1 // 面试题14：剪绳子
2 // 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）。
3 // 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
4 // 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
5 // 时得到最大的乘积18。
6
7 #include <iostream>
8 #include <cmath>
9
10 // ====================动态规划====================
11 int maxProductAfterCutting_solution1(int length)
12 {
13 if(length < 2)
14 return 0;
15 if(length == 2)
16 return 1;
17 if(length == 3)
18 return 2;
19
20 int* products = new int[length + 1];
21 products[0] = 0;
22 products[1] = 1;
23 products[2] = 2;
24 products[3] = 3;
25
26 int max = 0;
27 for(int i = 4; i <= length; ++i)
28 {
29 max = 0;
30 for(int j = 1; j <= i / 2; ++j)
31 {
32 int product = products[j] * products[i - j];
33 if(max < product)
34 max = product;
35
36 products[i] = max;
37 }
38 }
39
40 max = products[length];
41 delete[] products;
42
43 return max;
44 }
45
46 // ====================贪婪算法====================
47 int maxProductAfterCutting_solution2(int length)
48 {
49 if(length < 2)
50 return 0;
51 if(length == 2)
52 return 1;
53 if(length == 3)
54 return 2;
55
56 // 尽可能多地减去长度为3的绳子段
57 int timesOf3 = length / 3;
58
59 // 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。
60 // 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。
61 if(length - timesOf3 * 3 == 1)
62 timesOf3 -= 1;
63
64 int timesOf2 = (length - timesOf3 * 3) / 2;
65
66 return (int) (pow(3, timesOf3)) * (int) (pow(2, timesOf2));
67 }
68
69 // ====================测试代码====================
70 void test(const char* testName, int length, int expected)
71 {
72 int result1 = maxProductAfterCutting_solution1(length);
73 if(result1 == expected)
74 std::cout << "Solution1 for " << testName << " passed." << std::endl;
75 else
76 std::cout << "Solution1 for " << testName << " FAILED." << std::endl;
77
78 int result2 = maxProductAfterCutting_solution2(length);
79 if(result2 == expected)
80 std::cout << "Solution2 for " << testName << " passed." << std::endl;
81 else
82 std::cout << "Solution2 for " << testName << " FAILED." << std::endl;
83 }
84
85 void test1()
86 {
87 int length = 1;
88 int expected = 0;
89 test("test1", length, expected);
90 }
91
92 void test2()
93 {
94 int length = 2;
95 int expected = 1;
96 test("test2", length, expected);
97 }
98
99 void test3()
100 {
101 int length = 3;
102 int expected = 2;
103 test("test3", length, expected);
104 }
105
106 void test4()
107 {
108 int length = 4;
109 int expected = 4;
110 test("test4", length, expected);
111 }
112
113 void test5()
114 {
115 int length = 5;
116 int expected = 6;
117 test("test5", length, expected);
118 }
119
120 void test6()
121 {
122 int length = 6;
123 int expected = 9;
124 test("test6", length, expected);
125 }
126
127 void test7()
128 {
129 int length = 7;
130 int expected = 12;
131 test("test7", length, expected);
132 }
133
134 void test8()
135 {
136 int length = 8;
137 int expected = 18;
138 test("test8", length, expected);
139 }
140
141 void test9()
142 {
143 int length = 9;
144 int expected = 27;
145 test("test9", length, expected);
146 }
147
148 void test10()
149 {
150 int length = 10;
151 int expected = 36;
152 test("test10", length, expected);
153 }
154
155 void test11()
156 {
157 int length = 50;
158 int expected = 86093442;
159 test("test11", length, expected);
160 }
161
162 int main(int agrc, char* argv[])
163 {
164 test1();
165 test2();
166 test3();
167 test4();
168 test5();
169 test6();
170 test7();
171 test8();
172 test9();
173 test10();
174 test11();
175
176 return 0;
177 }
