1. Lecture 5 • Today, how to solve recurrences • We learned “guess and proved by induction” • We also learned “substitution” method • Today, we learn the “master theorem” • More divide and conquer: • closest pair problem • matrix multiplication

2. Master Theorem Theorem 4.1 (CLRS, Theorem 4.1) Let a ≥ 1 and b > 1 be constants. Let f(n) be a function and let T(n) be defined on the nonnegative integers by T(n) = aT(n/b) + f(n). Then

3. Note • Only apply to a particular family of recurrences. • f(n) is positive for large n. • Key is to compare f(n) with nlog_b a • Case 2, more general is f(n) = Θ( nlog_b a lgkn). Then the result is T(n) = Θ( nlog_b a lgk+1n). • Sometimes it does not apply. Ex. T(n) = 4T(n/2) + n2 /logn.

4. Proof ideas of Master Theorem • Consider a tree with T(n) at the root, and apply the recursion to each node, until we get down to T(1) at the leaves. The first recursion is T(n) = aT(n/b) + f(n), so assign a cost of f(n) to the root. At the next level we have “a” nodes, each with a cost of T(n/b). When we apply the recursion again, we get a cost of af(n/b) for all of these. At the next level we have a2 nodes, each with a cost of T(n/b2). We get a cost of a2f(n/b2). We continue down to T(1) at the leaves. There are alog_b n leaves and each costs Θ(1), which gives Θ(alog_b n). The total cost associated with f is Σ 0 ≤ i ≤ log_b n - 1 ai f(n/bi). • Thus T(n) = Θ(n log_b a) + Σ 0 ≤ i ≤ (log_b n) - 1 ai f(n/bi). • The three cases now come from deciding which term is dominant. In case (1), the Θ term is dominant. In case (2), the terms are roughly equal (but the second term has an extra lg n factor). In case (3), the f(n) term is dominant. The details are somewhat painful, but can be found in CLRS, pp. 76-84.

5. f(n) af(n/b) h = logbn a2f(n/b2) … #leaves = ah = alogbn = nlogba nlogbaT(1) Idea of master theorem Recursion tree: f(n) a … f(n/b) f(n/b) f(n/b) a … f(n/b2) f(n/b2) f(n/b2) … T(1)

6. Three common cases Compare f(n) with nlogba: • f(n) = O(nlogba – e) for some constant e > 0. • f(n) grows polynomially slower than nlogba (by an ne factor). • Solution: T(n) = Q(nlogba) .

7. These functions increase from top to bottom geometrically, hence we only need to have the last bottom term Idea of master theorem Recursion tree: f(n) f(n) a … af(n/b) f(n/b) f(n/b) f(n/b) a h = logbn … a2f(n/b2) f(n/b2) f(n/b2) f(n/b2) … … CASE 1: The weight increases geometrically from the root to the leaves. The leaves hold a constant fraction of the total weight. nlogbaT(1) T(1) Q(nlogba)

8. Case 2 Compare f(n) with nlogba: • f(n) = Q(nlogba lgkn) for some constant k³ 0. • f(n) and nlogba grow at similar rates. • This is clear for k=0. For k>0, the intuition is that lgk n factor remain for constant fraction of levels, hence sum to the following • Solution: T(n) = Q(nlogba lgk+1n) .

9. Idea of master theorem All levels same Recursion tree: f(n) f(n) a … af(n/b) f(n/b) f(n/b) f(n/b) a h = logbn … a2f(n/b2) f(n/b2) f(n/b2) f(n/b2) … … CASE 2: (k = 0) The weight is approximately the same on each of the logbn levels. nlogbaT(1) T(1) Q(nlogbalgn)

10. Case 3, c<1, akf(n/bk) geometrically decreases hence = Θ(f(n)) Compare f(n) with nlogba: • f(n) = W(nlogba + e) for some constant e > 0. • f(n) grows polynomially faster than nlogba (by an ne factor), • andf(n) satisfies the regularity conditionthat af(n/b) £cf(n) for some constant c< 1. • Solution: T(n) = Q(f(n)) .

11. Idea of master theorem af(n/b)<(1-ε)f(n) Recursion tree: f(n) f(n) a … af(n/b) f(n/b) f(n/b) f(n/b) a h = logbn … a2f(n/b2) f(n/b2) f(n/b2) f(n/b2) … … CASE 3: The weight decreases geometrically from the root to the leaves. The root holds a constant fraction of the total weight. nlogbaT(1) T(1) Q(f(n))

12. Examples for the Master Theorem • The Karatsuba recurrence has a = 3, b = 2, f(n) = cn. Then case 1 applies, and so T(n) = Θ(n l og_2 3 ), as we found. • The mergesort recurrence has a = 2, b = 2, f(n) = n. Then case 2 applies, and so T(n) = Θ(n lg n). • Finally, a recurrence like T(n) = 3T(n/2) + n2 gives rise to case 3. In this case f(n) = n2, so 3f(n/2) = 3 (n/2)2 = (3/4) n2 ≤ c n2 for c = 3/4, and so T(n) = Θ(n2). • Note that the master theorem does not cover all cases. In particular, it does not cover the case T(n) = 2 T(n/2) + n / lg n since then the only applicable case is case 3, but then the inequality involving f does not hold.

13. Closest pair problem • Input: • A set of points P = {p1,…, pn} in two dimensions • Output: • The pair of points pi, pj that minimize the Euclidean distance between them.

14. Distances • Euclidean distance

15. Closest Pair Problem

16. Closest Pair Problem

17. Divide and Conquer • O(n2) time algorithm is easy • Assumptions: • No two points have the same x-coordinates • No two points have the same y-coordinates • How do we solve this problem in 1 dimension? • Sort the number and walk from left to right to find minimum gap.

18. Divide and Conquer • Divide and conquer has a chance to do better than O(n2). • We can first sort the points by their x-coordinates and sort also by y-coordinates

19. Closest Pair Problem

20. Divide and Conquer for the Closest Pair Problem Divide by x-median

21. Divide L R Divide by x-median

22. Conquer L R Conquer: Recursively solve L and R

23. Combination I L R d2 Take the smaller one of d1 , d2 : d = min(d1 , d2 )

24. Combination IIIs there a point in L and a point in R whose distance is smaller than d ? L R d = min(d1 , d2 )

25. Combination II • If the answer is “no” then we are done!!! • If the answer is “yes” then the closest such pair forms the closest pair for the entire set • How do we determine this?

26. Combination IIIs there a point in L and a point in R whose distance is smaller than d ? L R

27. Combination IIIs there a point in L and a point in R whose distance is smaller than d ? L R Need only to consider the narrow band O(n) time

28. Combination IIIs there a point in L and a point in R whose distance is smaller than d ? L R Denote this set by S, assume Sy is the sorted list of S by the y-coordinates.

29. Combination II • There exists a point in L and a point in R whose distance is less than d if and only if there exist two points in S whose distance is less than d. • If S is the whole thing, did we gain anything? • CLAIM: If s and t in S have the property that ||s-t|| <d, then s and t are within 15 positions of each other in the sorted list Sy.

30. Combination IIIs there a point in L and a point in R whose distance is smaller than d ? L R There are at most one point in each box of size δ/2 by δ/2. Thus s and t cannot be too far apart.

31. Closest-Pair • Preprocessing: • Construct Pxand Pyas sorted-list by x- and y-coordinates • Closest-pair(P, Px,Py) • Divide • Construct L, Lx, Lyand R, Rx, Ry • Conquer • Let d1= Closest-Pair(L, Lx, Ly) • Let d2= Closest-Pair(R, Rx, Ry) • Combination • Let d = min(d1 , d2 ) • Construct S and Sy • For each point in Sy, check each of its next 15 points down the list • If the distance is less than d, update the das this smaller distance

32. Complexity Analysis • Preprocessing takes O(n lg n) time • Divide takes O(n) time • Conquer takes 2 T(n/2) time • Combination takes O(n) time T(n) = 2T(n/2) + cn • So totally takes O(n lg n) time.

33. Matrix Multiplication • Suppose we multiply two NxN matrices together. • Regular method is NxNxN = N3 multiplications • O(N3)

34. Can we Divide and Conquer? B = C= A*B = A = C11 = A11*B11 + A12*B21 C12 = A11*B12 + A12*B22 C21 = A21*B11 + A22*B21 C22 = A21*B12 + A22*B22 Complexity : T(N) = 8T(N/2) + O(N2) = O(Nlog28) = O(N3) No improvement

35. Strassen’s Matrix Multiplication P1 = (A11+ A22)(B11+B22) P2 = (A21 + A22) * B11P3 = A11 * (B12 - B22) P4 = A22 * (B21 - B11) P5 = (A11 + A12) * B22P6 = (A21 - A11) * (B11 + B12) P7 = (A12 - A22) * (B21 + B22) C11 = P1 + P4 - P5 + P7C12 = P3 + P5C21 = P2 + P4C22 = P1 + P3 - P2 + P6 Volker Strassen And do this recursively as usual.

36. Time analysis • T(n) = 7T(n/2) + O(n2 ) = 7logn by the Master Theorem =nlog7 =n2.81 • Best bound: O(n2.376) by Coppersmith-Winograd. • Best known (trivial) lower bound: Ω(n2). • Open: what is the true complexity of matrix multiplication?