Discrete Mathematics U. Waterloo ECE 103, Spring 2010 Ashwin Nayak May 17, 2010 Recursion
During the past week, we learnt about inductive reasoning, in which we broke down a problem of “size” n, into one or more problems of size smaller than n. For example, to establish a factorisation of an integer n into primes, we expressed it (if possible) as a product of two integers, a, b, both smaller than n. Assuming that we had a factorisation for a, b, we constructed one for n. If we were asked to produce a prime factori- sation for n, the same idea would work: we would test if n is a prime. If it is, our task is done. If not, we would ﬁnd some divisor a, with 1 < a < n, so that n = ab. Then, we would repeat the same procedure on a, to get a factorisation for a, and similarly for b. Together, these would give us the factorisation for n. This method of solution is called recursion. In the next few lectures, we will see several such instances of problems we can solve using recursion. In addition, we will see how induction helps prove properties (including correctness) of our solutions.
Suppose we are given numbers a, n, where n > 0 is an integer. We wish to calculate the number an. What is the quickest way to do this? How many multiplication operations do we have to perform? Of course, we may compute 198 by calculating 19 × 19 = 361, then calculating 193 = 361 × 19 = 6859, then 194 = 6859 × 19 = 130321, and so on, until we get 198. This takes seven multiplications in total. Is this the quickest possible? Note that 8 = 2 × 4, so we can also write 198 = 194 × 194. If we compute 194 ﬁrst, and then square it, we need only one more multiplication. The straightforward method would require four more multiplications: 198 = 194 ×19×19×19×19. Similarly, 194 = 192 ×192. So if we calculate 192 = 361 with one multiplication, 194 = 3612 = 130321 with one more, we get 198 = 1303212 = 16983563041 with the third multiplication. This cleverer method requires only three multiplications. The method above seems to work when the exponent n is even. What do we do when it is odd? Say, we would like to calculate 197. We may write 7 = 6+1, so 197 = 196 ×19, then 196 = 193 ×193, and ﬁnally 193 = 192 × 19. So 193 = 361 × 19 = 6859, 196 = 68592 = 47045881, and 197 = 47045881 × 19 = 893871739. The straightforward method of calculation requires 6 multiplications, and we needed only 4 here. Curiously, the number is more than we needed for 198. We will see the reason for this below. We can combine the ideas from the two examples above to get a procedure to calculate the power an for any pair a, n. This kind of procedure, which speciﬁes a sequence of steps to complete a given task is called an algorithm.
Algorithm 1: POWER(a,n), a recursive procedure to calculate a power. input : Real number a, integer n ≥ 1. output: The number an. if n = 1 then return a; if n is even then Let b = POWER(a, n/2) ; return b2; else // n is odd Let b = POWER(a, (n − 1)/2) ; return a × b2 ;
Let us see how this algorithm computes 213.
1 13 Example 1 Calculate 2 using algorithm POWER, and state the number of multiplications used.
Solution : The following table summarizes the calculations made by POWER(2, 13). Let a = 2, and b denote the value returned by the recursive call to POWER in one step.
Step n b calculation value 1 13 a6 a × (a6)2 8192 2 6 a3 (a3)2 64 3 3 a a × (a)2 8 4 1 2
The number of multiplications used is 5.
n The algorithm POWER(a, n) is correctly calculates a , as long as it correctly calculates the relevant smaller powers of a. We may prove by induction over n that this happens for any number a.
n Theorem 1 The algorithm POWER(a, n) returns a for every real number a and integer n ≥ 1.
Proof : The base case is n = 1, when the algorithm correctly returns a. Assume that the algorithm is correct for all k, with 1 ≤ k ≤ n − 1. Consider POWER(a, n). If n is even, the algorithm returns the 2 value b , where b = POWER(a, n/2). Since n > 1 and is even, n/2 is an integer, and 1 ≤ n/2 ≤ n − 1. n/2 2 n By induction hypothesis, POWER(a, n/2) correctly returns a , so the algorithm returns b = a . If n is odd (n > 1), then (n − 1)/2 is an integer, and 1 ≤ (n − 1)/2 ≤ n − 1. By the induction hypothesis, (n−1)/2 2 n b = POWER(a, (n − 1)/2) = a , and the algorithm correctly returns a × b = a in this case as well. So the algorithm is correct by the second principle of mathematical induction.
How many multiplications does the above procedure need to compute an? Running through our earlier examples, we see that the procedure calculates 198 and 197 just as we did. So it takes three and four multi- plications in these cases. It did ﬁve multiplications to calculate 213. How does the number of multiplications scale with n? In every step of the recursion, the exponent n decreases by a factor of at least 2, and we perform at most 2 multiplications. We stop when the exponent is 1. If there are k recursive steps in all, the total number of multiplications is at most 2k. Since the exponent decreases by a factor of two in every step, the ﬁnal expo- k k nent (i.e., 1) is at most n/2 . This gives us 1 ≤ n/2 , i.e., k ≤ log2 n, and that the number of multiplications is at most 2 log2 n. Let us prove all of this more rigorously. n Let M(n) be the number of multiplications that the procedure POWER(a, n) uses to compute a . We have
M(1) = 0 (1) M(n) = M(n/2) + 1, if n is even, (2) M(n) = M((n − 1)/2) + 2, if n is odd and n > 1. (3)
Theorem 2 M(n) ≤ 2 log2 n for all integers n ≥ 1.
Proof : We prove this using strong induction. The base case is n = 1, when M(1) = 0 = 2 log2 1. Assume that the statement is true for all k such that 1 ≤ k ≤ n−1. Consider n. If n is even, we have 1 ≤ n/2 ≤ n−1, so that M(n) = M(n/2) + 1 ≤ 2 log2(n/2) + 1 = 2 log2 n − 1. If n is odd, 1 ≤ (n − 1)/2 ≤ n − 1 and M(n) = M((n − 1)/2) + 2 ≤ 2 log2((n − 1)/2) + 2 ≤ 2 log2(n − 1). In both cases, M(n) ≤ 2 log2 n.
In fact, the recurrence for M(n) may be solved exactly to get
kn M(n) = kn + h(n − 2 ), (4)
2 k where kx = blog2 xc is the largest integer k such that 2 ≤ x, and h(x) denotes the number of 1s in the base 2 (i.e., binary) representation of the non-negative integer x. This latter term, which is the number of bits required to represent x, is bounded by log2 x, so M(n) ≤ 2 log2 n. We however do not attempt to derive this exact solution here. The straightforward (sequential) method for calculating an would take n−1 multiplications. The difference 500 between n and log2 n is enormous. If n = 2 , a 500-bit integer, the algorithm POWER(a, n) needs at most 1000 multiplications (actually at most 500), whereas the sequential method needs 2500 − 1. Even assuming the current generation of computers can multiply two numbers in one clock cycle, i.e., in 10−9s, the sequential algorithm will take more than the estimated lifetime of the universe! The recursive procedure however will give us the answer within a second. In the theory of computation, algorithms such as POWER are deemed to be efﬁcient, as they take time that is polynomial (in this case, linear) in the bit-representation of the input. Precisely which problems admit efﬁcient algorithms, and which may not, is a subject of intensive study in computer science.
In the our discussion on powering, we intentionally regarded multiplication as a basic operation, and esti- mated the “cost” of powering in terms of this operation. It is clear, though, even from the small examples we studied above (197, 198) that the numbers encountered by the algorithm grow rapidly in size. The re- sult of multiplying two n digit integers is up to 2n digits long. Evidently, the “cost” of multiplying these numbers grows as the number of digits increases, and should be taken into account. Recall the method for multiplication we learnt in school. Let us square the integer 130321 using this method:
130321 130321× 130321 260642 390963 000000 390963 130321 16983563041
We multiply the ﬁrst integer 130321 by the digits of the second, starting from the right, one at a time, writing the results in a staggered fashion, shifting each successive result one digit to the left. We then add up all the six integers to get our answer. In our example above, we started with two 6-digit integers, whose product is 12-digit long. To obtain the product, we added six integers, each up to 12-digit long. (The shifting to the left corresponds to extra 0- digits we append on the right.) In total, we added roughly 6 × 12 = 72 digits to get our answer. In general, this method involves n2 single digit multiplications and the addition of n integers with at most 2n digits to calculate the product of two n-digit integers. So its “cost” is 3n2. Can we do better? In 1960, Andrey Kolmogorov, an eminent mathematician at Moscow State University, posed this question in a seminar. (Among other famous pieces of work, he laid the foundations of modern probability theory.) He conjectured that no procedure could multiply two n-digit integers with signiﬁcantly fewer than n2 elementary operations. Within a week, Anatolii Karatsuba, then a 23-year old student in the seminar, found a recursive algorithm that accomplishes this in a constant times nlog2 3 ≈ n1.585 elementary operations, thus disproving Kolmogorov’s conjecture. Apparently, the term “divide and conquer”, which refers to a commonly used recursive technique, was ﬁrst used for this algorithm.
3 To understand the Karatsuba algorithm, let us look at another example. Suppose we would like to com- pute 130321 × 123131. We may write
In other words, we may express a 6 digit integer in terms of two 3-digit integers. (Note that multiplication by a power of 103 corresponds to shifting an integer to the left by three digits.) Now
130321 × 123131 = (130 × 103 + 321)(123 × 103 + 131) = (130 × 123)106 + (130 × 131 + 321 × 123)103 + 321 × 131 = (ac)106 + (ad + bc)103 + bd. (5)
In effect, we have reduced the problem to that of multiplying four pairs of 3-digit integers, and the addition of three pairs of integers with at most 12 digits. (Multiplying by a power of 10, say 10k, corresponds to simply shifting the integer by k digits.) In general, the multiplication of two n-digit integers is reduced to four multiplications of (n/2)-digit inte- gers, and three additions of 2n digit integers. If we work out the number of elementary operations in this scheme, using the estimate from the earlier method for multiplication, we get 4 × 3(n/2)2 operations for the multiplications, and 3 × (2n) for the additions, for a total of 3n2 + 6n. The dominant term here is 3n2 for large n, which gives us no advantage over the earlier method. Karatsuba’s idea was to reduce the number of multiplications of the smaller integers from four to three, while increasing the number of additions. He suggested computing the middle term in Equation (5) as follows:
ad + bc = (a + b)(c + d) − ac − bd, i.e., as 130 × 131 + 321 × 123 = (130 + 321)(123 + 131) − (130 × 123) − (321 × 131).
Since the products we subtract above are the ﬁrst and the last products in Equation (5), he effectively re- duced two multiplications and an addition to one multiplication and four additions. (Recall that subtraction is the addition of a negative number). If we use Karatsuba’s idea for multiplying n-digit integers, the number of elementary operations we need is 3 × 3(n/2)2 for the multiplications, and 6 × (2n) for the additions, for a total of (9/4)n2 + 12n. The dominant term here is (9/4)n2, a factor of 3/4 smaller than in the ﬁrst attempt. Evidently, this improvement comes about from our reduction of four multiplications to three. Strictly speaking, our estimate above is not entirely correct, as the sum of the two (n/2)-digit integers may have a carry over, and give us an (n/2 + 1)-digit integer. However, we can verify that even with this correction, the dominant term remains (9/4)n2. The rewards of this constant factor savings become clearer when we apply it recursively on the multiplication of smaller, (n/2)-digit integers. Let us see how this algorithm looks. Let x, y be two n-digit integers. For the moment, assume that n is even. We break each integer into a sum coming from the ﬁrst (n/2) and the last (n/2) digits. Call the integer represented by the ﬁrst (n/2) digits of x as xL and the integer represented by the last (n/2) digits as xR. Similarly we get the integers yL, yR from y. Then
n/2 x = xL × 10 + xR n/2 y = yL × 10 + yR,
4 the product
n/2 n/2 xy = (xL10 + xR)(yL10 + yR) n n/2 = (xLyL)10 + (xLyR + xRyL)10 + xRyR, and, as before, we may calculate the middle term using one multiplication as
xLyR + xRyL = (xL + xR)(yL + yR) − (xLyL) − (xRyR), so that we may compute xy with only three multiplications. The multiplication of the (n/2)-digit integers is done recursively, using the same kind of splitting, until we arrive at integers with a constant number of digits. (We choose this constant to be 3, so that we are guaranteed that the number of digits decreases in every recursive step.) At this point, we directly multiply the integers. If the integer n is not even, we split x, y into two integers each, with dn/2e = (n+1)/2 and bn/2c = (n−1)/2 digits, and the rest of the operations are modiﬁed appropriately. Here, bac denotes the result of rounding down the integer a to the closest integer, and dae denotes the result of rounding up a to the nearest integer. If n is even, bn/2c = dn/2e = n/2. Regardless of whether n is even or odd, we have bn/2c + dn/2e = n. For n ≥ 4, we have dn/2e + 1 < n, so that the algorithm computes xy by calculating the product of numbers with strictly fewer digits. The resulting recursive algorithm for multiplication is summarised below.
Algorithm 2: MULTIPLY(x, y), the Karatsuba multiplication algorithm input : Positive integers x, y, each with n ≥ 1 decimal digits. output: The product xy. if n ≤ 3 then return xy;
Let xL, xR be the leftmost dn/2e and the rightmost bn/2c digits of x, respectively; Let yL, yR be the leftmost dn/2e and the rightmost bn/2c digits of y, respectively;
Let P1 = MULTIPLY(xL, yL); Let P2 = MULTIPLY(xR, yR); Let P3 = MULTIPLY(xL + xR, yL + yR); 2bn/2c bn/2c return P1 × 10 + (P3 − P1 − P2) × 10 + P2;
As we did for the recursive algorithm for exponentiation, we may prove the Karatsuba algorithm to be correct using strong induction on the number of digits. We leave this as an exercise. An important difference between the recursion here and the recursion in POWER is that we get three smaller subproblems at every step, rather than one. So the algorithm cannot easily be unraveled into an iterative procedure.
Example 2 Multiply the numbers 1010203 and 3020101 using the Karatsuba multiplication algorithm.
Solution : We summarise the calculations made by the algorithm in a table:
Step x y P1 = xL × yL P2 = xR × yR (xL + xR) (yL + yR)
1 1010203 3020101 1010 × 3020 203 × 101 1213 3121 2 1010 3020 10 × 30 10 × 20 20 50 3 1213 3121 12 × 31 13 × 21 25 52
In this table, we have not included separate lines for steps in which the algorithm directly multiplies two numbers with at most 3 digits (e.g., for 203 × 101 or 10 × 30). Recall that P3 = (xL + xR)(yL + yR). The calculations we make for xy in every step are:
5 2bn/2c bn/2c Step P1 × 10 + (P3 − P1 − P2) × 10 + P2 xy
1 3050200 × 106 + (3785773 − 3050200 − 20503)103 + 20503 3050915090503 2 300 × 104 + (1000 − 300 − 200)102 + 200 3050200 3 372 × 104 + (1300 − 372 − 273)102 + 273 3785773
So we have 1010203 × 3020101 = 3050915090503.
Let us turn to the number of elementary operations performed by the algorithm. Let T (n) be (a bound on) the number of elementary operations performed by the algorithm MULTIPLY(x, y) on inputs x, y with n digits. We get the following recurrence for T (n): T (3) = 27 T (n) = 2 T (n/2) + T (n/2 + 1) + 12(n + 1), if n is even and n > 3 T (n) = 2 T ((n + 1)/2) + T ((n + 1)/2 + 1) + 12(n + 1), if n is odd and n > 3, where we have used estimates from the straightforward method of multiplication for 3-digit integers to get T (3) = 27, and obtained the term 12(n + 1) from the 6 additions of integers with at most 4dn/2e digits each in the recursive step. This is a rather complicated recurrence, but its solution is within a constant factor of the solution of a recurrence described below in which the number of digits is exactly halved at every step. (The proof of this connection is not difﬁcult, but is not illuminating. We leave it to the interested reader to verify.) Let n be a power of 2, and deﬁne T 0(n) by the following recurrence. T 0(1) = 1 T 0(n) = 3 T 0(n/2) + n, if n > 1. Unraveling this recurrence with n = 2k, for some k ≥ 0 gives us T 0(2k) = 3 T 0(2k−1) + 2k = 32 T 0(2k−2) + 3 × 2k−1 + 2k = 33 T 0(2k−3) + 32 × 2k−2 + 3 × 2k−1 + 2k . . = 3k × 20 + 3k−1 × 2 + 3k−2 × 22 + ··· + 30 × 2k " # 2 22 2k = 3k 1 + + + ··· 3 3 3
" 2 k # k 1 − 3 = 3 2 1 − 3 = 3k+1 − 2k+1. Had we been given this solution to the recurrence, we could have veriﬁed by induction that T 0(2k) = k+1 k+1 0 log2 n+1 log2 3 3 −2 for all k ≥ 0. We leave this as an exercise. Since k = log2 n, we have T (n) ≤ 3 = 3 n . As mentioned above, we can show that the solution T (n) to the original recurrence is at most a constant times T 0(n) ≤ 3 nlog2 3 ≈ 3 n1.585. This dependence on n holds even when n is not a power of 2. Thus, the Karatsuba algorithm uses many fewer elementary operations than 3n2, the “cost” of the method we learnt in school. This is however, not the end of the story. Yet more sophisticated multiplication algorithms have been discovered since. Andrei Toom and Stephen Cook found improvements to the Karatsuba algorithm shortly after it was published. In 1971 Arnold Schonhage¨ and Volker Strassen described a fundamentally new method that is subtantially faster, and in practice beats the previous ones for numbers with more than around 10, 000 decimal digits. The best known algorithm is due to Martin Furer,¨ and was discovered only very recently, in 2007.