CSE 101. Algorithm Design and Analysis Miles Jones Office 4208 CSE Building Lecture 17: Divide and Conquer Design examples.

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1 CSE 101 Algorithm Design and Analysis Miles Jones Office 4208 CSE Building Lecture 17: Divide and Conquer Design examples.

2 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. An interval [a,b] is a set of integers starting at a and ending at b. For example: [16,23] = {16,17,18,19,20,21,22,23} An overlap between two intervals [ a,b] and [c,d] is their intersection. Given two intervals [a,b] and [c,d], how would you compute the length of their overlap?

3 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given two intervals [a,b] and [c,d], how would you compute the length of their overlap? procedure overlap([a,b],[c,d]) [Assume that a c] if b < c: return????????? else: if b d: return?????????? if b > d: return??????????

4 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given two intervals [a,b] and [c,d], how would you compute the length of their overlap? procedure overlap([a,b],[c,d]) [Assume that a c] if b < c: return 0 else: if b d: return b c + 1 if b > d: return d c + 1

5 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. Example: What is the greatest overlap of the intervals: [45,57],[17,50],[10,29],[12,22],[23,51],[31,32],[10,44],[27,35]

6 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. Example: What is the greatest overlap of the intervals: [45,57],[17,50],[10,29],[12,22],[23,51],[31,32],[10,15],[23,35]

7 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. Brute Force algorithm:

8 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. Brute Force algorithm: olap:=0 for i from 1 to n-1 for j from i+1 to n if overlap([a_i,b_i],[a_j,b_j])>olap then olap:=overlap([a_i,b_i],[a_j,b_j]) return olap

9 DIVIDE AND CONQUER EXAMPLES (GREATEST OVERLAP.) Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. Brute Force algorithm olap:=0 for i from 1 to n-1 for j from i+1 to n if overlap([a_i,b_i],[a_j,b_j])>olap then olap:=overlap([a_i,b_i],[a_j,b_j]) return olap What is the runtime? Can we do better?

10 GREATEST OVERLAP. Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. Compose your base case Break the problem into smaller pieces Recursively call the algorithm on the smaller pieces Combine the results

11 COMPOSE YOUR BASE CASE What happens if there is only one interval?

12 COMPOSE YOUR BASE CASE What happens if there is only one interval? if n=1 then return 0

13 BREAK THE PROBLEM INTO SMALLER PIECES The question here is Would knowing the result on smaller problems help with knowing the solution on the original problem? (In this stage, let s keep the combine part in mind.) How would you break the problem into smaller pieces?

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15 BREAK THE PROBLEM INTO SMALLER PIECES Would it be helpful to break the problem into two depending on the starting value?

16 BREAK THE PROBLEM INTO SMALLER PIECES Sort the list and break it into lists each of size n/2. [10,15],[10,29],[12,22],[17,50],[23,51],[27,35],[31,32],[45,57]

17 BREAK THE PROBLEM INTO SMALLER PIECES Sort the list and break it into lists each of size n/2. [10,15],[10,29],[12,22],[17,50],[23,51],[27,35],[31,32],[45,57] Let s assume we can get a DC algorithm to work. Then what information would it give us to recursively call each subproblem?

18 BREAK THE PROBLEM INTO SMALLER PIECES Let s assume we can get a DC algorithm to work. Then what information would it give us to recursively call each subproblem? overlapdc([10,15],[10,29],[12,22],[17,50])=13 overlapdc([23,51],[27,35],[31,32],[45,57])= = =

19 BREAK THE PROBLEM INTO SMALLER PIECES overlapdc([10,15],[10,29],[12,22],[17,50])=13 overlapdc([23,51],[27,35],[31,32],[45,57])=9 Is this enough information to solve the problem? What else must we consider? = =

20 BREAK THE PROBLEM INTO SMALLER PIECES overlapdc([10,15],[10,29],[12,22],[17,50])=13 overlapdc([23,51],[27,35],[31,32],[45,57])=9 The greatest overlap overall may be contained entirely in one sublist or it may be an overlap of one interval from either side = =

21 COMBINE THE RESULTS So far we have split up the set of intervals and recursively called the algorithm on both sides. The runtime of this algorithm satisfies a recurrence that looks something like this. What goes into the O(??? )? T n = 2T n 2 + O(??? )

22 COMBINE THE RESULTS So far we have split up the set of intervals and recursively called the algorithm on both sides. The runtime of this algorithm satisfies a recurrence that looks something like this. What goes into the O(??? )? T n = 2T n 2 + O(??? ) How long does it take to combine. In other words, how long does it take to check if there is not a bigger overlap between sublists?

23 COMBINE What is an efficient way to determine the greatest overlap of intervals where one is red and the other is blue?

24 GREATEST OVERLAP BETWEEN SETS. Let s formalize our algorithm that finds the greatest overlap of two intervals such that they come from different sets sorted by starting point. procedure overlapbetween ([[a 1, b 1 ], [a l, b l ]], [[c 1, d 1 ], [c k, d k ]]) (a 1 a 2 a l c 1 c 2 c k ) if k=0 or l = 0 then return 0.

25 GREATEST OVERLAP BETWEEN SETS. Let s formalize our algorithm that finds the greatest overlap of two intervals such that they come from different sets sorted by starting point. procedure overlapbetween ([[a 1, b 1 ], [a l, b l ]], [[c 1, d 1 ], [c k, d k ]]) (a 1 a 2 a l c 1 c 2 c k ) if k==0 or l == 0 then return 0. minc = c 1 maxb = 0 olap = 0 for i from 1 to l: if maxb < b i : maxb = b i for j from 1 to k: if olap < overlap([minc,maxb],[c k, d k ]): olap = overlap([minc,maxb],[c k, d k ]) return olap runtime?

26 GREATEST OVERLAP. Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. procedure overlapdc ([a 1, b 1 ], [a n, b n ]) if n==1 then return 0.

27 GREATEST OVERLAP. Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. procedure overlapdc ([a 1, b 1 ], [a n, b n ]) if n==1 then return 0. sort([a 1, b 1 ], [a n, b n ]) by a values. mid = n/2 LS = [[a 1, b 1 ], [a mid, b mid ]] RS = [[a mid+1, b mid+1 ], [a n, b n ]] olap1 = overlapdc(ls) olap2 = overlapdc(rs) olap3 = overlapbetween(ls,rs) return max(olap1,olap2,olap3)

28 GREATEST OVERLAP RUNTIME. T n = 2T n 2 + O n log n

29 GREATEST OVERLAP. Given a list of intervals [a_1,b_1], [a_n,b_n] write pseudocode for a D/C algorithm that outputs the length of the greatest overlap between two intervals. procedure overlapdc (sort([a 1, b 1 ], [a n, b n ])) if n==1 then return 0. mid = n/2 LS = [[a 1, b 1 ], [a mid, b mid ]] RS = [[a mid+1, b mid+1 ], [a n, b n ]] olap1 = overlapdc(ls) olap2 = overlapdc(rs) olap3 = overlapbetween(ls,rs) return max(olap1,olap2,olap3)

30 GREATEST OVERLAP RUNTIME. If you sort first then run the overlapdc algorithm you will have Sorting first: T n = 2T n 2 S n = O n log n + O n T n = O n log n Total runtime = O n log n + O n log n = O(n log n)

31 GREATEST OVERLAP (CORRECTNESS) It is clear that the greatest overlap will be from two intervals in the left half, two from the right half, or one from the left and one from the right. Our algorithm finds all three of these values and outputs the max.

32 DIVIDE AND CONQUER EXAMPLES (132-MATCHES) Given a list a 1,, a n of distinct positive integers, write pseudocode for a D/C algorithm that outputs true if there exists a 132-pattern and false if there does not exist a 132-pattern. Example: 4,12,22,28,26,8,5,15,11,17 Brute Force:

33 DIVIDE AND CONQUER EXAMPLES (132-MATCHES) Given a list a 1,, a n of distinct positive integers, write pseudocode for a D/C algorithm that outputs true if there exists a 132-pattern and false if there does not exist a 132-pattern. Example: 4,12,22,28,26,8,5,15,11,17 Brute Force: for i from 1 to n-2: for j from i+1 to n-1: for k from j+1 to n: if a i < a k < a j : return true return false

34 DIVIDE AND CONQUER EXAMPLES (132-MATCHES) Given a list a 1,, a n of distinct positive integers, write pseudocode for a D/C algorithm that outputs true if there exists a 132-pattern and false if there does not exist a 132-pattern. Can anyone think of an example of a sequence of numbers that avoids 132?

35 DIVIDE AND CONQUER EXAMPLES (132-MATCHES) Given a list a_1,,a_n of distinct positive integers, write pseudocode for a D/C algorithm that outputs true if there exists a 132-pattern and false if there does not exist a 132-pattern. Can we do better than O(n 3 )?

36 COMPOSE YOUR BASE CASE What happens if there is less than three integers?

37 COMPOSE YOUR BASE CASE What happens if there is less than 3 integers? if n<3 then return false

38 BREAK THE PROBLEM INTO SMALLER PIECES Usually the smaller pieces are each of size n/2. The question here is Would knowing the result on smaller problems help with knowing the solution on the originial problem? (In this stage, let s keep the combine part in mind.)

39 BREAK THE PROBLEM INTO SMALLER PIECES This problem is a little different than most D/C problems. There is not a fixed size or position where we break the original. Instead we will break the list at the maximum element. break the list at the maximum element. if a m is the max then set L 1 = a 1,, a m 1 L 2 = a m+1,, a n

40 RECURSIVELY CALL THE ALGORITHM ON THE SMALLER PIECES Must we call on both sublists? How many times?

41 COMBINE THE RESULTS Let s assume that we can get the algorithm to work. Then if 132dc(L 1 ) or 132dc(L 2 ) is true then we are done!!! What else must we consider?

42 COMBINE THE RESULTS Let s assume that we can get the algorithm to work. Then if 132dc(L 1 ) or 132dc(L 2 ) is true then we are done!!! What else must we consider? What if there were a 132 occurrence with numbers from L 1 and L 2?

43 COMBINE THE RESULTS Let s assume that we can get the algorithm to work. Then if 132dc(L 1 ) or 132dc(L 2 ) is true then we are done!!! What else must we consider? What if there were a 132 occurrence with numbers from L 1 and L 2? Example: [24, 36, 20, 46, 13, 48, 25, 38, 12]

44 COMBINE THE RESULTS what must happen if there is not a 132-pattern between an element from L_1, a_m, and L_2?

45 COMBINE THE RESULTS if min(l_1)<max(l_2) then there cannot be a 132-pattern between the two lists.

46 132-PATTERNS procedure 132dc (a_1, a_n) if n<3 then return false m:=findmaxindex(a_1, a_n) L_1:=a_1,,a_(m-1) L_2:=a_(m+1),,a_n B1:=132dc(L_1) B2:=132dc(L_2) if min(l_1)<max(l_2) then return true else return (B1 or B2)

47 132-PATTERNS (RUNTIME) procedure 132dc (a_1, a_n) if n<3 then return false m:=findmaxindex(a_1, a_n) L_1:=a_1,,a_(m-1) L_2:=a_(m+1),,a_n B1:=132dc(L_1) B2:=132dc(L_2) if min(l_1)<max(l_2) then return true else return (B1 or B2)

48 132-PATTERNS (RUNTIME) Worst case runtime: Best case runtime: