The document summarizes a seminar on recursion, divide and conquer algorithms, and two specific algorithms: finding medians and closest pair of points. It introduces recursion as a basic concept. Divide and conquer is described as breaking a problem into smaller subproblems, solving them recursively, and combining the solutions. Finding medians and closest pair of points are then discussed as examples solved using divide and conquer. Finding medians uses random selection of a pivot value to partition the data set. Closest pair of points works by recursively finding the closest pairs in each half of the data set and using geometric constraints to limit the search space for the overall closest pair.

1. POSCAT Seminar 5 :
Recursion and Divide & Conquer
yougatup @ POSCAT
1

2. Topic
 Topic today
− Recursion
• Basic Concept
− Divide & Conquer
• Basic Concept
• Finding Medians
• Closest Pair
• Implementation
POSCAT Seminar 1-2
2 July 2014
yougatup
By the way, do you solve all the problems ?

3. You should already know
what recursion is !
POSCAT Seminar 1-3
2 July 2014
yougatup

4. Divide & Conquer
 Problem Solving Paradigm
1. Break the problem into subproblems : smaller instances
2. Solve them recursively
3. Combine the solutions to get the answer to the problem.
 Calculation of time complexity is not trivial
− We need Master theorem to calculate it
− Recurrence relation
− Not that important now
POSCAT Seminar 1-4
2 July 2014
yougatup

5. Finding Medians and Selection
 Problem
Find the k-th smallest element in S, where S is a set of numbers
POSCAT Seminar 1-5
2 July 2014
yougatup

6. Finding Medians and Selection
 Problem
Find the k-th smallest element in S, where S is a set of numbers
Naïve approach : Sort it ! O(𝑛 log 𝑛)
POSCAT Seminar 1-6
2 July 2014
yougatup

7. Finding Medians and Selection
 Problem
Find the k-th smallest element in S, where S is a set of numbers
Naïve approach : Sort it ! O(𝑛 log 𝑛)
Divide & Conquer : Choose 𝑣 and split S into 𝑆 𝐿, 𝑆 𝑣, 𝑆 𝑅 !
POSCAT Seminar 1-7
2 July 2014
yougatup

8. Finding Medians and Selection
 Problem
Find the k-th smallest element in S, where S is a set of numbers
Naïve approach : Sort it ! O(𝑛 log 𝑛)
Divide & Conquer : Choose 𝑣 and split S into 𝑆 𝐿, 𝑆 𝑣, 𝑆 𝑅 !
POSCAT Seminar 1-8
2 July 2014
yougatup

9. Finding Medians and Selection
 Problem
Find the k-th smallest element in S, where S is a set of numbers
Naïve approach : Sort it ! O(𝑛 log 𝑛)
Divide & Conquer : Choose 𝑣 and split S into 𝑆 𝐿, 𝑆 𝑣, 𝑆 𝑅 !
We have to choose 𝑣 nicely ! … How ?
POSCAT Seminar 1-9
2 July 2014
yougatup

10. Finding Medians and Selection
 Problem
Find the k-th smallest element in S, where S is a set of numbers
Naïve approach : Sort it ! O(𝑛 log 𝑛)
Divide & Conquer : Choose 𝑣 and split S into 𝑆 𝐿, 𝑆 𝑣, 𝑆 𝑅 !
We have to choose 𝑣 nicely ! … How ?
POSCAT Seminar 1-10
2 July 2014
yougatup
Randomly !

11. Finding Medians and Selection
 Problem
If we’re lucky, then the partition goes to small very fast. If not,
It will take O(𝑛2
). Can we trust random choice of 𝑣 ?
POSCAT Seminar 1-11
2 July 2014
yougatup

12. Finding Medians and Selection
 Problem
If we’re lucky, then the partition goes to small very fast. If not,
It will take O(𝑛2
). Can we trust random choice of 𝑣 ?
Yes! because the worst case is extremely unlike to occur !
There are 50% chance of being good, that is, the Sublists 𝑆 𝐿 and
𝑆 𝑅 have size at most ¾ of that of S.
POSCAT Seminar 1-12
2 July 2014
yougatup

13. Finding Medians and Selection
 Problem
If we’re lucky, then the partition goes to small very fast. If not,
It will take O(𝑛2
). Can we trust random choice of 𝑣 ?
Therefore, the expected running time T(𝑛) is
T(𝑛) ≤ T(3𝑛/4) + O(𝑐 ⋅ 𝑛)
where 𝑐 is the average number of consecutive splits to find a
good split. Clearly 𝑐 = 2. So we can conclude that T(𝑛) = O(𝑛)
POSCAT Seminar 1-13
2 July 2014
yougatup

14. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
POSCAT Seminar 1-14
2 July 2014
yougatup

15. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
Brute force approach : Check all pairs of points. O(𝑛2)
POSCAT Seminar 1-15
2 July 2014
yougatup

16. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
Divide & Conquer approach
1. Split points into two groups
2. Find closest pair of points for each group
3. Combine the solutions !
POSCAT Seminar 1-16
2 July 2014
yougatup

17. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
1. Split points into two group
POSCAT Seminar 1-17
2 July 2014
yougatup

18. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
1. Split points into two group
POSCAT Seminar 1-18
2 July 2014
yougatup

19. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
2. Find closest pair of points for each group
POSCAT Seminar 1-19
2 July 2014
yougatup

20. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
3. Combine the solutions ?
POSCAT Seminar 1-20
2 July 2014
yougatup

21. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
3. Combine the solutions ?
POSCAT Seminar 1-21
2 July 2014
yougatup
Is it enough to choose
small one between two solutions ?

22. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
3. Combine the solutions ?
POSCAT Seminar 1-22
2 July 2014
yougatup
Is it enough to choose
small one between two solutions ?
We didn’t consider it

23. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
3. Combine the solutions ?
POSCAT Seminar 1-23
2 July 2014
yougatup
Then, do we have to consider all the cases
when a pair consists of a point in the left
and a point in the right ?

24. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
3. Combine the solutions ?
POSCAT Seminar 1-24
2 July 2014
yougatup
Then, do we have to consider all the cases
when a pair consists of a point in the left
and a point in the right ?
No! because we have partial solution !

25. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
Idea : If a closer pair is exist, then it must be in the grey region
POSCAT Seminar 1-25
2 July 2014
yougatup
𝛿
𝛿 𝛿

26. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
Idea : If a closer pair is exist, then it must be in the grey region
POSCAT Seminar 1-26
2 July 2014
yougatup
𝛿
𝛿 𝛿 Still, worst case can happen

27. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
Claim. If 𝑖 − 𝑗 ≥ 12, then the distance
between 𝑝𝑖 and 𝑝𝑗 is at least 𝛿
• No two points lie in same 𝛿/2-by-𝛿/2 box.
• Two points at least 2 rows apart
have distance ≥ 2(𝛿/2)
Therefore, it is sufficient to consider 12 points
above me !
POSCAT Seminar 1-27
2 July 2014
yougatup

28. Closest pair of points
 Problem
Given 𝑛 points in the plane, find a pair with smallest Euclidean
distance between them.
Claim. If 𝑖 − 𝑗 ≥ 12, then the distance
between 𝑝𝑖 and 𝑝𝑗 is at least 𝛿
• No two points lie in same 𝛿/2-by-𝛿/2 box.
• Two points at least 2 rows apart
have distance ≥ 2(𝛿/2)
Therefore, it is sufficient to consider 12 points
above me ! ∴ T(𝑛) = 2T(𝑛/2) + O(𝑛) = O(𝒏 𝐥𝐨𝐠 𝒏)
POSCAT Seminar 1-28
2 July 2014
yougatup