The power of logarithmic computations. Recursive x y. Calculate x y. Can we reduce the number of multiplications?

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Let us consider the calculation of x 3 : Introduction to Programming Dept.

1 Calculate x y The power of logarithmic computations // Pre: x 0, y 0 // Returns x y int power(int x, int y) { int p = 1; for (int i = 0; i < y; ++i) p = p x; return p; Jordi Cortadella Department of Computer Science The algorithm performs y multiplications. Can we reduce the number of multiplications? Let us consider the calculation of x 3 : Introduction to Programming Dept. CS, UPC Recursive x y Basic case (y = 0): x 0 = 1 Recursive cases (y > 0): y is even: x y = x y y is odd: x y = x x y 1 = x x (y 1) For integer division: For x y we will need about log y multiplications y 1 = y when y is odd Introduction to Programming Dept. CS, UPC 3 Introduction to Programming Dept. CS, UPC 4

2 Recursive x y Example: 19 // Pre: x 0, y 0 // Returns x y int power(int x, int y) { if (y == 0) return 1; if (y% == 0) return power(x x, y/); return x power(x x, y/); (exponents are powers of ) Introduction to Programming Dept. CS, UPC 5 Iterative x y Key strategy: find a good invariant. Let r be the result of the computation. int power(int x, int y) { int z = 1; // Invariant: r = x y z while (y!= 0) { if (y% == 0) { x = x x; y = y/; else { z = z x; y = y 1; return z; Introduction to Programming Dept. CS, UPC 6 Iterative x y (improved version) // Pre: x 0, y 0 // Returns x y int power(int x, int y) { int z = 1; // Invariant: r = x y z while (y!= 0) { if (y% == 1) z = z x; x = x x; y = y/; return z; Example: 19 x y z Introduction to Programming Dept. CS, UPC 7 Introduction to Programming Dept. CS, UPC 8

The Fibonacci sequence is defined as follows: 0, 1,

mathematical terms, it is defined by the following

1170-150 Introduction to Programming Dept.

CS, UPC 10 Tiling with Fibonacci squares The

org/wiki/fibonacci_number https://en.wikipedia.

3 The Fibonacci sequence is defined as follows: 0, 1, 1,, 3, 5, 8, 13, 1, 34, 55, 89, 144, 33, 377, In mathematical terms, it is defined by the following recurrence relation: Leonardo Fibonacci Pisa, Introduction to Programming Dept. CS, UPC 9 Introduction to Programming Dept. CS, UPC 10 Tiling with Fibonacci squares The Fibonacci spiral Introduction to Programming Dept. CS, UPC 11 Introduction to Programming Dept. CS, UPC 1

Number of petals in flowers Introduction to

CS, UPC 13 Introduction to CS, UPC 14 // Pre: n 0

int fib(int n); Basic case: n = 0 return 0.

Shallow diagonals of Pascal s Triangle General

4 Number of petals in flowers Introduction to Programming Dept. CS, UPC 13 Introduction to Programming Dept. CS, UPC 14 // Pre: n 0 // Post: Returns the Fibonacci number of order n. int fib(int n); Basic case: n = 0 return 0. n = 1 return 1. Shallow diagonals of Pascal s Triangle General case: n > 1 return fib(n - 1) + fib(n ) Introduction to Programming Dept. CS, UPC 15 Introduction to Programming Dept. CS, UPC 16

: recursive version 8 // Pre: n 0 // Returns the Fibonacci number of order

n; else return fib(n - 1) + fib(n - ); 3 4 5 3 1 4 3 1 4 3 3 1 1 4 3 3 3 1

CS, UPC 17 8 7 6 6 5 5 4 5 4 4 3 4 3 3 4 3 3 3 1 3 1 1 3 1 1 1 1 1 For

CS, UPC 18 When fib(8) is calculated: fib(7) is called once fib(6) is

called 8 times fib() is called 13 times fib(1) is called 1 times fib(0) is

fib(0) be called? Example: fib(50) calls fib(1) and fib(0) about.

5 : recursive version 8 // Pre: n 0 // Returns the Fibonacci number of order n int fib(int n) { // Recursive solution if (n <= 1) return n; else return fib(n - 1) + fib(n - ); How many recursive calls? Introduction to Programming Dept. CS, UPC For example, fib(5) is re-calculated 3 times. Introduction to Programming Dept. CS, UPC 18 When fib(8) is calculated: fib(7) is called once fib(6) is called twice fib(5) is called 3 times fib(4) is called 5 times fib(3) is called 8 times fib() is called 13 times fib(1) is called 1 times fib(0) is called 13 times When fib(n) is calculated, how many times will fib(1) and fib(0) be called? Example: fib(50) calls fib(1) and fib(0) about times Introduction to Programming Dept. CS, UPC 19 Introduction to Programming Dept. CS, UPC 0

$int fib(int n) { // iterative solution int f_i = 0; int f_i1 = 1; // Inv:$ f_i is the Fibonacci number of order i.

$for (int i = 0; i < n; ++i) { int f = f_i + f_i1; f_i = f_i1; f_i1 = f;$ Introduction to Programming Dept. CS, UPC 1 Introduction to Programming Dept.

B are x integer matrices // Returns A B Mx MatrixMul(const Mx& A, const Mx&

C[0][1] = A[0][0] B[0][1] + A[0][1] B[1][1]; C[1][0] = A[1][0] B[0][0] +

6 : iterative version // Pre: n 0 // Returns the Fibonacci number of order n. int fib(int n) { // iterative solution int f_i = 0; int f_i1 = 1; // Inv: f_i is the Fibonacci number of order i. // f_i1 is the Fibonacci number of order i+1. for (int i = 0; i < n; ++i) { int f = f_i + f_i1; f_i = f_i1; f_i1 = f; return f_i; Complexity: O(n) Algebraic solution: find matrix A such that Introduction to Programming Dept. CS, UPC 1 Introduction to Programming Dept. CS, UPC typedef vector< vector<int> > Mx; Complexity: O(log n) // Pre: A and B are x integer matrices // Returns A B Mx MatrixMul(const Mx& A, const Mx& B) { Mx C(, vector<int>()); C[0][0] = A[0][0] B[0][0] + A[0][1] B[1][0]; C[0][1] = A[0][0] B[0][1] + A[0][1] B[1][1]; C[1][0] = A[1][0] B[0][0] + A[1][1] B[1][0];; C[1][1] = A[1][0] B[0][1] + A[1][1] B[1][1]; return C; Introduction to Programming Dept. CS, UPC 3 Introduction to Programming Dept. CS, UPC 4

7 // Pre: A is a x integer matrix // Returns A n Mx power(const Mx& A, int n) { if (n == 0) return Identity(); // returns I if (n% == 0) return power(matrixmul(a, A), n/); return MatrixMul(A, power(matrixmul(a, A), n/)); Complexity: O(log n) // Pre: n 0 // Returns the Fibonacci number of order n. int fib(int n) { if (n <= 1) return n; // Creates the Fibonacci matrix [[1,1],[1,0]] Mx A(, vector<int>(, 1)); A[1][1] = 0; Mx Fn = power(a, n - 1); // Complexity O(log n) return Fn[0][0]; Introduction to Programming Dept. CS, UPC 5 and golden ratio Two quantities a > b > 0 are in the golden ratio if The golden ratio is: a + b a φ = = a b = φ = The ratio of two consecutives converges to the golden ratio: F n+1 lim = φ F n F n+1 φ F n n Introduction to Programming Dept. CS, UPC 6 and golden ratio Let A be the Fibonacci matrix A = 1 1 The eigenvalues of A are (Ax= x): 1 = φ, = 1 φ Example (F 15 = 610, F 16 = 987, F 17 = 1597): φ F 15 = F 16 φ F 16 = F 17 Introduction to Programming Dept. CS, UPC 7 Introduction to Programming Dept. CS, UPC 8

8 1.9 and golden ratio Conclusions Many naïve algorithms perform repeated computations, often hidden behind the natural computations Ratio F n /F n Identify repeated computations and re-design algorithms accordingly. A deep knowledge of the problem is required. Doubly-recursive functions usually generate an explosion of computations (see Fibonacci). Try to avoid them whenever possible. Introduction to Programming Dept. CS, UPC 9 Introduction to Programming Dept. CS, UPC 30

Algorithm Analysis. Jordi Cortadella and Jordi Petit Department of Computer Science

Algorithm Analysis. Jordi Cortadella and Jordi Petit Department of Computer Science Algorithm Analysis Jordi Cortadella and Jordi Petit Department of Computer Science What do we expect from an algorithm? Correct Easy to understand Easy to implement Efficient: Every algorithm requires