public static int KnapSack(int capacity, Item[] items, int numItems) {
if (numItems == 0 || capacity == 0)
return 0;
if (items[numItems-1].weight > capacity)
return KnapSack(capacity, items, numItems-1);
else {
int took = items[numItems-1].value + KnapSack(capacity - items[numItems-1].weight, items, numItems-1);
int left = KnapSack(capacity, items, numItems-1);
return Math.max(took, left);
}
}
所以我有一个工作0/1递归强力算法为KnapSack问题工作。我想知道打印出工作解决方案的方法是什么（即从项目集中收集到背包里的物品）。我已经尝试了很多东西，例如添加到列表中并试图跟踪我添加的内容，但是没有一个可以解决实现或设计问题。所以我来这里寻求帮助，谢谢！
答案 0 :(得分：4)
要跟踪拍摄的项目，请尝试以下操作：
public static int KnapSack(int capacity, Item[] items, int numItems, ArrayList<Integer> taken) {
if (numItems == 0 || capacity == 0)
return 0;
if (items[numItems-1].weight > capacity)
return KnapSack(capacity, items, numItems-1, taken);
else {
final int preTookSize = taken.size();
final int took = items[numItems-1].value + KnapSack(capacity - items[numItems-1].weight, items, numItems-1, taken);
final int preLeftSize = taken.size();
final int left = KnapSack(capacity, items, numItems-1, taken);
if (took > left) {
if (taken.size() > preLeftSize)
taken.removeRange(preLeftSize, taken.size());
taken.add(Integer.valueOf(numItems - 1));
return took;
}
else {
if (preLeftSize > preTookSize)
taken.removeRange(preTookSize, preLeftSize);
return left;
}
}
}
这可能不是最有效的，但我认为它应该有效。
（为了提高效率，您可以尝试将所采用的ArrayList预先分配为“足够大”，以便在递归循环期间不需要进行分配。）
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）
版本4。暴力（通过所有组合） - 捕获字段中的最大值和值索引（this._valueIndices，this._maxValue）（请参阅tests2 - 应显示调用者和使用这些捕获的值）
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);
}