// 面试题14:剪绳子 // 题目:给你一根长度为n绳子,请把绳子剪成m段(m、n都是整数,n>1并且m≥1)。 // 每段的绳子的长度记为k[0]、k[1]、……、k[m]。k[0]*k[1]*…*k[m]可能的最大乘 // 积是多少?例如当绳子的长度是8时,我们把它剪成长度分别为2、3、3的三段,此 // 时得到最大的乘积18。 #include <iostream> #include <cmath> // ====================动态规划==================== //四个特点: //1.求问题最优解 //2.问题最优解可以分解为子问题的最优解 //3.可以分解为具有重复的子问题 //4.从上向下分析问题,从下向上计算问题 //主要思路:自顶向下分析问题, 自底向上解决问题 int maxProductAfterCutting_solution1(int length) { if (length < 2) return 0; if (length == 2) return 1; if (length == 3) return 2; //子问题最优解 int* products = new int[length + 1]; products[0] = 0; products[1] = 1; products[2] = 2; products[3] = 3; int max = 0; for (int i = 4; i <= length; ++i) //自底向上求解 { max = 0; //重置最大值 for (int j = 1; j <= i / 2; ++j) //寻找最优解 { int product = products[j] * products[i - j]; if (product > max) max = product; } products[i] = max; //放外头试试 } max = products[length]; delete[] products; return max; } // ====================贪婪算法==================== int maxProductAfterCutting_solution2(int length) { if (length < 2) return 0; if (length == 2) return 1; if (length == 3) return 2; // 尽可能多地减去长度为3的绳子段 int timeOf3 = length / 3; // 当绳子最后剩下的长度为4的时候,不能再剪去长度为3的绳子段。 // 此时更好的方法是把绳子剪成长度为2的两段,因为2*2 > 3*1。 if (length - timeOf3 * 3 == 1) timeOf3 -= 1; int timeOf2 = (length - timeOf3 * 3) / 2; return (int)(pow(3, timeOf3)) * (int)(pow(2, timeOf2)); }
![](https://images.cnblogs.com/OutliningIndicators/ContractedBlock.gif)
// ====================测试代码==================== void test(const char* testName, int length, int expected) { int result1 = maxProductAfterCutting_solution1(length); if (result1 == expected) std::cout << "Solution1 for " << testName << " passed." << std::endl; else std::cout << "Solution1 for " << testName << " FAILED." << std::endl; int result2 = maxProductAfterCutting_solution2(length); if (result2 == expected) std::cout << "Solution2 for " << testName << " passed." << std::endl; else std::cout << "Solution2 for " << testName << " FAILED." << std::endl; } void test1() { int length = 1; int expected = 0; test("test1", length, expected); } void test2() { int length = 2; int expected = 1; test("test2", length, expected); } void test3() { int length = 3; int expected = 2; test("test3", length, expected); } void test4() { int length = 4; int expected = 4; test("test4", length, expected); } void test5() { int length = 5; int expected = 6; test("test5", length, expected); } void test6() { int length = 6; int expected = 9; test("test6", length, expected); } void test7() { int length = 7; int expected = 12; test("test7", length, expected); } void test8() { int length = 8; int expected = 18; test("test8", length, expected); } void test9() { int length = 9; int expected = 27; test("test9", length, expected); } void test10() { int length = 10; int expected = 36; test("test10", length, expected); } void test11() { int length = 50; int expected = 86093442; test("test11", length, expected); } int main(int agrc, char* argv[]) { test1(); test2(); test3(); test4(); test5(); test6(); test7(); test8(); test9(); test10(); test11(); return 0; }
分析:动态规划:自顶向下分析问题, 自底向上解决问题。贪婪算法:需证明思路正确性。