI11 - Sprenkle 18

19 Analysis Recurrence Rela&on: T(n) T(n/) + T(n/) + O(n) èt(n) O(n log n) Sort-and-Count(L) if list L has one element return 0 and the list L Divide the list into two halves A and B (i A, A) = Sort-and-Count(A) T(n/) (i B, B) = Sort-and-Count(B) T(n/) (i, L) = Merge-and-Count(A, B) O(n) total_inversions = i A + i B + i return total_inversions and the sorted list L March 1, 018 CSCI11 - Sprenkle 7 CLOSEST PAIR OF POINTS March 1, 018 CSCI11 - Sprenkle 8 19

20 Computa&onal Geometry Algorithms and data structures for geometrical objects Ø Points, line segments, polygons, etc. Ø Common mo&vator: large data sets à efficiency Some Applica&ons Ø Graphics Ø Robo&cs mo&on planning and visibility problems Ø Geographic informa&on systems (GIS) geometrical loca&on and search, route planning March 1, 018 CSCI11 - Sprenkle 9 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Ø Special case of nearest neighbor, Euclidean MST, Voronoi. Brute force? fast closest pair inspired fast algorithms for these problems March 1, 018 CSCI11 - Sprenkle 40 0

21 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Ø Special case of nearest neighbor, Euclidean MST, Voronoi. Brute force. Check all pairs of points p and q with Θ(n ) comparisons March 1, 018 CSCI11 - Sprenkle 41 Simplify: All Points on a Line How could we solve this problem? What is its running &me? March 1, 018 CSCI11 - Sprenkle 4 1

22 Simplify: All Points on a Line How could we solve this problem? Ø Sort the points Monotonically increasing x/y coordinates No closer points than neighbors in sorted list Ø Step through, looking at the distances between each pair What is its running &me? Ø O(n logn) Why won t this work for D? March 1, 018 CSCI11 - Sprenkle 4 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Ø Special case of nearest neighbor, Euclidean MST, Voronoi. Brute force. Check all pairs of points p and q with Θ(n ) comparisons 1-D version. O(n log n) Ø Easy if points are on a line Assump&on. No two points have same x coordinate to make presenta&on cleaner March 1, 018 CSCI11 - Sprenkle 44

23 Closest Pair of Points: First Aiempt Divide. Sub-divide region into 4 quadrants L Why does this seem to be a natural first step? Any problems with implementing this approach? March 1, 018 CSCI11 - Sprenkle 45 Closest Pair of Points: First Aiempt Divide. Sub-divide region into 4 quadrants Obstacle. Impossible to ensure n/4 points in each piece L March 1, 018 CSCI11 - Sprenkle 4

24 Closest Pair of Points Divide: draw ver&cal line L so that roughly ½n points on each side How do we implement this? L March 1, 018 CSCI11 - Sprenkle 47 Closest Pair of Points Divide: draw ver&cal line L so that roughly ½n points on each side Conquer: find closest pair in each side recursively L 1 1 March 1, 018 CSCI11 - Sprenkle 48 4

25 Closest Pair of Points Divide: draw ver&cal 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 solu&ons seems like Θ(n ) Do we need to check all pairs? L March 1, 018 CSCI11 - Sprenkle 49 Closest Pair of Points Find closest pair with one point in each side, assuming that distance < δ where δ = min(lel_min_dist, right_min_dist) L 1 1 δ = min(1, 1) March 1, 018 CSCI11 - Sprenkle 50 5

26 Closest Pair of Points Find closest pair with one point in each side, assuming that distance < δ. Ø Observa&on: only need to consider points within δ of line L. δ L 1 1 δ = min(1, 1) March 1, 018 CSCI11 - Sprenkle 51 Closest Pair of Points Find closest pair w/ 1 point in each side, assuming that distance < δ. Ø Observa&on: only consider points within δ of line L Ø Sort points in δ-strip by their y coordinate How many points are within that region? 7 L δ = min(1, 1) 1 March 1, 018 CSCI11 - Sprenkle δ 5

27 Closest Pair of Points Find closest pair w/ 1 point in each side, assuming that distance < δ Ø Observa&on: only consider points within δ of line L Ø Sort points in δ-strip by their y coordinate Only checks distances of those within 11 posi&ons in sorted list! 7 L δ = min(1, 1) 1 March 1, 018 CSCI11 - Sprenkle 5 δ Analyzing Cost of Combining Prepare minds to be blown Def. Let s i be the point in the δ-strip, with the i th smallest y- coordinate Claim. If i j 1, then the distance between s i and s j is at least δ Ø What is the distance of the box? Ø How many points can be in a box? Ø When do we know that points are > δ apart? i j ½δ ½δ ½δ March 1, 018 CSCI11 - Sprenkle δ δ 54 7

28 Looking Ahead Wiki: 4.8, PS 7 due Friday Exam handed out on Friday Ø Greedy and D&C Ø Due following Friday March 1, 018 CSCI11 - Sprenkle 55 8