Divide and Conquer Approach
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1 Divide and Conquer Approac
2 Deiverabes Divide and Conquer Paradigm nteger Mutipication Strassen Matrix Mutipication Cosest Pair of points nfinite Wa Probem 6/7/01 8:58 PM gdeepak.com
3 Divide and Conquer 3 steps First step is caed Divide wic is dividing te given probems into smaer sub probems wic are identica to te origina probem and aso tese sub probems may or may not be of same size. Second step is caed Conquer were we sove tese sub probems recursivey. Tird step is caed Combine were we combine soutions of te sub probems to get soution for te origina probem 6/7/01 8:58 PM gdeepak.com 3
4 Points to be taken care of Tresod ower imit, beow wic te probem can t be subdivided or were you want to ande te probem witout furter divisions. Te size of te sub instances into wic an instance is spit. Number of suc nstances Te Agoritm used to combine sub-soutions. 6/7/01 8:58 PM gdeepak.com 4
5 Divide and Conquer- few exampes n Binary searc Combine step is not tere because eement is found during recursive iterations to find te eement and it is aso a specia case were af of te probem vanises after eac iteration. Quick sort aso does not ave a combine step because te eements are sorted during recursive impementation of divide step. Merge Sort does not ave a Divide step because eements are simpy compared by array index. 6/7/01 8:58 PM gdeepak.com 5
6 Genera Agoritm Divide-conq(p, q) if Sma(p, q) ten return G(p, q); ese m Divide(p, q); [ p m < q ] return Combine(Divide-conq(p, m), Divide-conq(m+1, q)); end 6/7/01 8:58 PM gdeepak.com 6
7 Suitabiity of D&C Divide and conquer is not suitabe were te soution of size n depends upon n sub-soutions, eac of size (n-1). Overapping Sub probems - Were sub probems ave a dependency on eac oter and tere is no easy way to merge tose sub probems and merging wi take significant time as soving te origina probem. 6/7/01 8:58 PM gdeepak.com 7
8 nteger Mutipication Given two n-digit integers a and b, compute a b. Brute force soution: O(n ) bit operations 6/7/01 8:58 PM gdeepak.com 8
9 nteger Mutipication Divide step: Spit and into ig-order and ow-order bits Define * by mutipying te parts and adding: T(n) = 4T(n/) + n, wic impies T(n) is O(n ). gdeepak.com 9 n n / / n n n n n / / / / ) )*( ( * 6/7/01 8:58 PM
10 mproved nteger Mutipication T(n) = 3T(n/) + n, wic impies T(n) is O(n og 3 ), by te Master Teorem. Tus, T(n) is O(n ). gdeepak.com 10 n n n n n n / / / ) ( ] ) [( ] ) )( [( * 6/7/01 8:58 PM
11 Matrix Mutipication C 11 C 1 = A 11 A 1 * B 11 B 1 C 1 C A 1 A B 1 B C i, j N k 1 a i, k b k, j TusT ( N) N N N i1 j1 k 1 c cn 3 O( N 3 ) 6/7/01 8:58 PM gdeepak.com 11
12 Strassen Mutipication C 11 = a 11 b 11 + a 1 b 1 C 1 = a 11 b 1 + a 1 b C 1 = a 1 b 11 + a b 1 C = a 1 b 1 + a b x matrix mutipication can be done in 8 mutipications T(n)= 8T(n/) + n = ( og 8 = 3 ) Strassen sowed tat x matrix mutipication can be accompised in 7 mutipication and 18 additions or subtractions T(n) = 7T(n/) + n = ( og 7 =.807 ) Best Known is Coppersmit-Winoguard, 1987 O(n.376 ) 6/7/01 8:58 PM gdeepak.com 1
13 Strassen Matrix Mutipication P1 = (A11+ A)(B11+B) P = (A1 + A) * B11 P3 = A11 * (B1 - B) P4 = A * (B1 - B11) P5 = (A11 + A1) * B P6 = (A1 - A11) * (B11 + B1) P7 = (A1 - A) * (B1 + B) 6/7/01 8:59 PM gdeepak.com 13
14 C11 = P1 + P4 - P5 + P7 C1 = P3 + P5 C1 = P + P4 C = P1 + P3 - P + P6 Strassen Mutipication C 11 = P 1 + P 4 - P 5 + P 7 =(A 11 +A )(B 11 +B )+A *(B 1 -B 11 )-(A 11 +A 1 )*B +(A 1 -A )*(B 1 +B ) = A 11 B 11 + A 11 B + A B 11 + A B + A B 1 A B 11 -A 11 B -A 1 B + A 1 B 1 + A 1 B A B 1 A B = A 11 B 11 + A 1 B 1 6/7/01 8:59 PM gdeepak.com 14
15 Deeper ook A11 A X C 11 = X X /7/01 8:59 PM gdeepak.com 15
16 Matrix Mutipication f te matrices A, B are not of type n x n we fi te missing rows and coumns wit zeros. 6/7/01 8:59 PM gdeepak.com 16
17 Counting nversions Many mportant Appications ike Page Ranking etc. Brute Force is O(n ) How to combine. Witout any trick it wi not give any better compexity. Five inversions in First af, 8 inversions in nd af and 9 inversions between two aves. 6/7/01 8:59 PM gdeepak.com 17
18 Divide and Conquer-How it wi work Divide: separate ist into two pieces. O(1) Conquer: recursivey count inversions in eac af. T(n/) Combine: count inversions were a i and a j are in different aves, and return sum of tree quantities. 6/7/01 8:59 PM gdeepak.com 18
19 Beautifu dea Combine: Assume eac af is sorted. Count inversions were a i and a j are in different aves. Merge two sorted aves into sorted woe in te nd ist comes before 6 eements of te first ist so it as 6 inversions and so on /7/01 8:59 PM gdeepak.com 19
20 Cosest Pair Probem Given a set of n points in a pane were P(i)= (x(i),y(i)) Distance between two points is (xi xj) +(y i yj) Brute force agoritm wi be O(n ) if we cacuate distance between a te points and ten find te cosest pair. 6/7/01 8:59 PM gdeepak.com 0
21 Cosest Pair tecnique Divide: draw vertica ine L so tat rougy ½(N) points on eac side. Conquer: find cosest pair in eac side recursivey. Combine: find cosest pair wit one point in eac side. Return best of 3 soutions. 6/7/01 8:59 PM gdeepak.com 1
22 Cosest Pair Find cosest pair wit one point in eac side, assuming tat distance <δ. Observation: ony need to consider points witin δ of ine L. Sort points in δ -strip by teir y coordinate. Ony ceck distances of tose witin 11 positions in sorted ist. 6/7/01 8:59 PM gdeepak.com
23 Cosest Pair Probem Let S i be te point in te δ strip, wit te it smaest y- coordinate. f i-j 1 ten distance between S i and So is at east δ No two points ie in same 1 δ by 1 δ box. Two points at east two rows apart ave distance ( 1 δ) i 1 δ 6/7/01 8:59 PM gdeepak.com 3
24 Cosest Pair agoritm Cosest-Pair(p 1,, p n ) { Compute separation ine L suc tat af te points are on one side and af on te oter side. δ 1 = Cosest-Pair(eft af) δ = Cosest-Pair(rigt af) δ = min(δ 1, δ ) Deete a points furter tan δ from separation ine L Sort remaining points by y-coordinate. Scan points in y-order and compare distance between eac point and next δ neigbours. f any of tese distances is ess tan δ, update δ. return δ. } T(n) Tn/ O(n og n) T(n) O(n og n) 6/7/01 8:59 PM gdeepak.com 4
25 Furter improvement Sort te set of points by y-coordinate before we start. Wenever we spit a point set, we can run troug ist sorted by y- coordinate and create a new ist for eac part, sorted by y-coordinates. Recurrence becomes T (n) = T (n/) + n and T (1) = 1. Soution: T (n) = O(ngn). 6/7/01 8:59 PM gdeepak.com 5
26 nfinite Wa Probem You ave an infinite wa on bot sides were you are standing and it as a gate somewere in one direction, you ave to find out te gate, No design-nfinite time ncrementa design- you go one step in one direction, come back go to oter direction one step, come back and ten go steps in oter direction. 6/7/01 8:59 PM gdeepak.com 6
27 : : n-1+.(n-1)+n-1 n+n 4 i+3n 4n(n-1)/+3n=n +n Quadratic time soution 6/7/01 8:59 PM gdeepak.com 7
28 Linear time soution You go 0 step in one direction, come back go 1 step in oter ten 1 direction in one way ten come back and ten up to k : K-1 +. K-1 + K-1 3. K 4( K-1 + K- + 1)+ 3. K 4( K -1)+ 3. K = 7. K -4 = 7N-4 6/7/01 8:59 PM gdeepak.com 8
29 Questions, Comments and Suggestions 6/7/01 8:59 PM gdeepak.com 9
30 Question 1 Divide step is te dominating operation and Combine step is te dominant operation respectivey in foowing A) Merge Sort, Quick Sort B) Quick Sort, Merge Sort C) Bubbe Sort, Counting Sort D) Radix Sort, Seection Sort 6/7/01 8:59 PM gdeepak.com 30
31 Question Wy Seection Sort can t be converted to Divide and Conquer Mecanism 6/7/01 8:59 PM gdeepak.com 31
32 Question 3 Wat are te deciding factors in determining te termination of furter divisions of te probem into sub probems. (Give two at east) 6/7/01 8:59 PM gdeepak.com 3
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