以下是en.wikipedia关于背包问题的文章中的代码：
// Input:
// Values (stored in array v)
// Weights (stored in array w)
// Number of distinct items (n)
// Knapsack capacity (W)
for w from 0 to W do
m[0, w] := 0
end for
for i from 1 to n do
for j from 0 to W do
if j >= w[i] then
m[i, j] := max(m[i-1, j], m[i-1, j-w[i]] + v[i])
else
m[i, j] := m[i-1, j]
end if
end for
end for
我有两点让我疲惫的大脑无法解决。他们很小，我敢肯定，但我真的可以使用帮助。
•m []数组的大小是多少？ M [N，W]？如果是，则伪代码忽略最后一行和最后一列，因为它用零填充整个第一行（for（）循环用m [0，w]：= 0），然后从1循环到n，例如，3个不同的项目（n == 3）和容量4（W == 3），是m [3,4]或m [4,5]？
•在某个地方是否有更好的动态背包算法示例？
答案 0 :(得分：4)
数组的大小为（n + 1）×（W + 1），因为值的范围从[0,0]到[n，W]包括在内。
网格的解释如下：position [k，w]表示使用前k个项目可以获得的最大值（假设项目编号为1,2，...，n）并且携带的总重量不超过w。
第一行完全设置为0的原因是因为形式[0，w]的任何条目对应于使用前0项并且最多可以承载w的最大值。这始终为零，因为您不会通过不选择任何项目来获得任何价值。这对应于递归的基本情况。
使用以下想法填写第一行后的行：如果你想尝试选择第k项，你首先需要确保你能够持有它（意味着w必须至少为w） [K]）。如果你不能握住它，你最好的选择是充分利用你当前体重限制的第一个k - 1项目（所以你最好取相应于m [k - 1，w]的值如果您可以持有该项目，您的选择要么不接受它（和以前一样，要充分利用其他项目，产生m [k-1，w]），要么接受它并最大化剩余项目剩余项目的承载能力。这给你的值v [k] + m [k - 1，w - w [k]]。
希望这有帮助！
答案 1 :(得分：2)
我知道它已经得到解答，只是在c＃中添加代码版本，仅供参考（对于简化的背包你可以参考：How do I solve the 'classic' knapsack algorithm recursively?）：
版本1。使用动态编程解决（类似于wiki） - 自下而上（制表方法）
版本2. 使用动态编程解决但从上到下（Memoization - lazy）
版本3. 递归（只是递归解决方案，不使用重叠子问题或能够使用DP的最佳子结构属性）
第4版。蛮力（通过所有组合）
参考文献：http://en.wikipedia.org/wiki/Knapsack_problem，How do I solve the 'classic' knapsack algorithm recursively?
表格 - DP - 版本（O（n W） - 伪多项式） - O（n W）内存 - 自下而上
public int Knapsack_0_1_DP_Tabular_Bottom_Up(int[] weights, int[] values, int maxWeight)
{
this.ValidataInput_Knapsack_0_1(weights, values, maxWeight);
int[][] DP_Memoization_Max_Value_Cache = new int[values.Length + 1][];
for (int i = 0; i <= values.Length; i++)
{
DP_Memoization_Max_Value_Cache[i] = new int[maxWeight + 1];
for (int j = 0; j <= maxWeight; j++)
{
DP_Memoization_Max_Value_Cache[i][j] = 0; //yes, its default -
}
}
/// f(i, w) = f(i-1, w) if Wi > w
/// Or, max (f(i-1, w), f(i-1, w-Wi) + Vi
/// Or 0 if i < 0
for(int i = 1; i<=values.Length; i++)
{
for(int w = 1; w <= maxWeight; w++)
{
//below code can be refined - intentional as i just want it
//look similar to other 2 versions (top_to_bottom_momoization
//and recursive_without_resuing_subproblem_solution
int maxValueWithoutIncludingCurrentItem =
DP_Memoization_Max_Value_Cache[i - 1][w];
if (weights[i - 1] > w)
{
DP_Memoization_Max_Value_Cache[i][w] = maxValueWithoutIncludingCurrentItem;
}
else
{
int maxValueByIncludingCurrentItem =
DP_Memoization_Max_Value_Cache[i - 1][w - weights[i - 1]]
+ values[i-1];
int overAllMax = maxValueWithoutIncludingCurrentItem;
if(maxValueByIncludingCurrentItem > overAllMax)
{
overAllMax = maxValueByIncludingCurrentItem;
}
DP_Memoization_Max_Value_Cache[i][w] = overAllMax;
}
}
}
return DP_Memoization_Max_Value_Cache[values.Length][maxWeight];
}
DP - 记忆 - 从上到下 - 懒惰评估
/// <summary>
/// f(i, w) = f(i-1, w) if Wi > w
/// Or, max (f(i-1, w), f(i-1, w-Wi) + Vi
/// Or 0 if i < 0
/// </summary>
int Knapsack_0_1_DP_Memoization_Top_To_Bottom_Lazy_Recursive(int[] weights, int[] values,
int index, int weight, int?[][] DP_Memoization_Max_Value_Cache)
{
if (index < 0)
{
return 0;
}
Debug.Assert(weight >= 0);
#if DEBUG
if ((index == 0) || (weight == 0))
{
Debug.Assert(DP_Memoization_Max_Value_Cache[index][weight] == 0);
}
#endif
//value is cached, so return
if(DP_Memoization_Max_Value_Cache[index][weight] != null)
{
return DP_Memoization_Max_Value_Cache[index][weight].Value;
}
Debug.Assert(index > 0);
Debug.Assert(weight > 0);
int maxValueWithoutIncludingCurrentItem = this.Knapsack_0_1_DP_Memoization_Top_To_Bottom_Lazy_Recursive
(weights, values, index - 1, weight, DP_Memoization_Max_Value_Cache);
if (weights[index-1] > weight)
{
DP_Memoization_Max_Value_Cache[index][weight] = maxValueWithoutIncludingCurrentItem;
//current item weight is more, so we cant include - so, just return
return maxValueWithoutIncludingCurrentItem;
}
int overallMaxValue = maxValueWithoutIncludingCurrentItem;
int maxValueIncludingCurrentItem = this.Knapsack_0_1_DP_Memoization_Top_To_Bottom_Lazy_Recursive
(weights, values, index - 1, weight - weights[index-1],
DP_Memoization_Max_Value_Cache) + values[index - 1];
if(maxValueIncludingCurrentItem > overallMaxValue)
{
overallMaxValue = maxValueIncludingCurrentItem;
}
DP_Memoization_Max_Value_Cache[index][weight] = overallMaxValue;
return overallMaxValue;
}
公共方法致电（有关来电者的详情，请参阅以下单位测试）
public int Knapsack_0_1_DP_Tabular_Bottom_Up(int[] weights, int[] values, int maxWeight)
{
this.ValidataInput_Knapsack_0_1(weights, values, maxWeight);
int[][] DP_Memoization_Max_Value_Cache = new int[values.Length + 1][];
for (int i = 0; i <= values.Length; i++)
{
DP_Memoization_Max_Value_Cache[i] = new int[maxWeight + 1];
for (int j = 0; j <= maxWeight; j++)
{
DP_Memoization_Max_Value_Cache[i][j] = 0; //yes, its default -
}
}
/// f(i, w) = f(i-1, w) if Wi > w
/// Or, max (f(i-1, w), f(i-1, w-Wi) + Vi
/// Or 0 if i < 0
for(int i = 1; i<=values.Length; i++)
{
for(int w = 1; w <= maxWeight; w++)
{
//below code can be refined - intentional as i just want it
//look similar to other 2 versions (top_to_bottom_momoization
//and recursive_without_resuing_subproblem_solution
int maxValueWithoutIncludingCurrentItem =
DP_Memoization_Max_Value_Cache[i - 1][w];
if (weights[i - 1] > w)
{
DP_Memoization_Max_Value_Cache[i][w] = maxValueWithoutIncludingCurrentItem;
}
else
{
int maxValueByIncludingCurrentItem =
DP_Memoization_Max_Value_Cache[i - 1][w - weights[i - 1]]
+ values[i-1];
int overAllMax = maxValueWithoutIncludingCurrentItem;
if(maxValueByIncludingCurrentItem > overAllMax)
{
overAllMax = maxValueByIncludingCurrentItem;
}
DP_Memoization_Max_Value_Cache[i][w] = overAllMax;
}
}
}
return DP_Memoization_Max_Value_Cache[values.Length][maxWeight];
}
递归 - 有DP子问题 - 但没有重复使用重叠子问题（这应该描述如何更容易将递归版本更改为DP从上到下（Memoization版本）
public int Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure(int[] weights, int[] values, int maxWeight)
{
this.ValidataInput_Knapsack_0_1(weights, values, maxWeight);
int v = this.Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure_Recursive(weights, values, index: weights.Length-1, weight: maxWeight);
return v;
}
/// <summary>
/// f(i, w) = f(i-1, w) if Wi > w
/// Or, max (f(i-1, w), f(i-1, w-Wi) + Vi
/// Or 0 if i < 0
/// </summary>
int Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure_Recursive(int[] weights, int[] values, int index, int weight)
{
if (index < 0)
{
return 0;
}
Debug.Assert(weight >= 0);
int maxValueWithoutIncludingCurrentItem = this.Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure_Recursive(weights,
values, index - 1, weight);
if(weights[index] > weight)
{
//current item weight is more, so we cant include - so, just return
return maxValueWithoutIncludingCurrentItem;
}
int overallMaxValue = maxValueWithoutIncludingCurrentItem;
int maxValueIncludingCurrentItem = this.Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure_Recursive(weights,
values, index - 1, weight - weights[index]) + values[index];
if(maxValueIncludingCurrentItem > overallMaxValue)
{
overallMaxValue = maxValueIncludingCurrentItem;
}
return overallMaxValue;
}
暴力（通过所有组合）
private int _maxValue = int.MinValue;
private int[] _valueIndices = null;
public void Knapsack_0_1_BruteForce_2_Power_N(int[] weights, int[] values, int maxWeight)
{
this.ValidataInput_Knapsack_0_1(weights, values, maxWeight);
this._maxValue = int.MinValue;
this._valueIndices = null;
this.Knapsack_0_1_BruteForce_2_Power_N_Rcursive(weights, values, maxWeight, 0, 0, 0, new List<int>());
}
private void Knapsack_0_1_BruteForce_2_Power_N_Rcursive(int[] weights, int[] values, int maxWeight, int index, int currentWeight, int currentValue, List<int> currentValueIndices)
{
if(currentWeight > maxWeight)
{
return;
}
if(currentValue > this._maxValue)
{
this._maxValue = currentValue;
this._valueIndices = currentValueIndices.ToArray();
}
if(index == weights.Length)
{
return;
}
Debug.Assert(index < weights.Length);
var w = weights[index];
var v = values[index];
//see if the current value, conributes to max value
currentValueIndices.Add(index);
Knapsack_0_1_BruteForce_2_Power_N_Rcursive(weights, values, maxWeight, index + 1, currentWeight + w,
currentValue + v, currentValueIndices);
currentValueIndices.Remove(index);
//see if current value, does not contribute to max value
Knapsack_0_1_BruteForce_2_Power_N_Rcursive(weights, values, maxWeight, index + 1, currentWeight, currentValue,
currentValueIndices);
}
单元测试1
[TestMethod]
public void Knapsack_0_1_Tests()
{
int[] benefits = new int[] { 60, 100, 120 };
int[] weights = new int[] { 10, 20, 30 };
this.Knapsack_0_1_BruteForce_2_Power_N(weights, values: benefits, maxWeight: 50);
Assert.IsTrue(this._maxValue == 220);
int v = this.Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure(weights,
values: benefits, maxWeight: 50);
Assert.IsTrue(v == 220);
v = this.Knapsack_0_1_DP_Memoization_Top_To_Bottom_Lazy(weights,
values: benefits, maxWeight: 50);
Assert.IsTrue(v == 220);
v = this.Knapsack_0_1_DP_Tabular_Bottom_Up(weights,
values: benefits, maxWeight: 50);
Assert.IsTrue(v == 220);
benefits = new int[] { 3, 4, 5, 8, 10 };
weights = new int[] { 2, 3, 4, 5, 9 };
this.Knapsack_0_1_BruteForce_2_Power_N(weights, values: benefits, maxWeight: 20);
Assert.IsTrue(this._maxValue == 26);
v = this.Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure(weights, values: benefits,
maxWeight: 20);
Assert.IsTrue(v == 26);
v = this.Knapsack_0_1_DP_Memoization_Top_To_Bottom_Lazy(weights,
values: benefits, maxWeight: 20);
Assert.IsTrue(v == 26);
v = this.Knapsack_0_1_DP_Tabular_Bottom_Up(weights,
values: benefits, maxWeight: 20);
Assert.IsTrue(v == 26);
benefits = new int[] { 3, 4, 5, 6};
weights = new int[] { 2, 3, 4, 5 };
this.Knapsack_0_1_BruteForce_2_Power_N(weights, values: benefits, maxWeight: 5);
Assert.IsTrue(this._maxValue == 7);
v = this.Knapsack_0_1_OverlappedSubPromblems_OptimalSubStructure(weights, values: benefits,
maxWeight: 5);
Assert.IsTrue(v == 7);
v = this.Knapsack_0_1_DP_Memoization_Top_To_Bottom_Lazy(weights,
values: benefits, maxWeight: 5);
Assert.IsTrue(v == 7);
v = this.Knapsack_0_1_DP_Tabular_Bottom_Up(weights,
values: benefits, maxWeight: 5);
Assert.IsTrue(v == 7);
}
单元测试2
[TestMethod]
public void Knapsack_0_1_Brute_Force_Tests()
{
int[] benefits = new int[] { 60, 100, 120 };
int[] weights = new int[] { 10, 20, 30 };
this.Knapsack_0_1_BruteForce_2_Power_N(weights, values: benefits, maxWeight: 50);
Assert.IsTrue(this._maxValue == 220);
Assert.IsTrue(this._valueIndices.Contains(1));
Assert.IsTrue(this._valueIndices.Contains(2));
Assert.IsTrue(this._valueIndices.Length == 2);
this._maxValue = int.MinValue;
this._valueIndices = null;
benefits = new int[] { 3, 4, 5, 8, 10 };
weights = new int[] { 2, 3, 4, 5, 9 };
this.Knapsack_0_1_BruteForce_2_Power_N(weights, values: benefits, maxWeight: 20);
Assert.IsTrue(this._maxValue == 26);
Assert.IsTrue(this._valueIndices.Contains(0));
Assert.IsTrue(this._valueIndices.Contains(2));
Assert.IsTrue(this._valueIndices.Contains(3));
Assert.IsTrue(this._valueIndices.Contains(4));
Assert.IsTrue(this._valueIndices.Length == 4);
}