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The French mathematician Blaise Pascal (1623 1662) discovered a recursive relationship among the binomial coefficients. By arranging them in a triangle, he found that each interior number is the sum of the two directly above it. (See Figure 9.3.) For example, 15 = 5 + 10. Let c(n,k) denote the coefficient in row number n and column number k (counting from 0). For example, c(6,2) = 15. Then Pascal s recurrence relation can be expressed as c(n, k) = c(n 1, k 1) + c(n 1, k), for 0 < k < n For example, when n = 6 and k = 2, c(6,2) = c(5,1) + c(5,2). EXAMPLE 9.11 Recursive Implementation of the Binomial Coefficient Function
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public static int c(int n, int k) { if (k==0 || k==n) { return 1; // basis } return c(n-1,k-1) + c(n-1,k); // recursion }
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The basis for the recursion covers the left and right sides of the triangle, where k = 0 and where k = n.
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Figure 9.3 Pascal s triangle
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The binomial coefficients are the same as the combination numbers used in combinatorial mathematics and computed explicitly by the formula n k+1 n 2 -----------------------------k 3 n In this context, the combination is often written c n k = and is pronounced n choose k. k For example, 8 choose 3 is 8 = (8/1)(7/2)(6/3) = 56. 3 n n! c n k = ---------------------- = -1 k! n k ! n 1 ----------2 EXAMPLE 9.12 Iterative Implementation of the Binomial Coefficient Function
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This version implements the explicit formula given above. The expression on the right consists of k factors, so it is computed by a loop iterating k times:
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public static int c(int n, int k) { if (n < 2 || k == 0 || k == n) { return 1; } int c = 1;
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RECURSION
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for (int j = 1; j <= k; j++) { c = c*(n-j+1)/j; } return c; }
THE EUCLIDEAN ALGORITHM The Euclidean Algorithm computes the greatest common divisor of two positive integers. Appearing as Proposition 2 in Book VII of Euclid s Elements (c. 300 B.C.), it is probably the oldest recursive algorithm. As originally formulated by Euclid, it says to subtract repeatedly the smaller number n from the larger number m until the resulting difference d is smaller than n. Then repeat the same steps with d in place of n and with n in place of m. Continue until the two numbers are equal. Then that number will be the greatest common divisor of the original two numbers. Figure 9.4 applies this algorithm to find the greatest common divisor of 494 and 130 to be 26. This is correct because 494 = 26 19 and 130 = 26 5.
Figure 9.4 The Euclidean algorithm
EXAMPLE 9.13 Recursive Implementation of the Euclidean Algorithm
Each step in the algorithm simply subtracts the smaller number from the larger. This is done recursively by calling either gcd(m,n-m) or gcd(m-n,n):
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public static int gcd(int if (m==n) { return n; } else if (m<n) { return gcd(m,n-m); } else { return gcd(m-n,n); } }
m, int n) { // basis // recursion // recursion
For example, the call gcd(494,130) makes the recursive call gcd(364,130), which makes the recursive call gcd(234,130), which makes the recursive call gcd(104,130), which makes the recursive call gcd(104,26), which makes the recursive call gcd(78,26), which makes the recursive call gcd(52,26), which makes the recursive call gcd(26,26), which returns 26. The value 26 is then successively returned all the way back up the chain to the original call gcd(494,130), which returns it to its caller.
INDUCTIVE PROOF OF CORRECTNESS Recursive functions are usually proved correct by the principle of mathematical induction. This principle states that an infinite sequence of propositions can be proved to be true by verifying that (i) the first statement is true, and (ii) the truth of every other statement in the sequence can be derived from the assumption that its preceding statements are true. Part (i) is called the