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1 | | -/** Print all the Catalan numbers from 0 to n, n being the user input. |
2 | | - |
3 | | - * A Catalan number satifies the following two properties: |
4 | | - * C(0) = C(1) = 1; C(n) = sum(C(i).C(n-i-1)), from i = 0 to n-1 |
| 1 | +/** |
| 2 | + * @file |
| 3 | + * @brief Provides utilities to compute Catalan numbers using dynamic |
| 4 | + programming. |
| 5 | + * A Catalan numbers satisfy these recurrence relations: |
| 6 | + * C(0) = C(1) = 1; C(n) = sum(C(i).C(n-i-1)), for i = 0 to n-1 |
5 | 7 | * Read more about Catalan numbers here: |
6 | 8 | https://en.wikipedia.org/wiki/Catalan_number |
| 9 | + https://oeis.org/A000108/ |
7 | 10 | */ |
8 | 11 |
|
9 | | -#include <iostream> |
10 | | -using namespace std; |
11 | | - |
12 | | -int *cat; // global array to hold catalan numbers |
13 | | - |
14 | | -unsigned long int catalan_dp(int n) { |
15 | | - /** Using the tabulation technique in dynamic programming, |
16 | | - this function computes the first `n+1` Catalan numbers |
17 | | - |
18 | | - Parameter |
19 | | - --------- |
20 | | - n: The number of catalan numbers to be computed. |
| 12 | +#include <cassert> /// for assert |
| 13 | +#include <cstdint> /// for std::uint64_t |
| 14 | +#include <cstdlib> /// for std::size_t |
| 15 | +#include <numeric> /// for std::transform_reduce |
| 16 | +#include <vector> /// for std::vector |
21 | 17 |
|
22 | | - Returns |
23 | | - ------- |
24 | | - cat[n]: An array containing the first `n+1` Catalan numbers |
25 | | - */ |
26 | | - |
27 | | - // By definition, the first two Catalan numbers are 1 |
28 | | - cat[0] = cat[1] = 1; |
29 | | - |
30 | | - // Compute the remaining numbers from index 2 to index n, using tabulation |
31 | | - for (int i = 2; i <= n; i++) { |
32 | | - cat[i] = 0; |
33 | | - for (int j = 0; j < i; j++) |
34 | | - cat[i] += cat[j] * cat[i - j - 1]; // applying the definition here |
| 18 | +/** |
| 19 | + * @brief computes and caches Catalan numbers |
| 20 | + */ |
| 21 | +class catalan_numbers { |
| 22 | + using value_type = std::uint64_t; |
| 23 | + std::vector<value_type> known{1, 1}; |
| 24 | + |
| 25 | + value_type compute_next() { |
| 26 | + return std::transform_reduce(known.begin(), known.end(), known.rbegin(), |
| 27 | + static_cast<value_type>(), std::plus<>(), |
| 28 | + std::multiplies<>()); |
35 | 29 | } |
36 | 30 |
|
37 | | - // Return the result |
38 | | - return cat[n]; |
39 | | -} |
40 | | - |
41 | | -int main(int argc, char *argv[]) { |
42 | | - int n; |
43 | | - cout << "Enter n: "; |
44 | | - cin >> n; |
45 | | - |
46 | | - cat = new int[n + 1]; |
47 | | - |
48 | | - cout << "Catalan numbers from 0 to " << n << " are:\n"; |
49 | | - for (int i = 0; i <= n; i++) { |
50 | | - cout << "catalan (" << i << ") = " << catalan_dp(i) << endl; |
51 | | - // NOTE: Since `cat` is a global array, calling `catalan_dp` |
52 | | - // repeatedly will not recompute the the values already computed |
53 | | - // as in case of pre-computed values, the array will simply return them, |
54 | | - // instead of recomputing them. |
| 31 | + void add() { known.push_back(this->compute_next()); } |
| 32 | + |
| 33 | + public: |
| 34 | + /** |
| 35 | + * @brief computes the n-th Catalan number and updates the cache. |
| 36 | + * @return the n-th Catalan number |
| 37 | + */ |
| 38 | + value_type get(std::size_t n) { |
| 39 | + while (known.size() <= n) { |
| 40 | + this->add(); |
| 41 | + } |
| 42 | + return known[n]; |
55 | 43 | } |
56 | | - |
57 | | - return 0; |
| 44 | +}; |
| 45 | + |
| 46 | +void test_catalan_numbers_up_to_20() { |
| 47 | + // data verified with https://oeis.org/A000108/ |
| 48 | + catalan_numbers cn; |
| 49 | + assert(cn.get(0) == 1ULL); |
| 50 | + assert(cn.get(1) == 1ULL); |
| 51 | + assert(cn.get(2) == 2ULL); |
| 52 | + assert(cn.get(3) == 5ULL); |
| 53 | + assert(cn.get(4) == 14ULL); |
| 54 | + assert(cn.get(5) == 42ULL); |
| 55 | + assert(cn.get(6) == 132ULL); |
| 56 | + assert(cn.get(7) == 429ULL); |
| 57 | + assert(cn.get(8) == 1430ULL); |
| 58 | + assert(cn.get(9) == 4862ULL); |
| 59 | + assert(cn.get(10) == 16796ULL); |
| 60 | + assert(cn.get(11) == 58786ULL); |
| 61 | + assert(cn.get(12) == 208012ULL); |
| 62 | + assert(cn.get(13) == 742900ULL); |
| 63 | + assert(cn.get(14) == 2674440ULL); |
| 64 | + assert(cn.get(15) == 9694845ULL); |
| 65 | + assert(cn.get(16) == 35357670ULL); |
| 66 | + assert(cn.get(17) == 129644790ULL); |
| 67 | + assert(cn.get(18) == 477638700ULL); |
| 68 | + assert(cn.get(19) == 1767263190ULL); |
| 69 | + assert(cn.get(20) == 6564120420ULL); |
58 | 70 | } |
59 | 71 |
|
60 | | -/** Sample Test Case: |
61 | | - |
62 | | -$ cd "Dynamic Programming" |
63 | | -$ g++ Catalan-Numbers.cpp |
64 | | -$ ./a.exe |
65 | | - |
66 | | -Enter n: 5 |
67 | | -Catalan numbers from 0 to 5 are: |
68 | | -catalan (0) = 1 |
69 | | -catalan (1) = 1 |
70 | | -catalan (2) = 2 |
71 | | -catalan (3) = 5 |
72 | | -catalan (4) = 14 |
73 | | -catalan (5) = 42 |
| 72 | +void test_catalan_numbers_25() { |
| 73 | + // data verified with https://oeis.org/A000108/ |
| 74 | + catalan_numbers cn; |
| 75 | + assert(cn.get(25) == 4861946401452ULL); |
| 76 | +} |
74 | 77 |
|
75 | | -*/ |
| 78 | +int main() { |
| 79 | + test_catalan_numbers_up_to_20(); |
| 80 | + test_catalan_numbers_25(); |
| 81 | +} |
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