Lecture 9: Recursion and Fibonacci Numbers PDF

15. Recursion II Decrease and Conquer vs. Divide and Conquer, Recursion vs. Iteration 397 Sum of Vector Well-known problem: sum of all entries of a vector std::vector v Simple iterative solution: int sum(const std::vector& v)...

15. Recursion II Decrease and Conquer vs. Divide and Conquer, Recursion vs. Iteration 397 Sum of Vector Well-known problem: sum of all entries of a vector std::vector v Simple iterative solution: int sum(const std::vector& v) { int s = 0; for (int i = 0; i < int(v.size()); ++i) { s += v.at(i); } return s; } 398 Sum of Vector: Variant 1 sum({}) 0 sum({7}) 7+ 7 sum({6, 7}) 6+ 13 sum({4, 6, 7}) 4+ 17 sum({-1, 4, 6, 7}) −1+ 16 399 Sum of Vector: Variant 1 int sum(const std::vector& v, int from) { if (from >= int(v.size())) return 0; else return v.at(from) + sum(v, from + 1); } call with e.g. sum({-1, 4, 6, 7}, 0); 400 Sum of Vector: Variant 2 sum({-1}) sum({4}) sum({6}) sum({7}) −1 6 + 4 + 7 sum({-1, 4}) sum({6, 7}) 3 + 13 sum({-1, 4, 6, 7}) 16 401 Sum of Vector: Variant 2 int sum(const std::vector& v, int from, int to) { if (from > to) return 0; else if (from == to) return v.at(from); else { int middle = (from + to) / 2; return sum(v, from, middle) + sum(v, middle + 1, to); } } call with e.g. sum({-1, 4, 6, 7}, 0, 3); 402 Recursive Strategies......... Decrease and Conquer Divide and Conquer remove one element, halving the elements, one recursive call on remainder one recursive call per half limitedly parallelizable oftentimes well-parallelizable large recursion depth smaller recursion depth 403 Fibonacci Numbers  0, if n = 0         Fn :=   1, if n = 1   Fn−1 + Fn−2, if n > 1     0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... 404 Fibonacci Numbers in C++ Laufzeit fib(50) takes “forever” because it computes F48 two times, F47 3 times, F46 5 times, F45 8 times, F44 13 times, F43 21 times... F1 ca. 109 times (!) int fib(int n) { if (n == 0) return 0; if (n == 1) return 1; return fib(n-1) + fib(n-2); // n > 1 } Fibonacci Visualised................................................ F47 F46 F46 F45 F46 F45 F45 F44 F48 F47 F47 F46 F49 F48 F50 Recomputing same values = re-solving same subproblems 406 Fast (Iterative) Fibonacci Numbers Idea: Compute each Fibonacci number only once, in the order F0 , F1 , F2 ,... , Fn Memorize the most recent two Fibonacci numbers (variables fib1 and fib2) Compute the next number as a sum of fib1 and fib2 Can be implemented recursively and iteratively, the latter is easier/more direct 407 Fast Fibonacci Numbers in C++ int fibonacci(int n) { if (n == 0) return 0; if (n == 1) return 1; int fib2 = 0; // value two before int fib1 = 1; // value one before int fib; // current value very fast, also for fib(50) for (int i = 2; i r; std::cout > s; // computation and output std::cout