15.2   Fractional knapsack problem

Question

Given \(n\) items, the weight of the \(i\)-th item is \(wgt[i-1]\) and its value is \(val[i-1]\), and a knapsack with a capacity of \(cap\). Each item can be chosen only once, but a part of the item can be selected, with its value calculated based on the proportion of the weight chosen, what is the maximum value of the items in the knapsack under the limited capacity? An example is shown in Figure 15-3.

Figure 15-3   Example data of the fractional knapsack problem

The fractional knapsack problem is very similar overall to the 0-1 knapsack problem, involving the current item \(i\) and capacity \(c\), aiming to maximize the value within the limited capacity of the knapsack.

The difference is that, in this problem, only a part of an item can be chosen. As shown in Figure 15-4, we can arbitrarily split the items and calculate the corresponding value based on the weight proportion.

1. For item \(i\), its value per unit weight is \(val[i-1] / wgt[i-1]\), referred to as the unit value.
2. Suppose we put a part of item \(i\) with weight \(w\) into the knapsack, then the value added to the knapsack is \(w \times val[i-1] / wgt[i-1]\).

Figure 15-4   Value per unit weight of the item

1.   Greedy strategy determination

Maximizing the total value of the items in the knapsack essentially means maximizing the value per unit weight. From this, the greedy strategy shown in Figure 15-5 can be deduced.

1. Sort the items by their unit value from high to low.
2. Iterate over all items, greedily choosing the item with the highest unit value in each round.
3. If the remaining capacity of the knapsack is insufficient, use part of the current item to fill the knapsack.

Figure 15-5   Greedy strategy of the fractional knapsack problem

2.   Code implementation

We have created an Item class in order to sort the items by their unit value. We loop and make greedy choices until the knapsack is full, then exit and return the solution:

PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig

fractional_knapsack.py

class Item:
    """物品"""

    def __init__(self, w: int, v: int):
        self.w = w  # 物品重量
        self.v = v  # 物品价值

def fractional_knapsack(wgt: list[int], val: list[int], cap: int) -> int:
    """分数背包：贪心"""
    # 创建物品列表，包含两个属性：重量、价值
    items = [Item(w, v) for w, v in zip(wgt, val)]
    # 按照单位价值 item.v / item.w 从高到低进行排序
    items.sort(key=lambda item: item.v / item.w, reverse=True)
    # 循环贪心选择
    res = 0
    for item in items:
        if item.w <= cap:
            # 若剩余容量充足，则将当前物品整个装进背包
            res += item.v
            cap -= item.w
        else:
            # 若剩余容量不足，则将当前物品的一部分装进背包
            res += (item.v / item.w) * cap
            # 已无剩余容量，因此跳出循环
            break
    return res

fractional_knapsack.cpp

/* 物品 */
class Item {
  public:
    int w; // 物品重量
    int v; // 物品价值

    Item(int w, int v) : w(w), v(v) {
    }
};

/* 分数背包：贪心 */
double fractionalKnapsack(vector<int> &wgt, vector<int> &val, int cap) {
    // 创建物品列表，包含两个属性：重量、价值
    vector<Item> items;
    for (int i = 0; i < wgt.size(); i++) {
        items.push_back(Item(wgt[i], val[i]));
    }
    // 按照单位价值 item.v / item.w 从高到低进行排序
    sort(items.begin(), items.end(), [](Item &a, Item &b) { return (double)a.v / a.w > (double)b.v / b.w; });
    // 循环贪心选择
    double res = 0;
    for (auto &item : items) {
        if (item.w <= cap) {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += item.v;
            cap -= item.w;
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += (double)item.v / item.w * cap;
            // 已无剩余容量，因此跳出循环
            break;
        }
    }
    return res;
}

fractional_knapsack.java

/* 物品 */
class Item {
    int w; // 物品重量
    int v; // 物品价值

    public Item(int w, int v) {
        this.w = w;
        this.v = v;
    }
}

/* 分数背包：贪心 */
double fractionalKnapsack(int[] wgt, int[] val, int cap) {
    // 创建物品列表，包含两个属性：重量、价值
    Item[] items = new Item[wgt.length];
    for (int i = 0; i < wgt.length; i++) {
        items[i] = new Item(wgt[i], val[i]);
    }
    // 按照单位价值 item.v / item.w 从高到低进行排序
    Arrays.sort(items, Comparator.comparingDouble(item -> -((double) item.v / item.w)));
    // 循环贪心选择
    double res = 0;
    for (Item item : items) {
        if (item.w <= cap) {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += item.v;
            cap -= item.w;
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += (double) item.v / item.w * cap;
            // 已无剩余容量，因此跳出循环
            break;
        }
    }
    return res;
}

fractional_knapsack.cs

/* 物品 */
class Item(int w, int v) {
    public int w = w; // 物品重量
    public int v = v; // 物品价值
}

/* 分数背包：贪心 */
double FractionalKnapsack(int[] wgt, int[] val, int cap) {
    // 创建物品列表，包含两个属性：重量、价值
    Item[] items = new Item[wgt.Length];
    for (int i = 0; i < wgt.Length; i++) {
        items[i] = new Item(wgt[i], val[i]);
    }
    // 按照单位价值 item.v / item.w 从高到低进行排序
    Array.Sort(items, (x, y) => (y.v / y.w).CompareTo(x.v / x.w));
    // 循环贪心选择
    double res = 0;
    foreach (Item item in items) {
        if (item.w <= cap) {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += item.v;
            cap -= item.w;
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += (double)item.v / item.w * cap;
            // 已无剩余容量，因此跳出循环
            break;
        }
    }
    return res;
}

fractional_knapsack.go

/* 物品 */
type Item struct {
    w int // 物品重量
    v int // 物品价值
}

/* 分数背包：贪心 */
func fractionalKnapsack(wgt []int, val []int, cap int) float64 {
    // 创建物品列表，包含两个属性：重量、价值
    items := make([]Item, len(wgt))
    for i := 0; i < len(wgt); i++ {
        items[i] = Item{wgt[i], val[i]}
    }
    // 按照单位价值 item.v / item.w 从高到低进行排序
    sort.Slice(items, func(i, j int) bool {
        return float64(items[i].v)/float64(items[i].w) > float64(items[j].v)/float64(items[j].w)
    })
    // 循环贪心选择
    res := 0.0
    for _, item := range items {
        if item.w <= cap {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += float64(item.v)
            cap -= item.w
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += float64(item.v) / float64(item.w) * float64(cap)
            // 已无剩余容量，因此跳出循环
            break
        }
    }
    return res
}

fractional_knapsack.swift

/* 物品 */
class Item {
    var w: Int // 物品重量
    var v: Int // 物品价值

    init(w: Int, v: Int) {
        self.w = w
        self.v = v
    }
}

/* 分数背包：贪心 */
func fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -> Double {
    // 创建物品列表，包含两个属性：重量、价值
    var items = zip(wgt, val).map { Item(w: $0, v: $1) }
    // 按照单位价值 item.v / item.w 从高到低进行排序
    items.sort { -(Double($0.v) / Double($0.w)) < -(Double($1.v) / Double($1.w)) }
    // 循环贪心选择
    var res = 0.0
    var cap = cap
    for item in items {
        if item.w <= cap {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += Double(item.v)
            cap -= item.w
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += Double(item.v) / Double(item.w) * Double(cap)
            // 已无剩余容量，因此跳出循环
            break
        }
    }
    return res
}

fractional_knapsack.js

/* 物品 */
class Item {
    constructor(w, v) {
        this.w = w; // 物品重量
        this.v = v; // 物品价值
    }
}

/* 分数背包：贪心 */
function fractionalKnapsack(wgt, val, cap) {
    // 创建物品列表，包含两个属性：重量、价值
    const items = wgt.map((w, i) => new Item(w, val[i]));
    // 按照单位价值 item.v / item.w 从高到低进行排序
    items.sort((a, b) => b.v / b.w - a.v / a.w);
    // 循环贪心选择
    let res = 0;
    for (const item of items) {
        if (item.w <= cap) {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += item.v;
            cap -= item.w;
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += (item.v / item.w) * cap;
            // 已无剩余容量，因此跳出循环
            break;
        }
    }
    return res;
}

fractional_knapsack.ts

/* 物品 */
class Item {
    w: number; // 物品重量
    v: number; // 物品价值

    constructor(w: number, v: number) {
        this.w = w;
        this.v = v;
    }
}

/* 分数背包：贪心 */
function fractionalKnapsack(wgt: number[], val: number[], cap: number): number {
    // 创建物品列表，包含两个属性：重量、价值
    const items: Item[] = wgt.map((w, i) => new Item(w, val[i]));
    // 按照单位价值 item.v / item.w 从高到低进行排序
    items.sort((a, b) => b.v / b.w - a.v / a.w);
    // 循环贪心选择
    let res = 0;
    for (const item of items) {
        if (item.w <= cap) {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += item.v;
            cap -= item.w;
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += (item.v / item.w) * cap;
            // 已无剩余容量，因此跳出循环
            break;
        }
    }
    return res;
}

fractional_knapsack.dart

/* 物品 */
class Item {
  int w; // 物品重量
  int v; // 物品价值

  Item(this.w, this.v);
}

/* 分数背包：贪心 */
double fractionalKnapsack(List<int> wgt, List<int> val, int cap) {
  // 创建物品列表，包含两个属性：重量、价值
  List<Item> items = List.generate(wgt.length, (i) => Item(wgt[i], val[i]));
  // 按照单位价值 item.v / item.w 从高到低进行排序
  items.sort((a, b) => (b.v / b.w).compareTo(a.v / a.w));
  // 循环贪心选择
  double res = 0;
  for (Item item in items) {
    if (item.w <= cap) {
      // 若剩余容量充足，则将当前物品整个装进背包
      res += item.v;
      cap -= item.w;
    } else {
      // 若剩余容量不足，则将当前物品的一部分装进背包
      res += item.v / item.w * cap;
      // 已无剩余容量，因此跳出循环
      break;
    }
  }
  return res;
}

fractional_knapsack.rs

/* 物品 */
struct Item {
    w: i32, // 物品重量
    v: i32, // 物品价值
}

impl Item {
    fn new(w: i32, v: i32) -> Self {
        Self { w, v }
    }
}

/* 分数背包：贪心 */
fn fractional_knapsack(wgt: &[i32], val: &[i32], mut cap: i32) -> f64 {
    // 创建物品列表，包含两个属性：重量、价值
    let mut items = wgt
        .iter()
        .zip(val.iter())
        .map(|(&w, &v)| Item::new(w, v))
        .collect::<Vec<Item>>();
    // 按照单位价值 item.v / item.w 从高到低进行排序
    items.sort_by(|a, b| {
        (b.v as f64 / b.w as f64)
            .partial_cmp(&(a.v as f64 / a.w as f64))
            .unwrap()
    });
    // 循环贪心选择
    let mut res = 0.0;
    for item in &items {
        if item.w <= cap {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += item.v as f64;
            cap -= item.w;
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += item.v as f64 / item.w as f64 * cap as f64;
            // 已无剩余容量，因此跳出循环
            break;
        }
    }
    res
}

fractional_knapsack.c

/* 物品 */
typedef struct {
    int w; // 物品重量
    int v; // 物品价值
} Item;

/* 分数背包：贪心 */
float fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {
    // 创建物品列表，包含两个属性：重量、价值
    Item *items = malloc(sizeof(Item) * itemCount);
    for (int i = 0; i < itemCount; i++) {
        items[i] = (Item){.w = wgt[i], .v = val[i]};
    }
    // 按照单位价值 item.v / item.w 从高到低进行排序
    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);
    // 循环贪心选择
    float res = 0.0;
    for (int i = 0; i < itemCount; i++) {
        if (items[i].w <= cap) {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += items[i].v;
            cap -= items[i].w;
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += (float)cap / items[i].w * items[i].v;
            cap = 0;
            break;
        }
    }
    free(items);
    return res;
}

fractional_knapsack.kt

/* 物品 */
class Item(
    val w: Int, // 物品
    val v: Int  // 物品价值
)

/* 分数背包：贪心 */
fun fractionalKnapsack(wgt: IntArray, _val: IntArray, c: Int): Double {
    // 创建物品列表，包含两个属性：重量、价值
    var cap = c
    val items = arrayOfNulls<Item>(wgt.size)
    for (i in wgt.indices) {
        items[i] = Item(wgt[i], _val[i])
    }
    // 按照单位价值 item.v / item.w 从高到低进行排序
    items.sortBy { item: Item? -> -(item!!.v.toDouble() / item.w) }
    // 循环贪心选择
    var res = 0.0
    for (item in items) {
        if (item!!.w <= cap) {
            // 若剩余容量充足，则将当前物品整个装进背包
            res += item.v
            cap -= item.w
        } else {
            // 若剩余容量不足，则将当前物品的一部分装进背包
            res += item.v.toDouble() / item.w * cap
            // 已无剩余容量，因此跳出循环
            break
        }
    }
    return res
}

fractional_knapsack.rb

[class]{Item}-[func]{}

[class]{}-[func]{fractional_knapsack}

fractional_knapsack.zig

[class]{Item}-[func]{}

[class]{}-[func]{fractionalKnapsack}

Code Visualization

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Apart from sorting, in the worst case, the entire list of items needs to be traversed, hence the time complexity is \(O(n)\), where \(n\) is the number of items.

Since an Item object list is initialized, the space complexity is \(O(n)\).

3.   Correctness proof

Using proof by contradiction. Suppose item \(x\) has the highest unit value, and some algorithm yields a maximum value res, but the solution does not include item \(x\).

Now remove a unit weight of any item from the knapsack and replace it with a unit weight of item \(x\). Since the unit value of item \(x\) is the highest, the total value after replacement will definitely be greater than res. This contradicts the assumption that res is the optimal solution, proving that the optimal solution must include item \(x\).

For other items in this solution, we can also construct the above contradiction. Overall, items with greater unit value are always better choices, proving that the greedy strategy is effective.

As shown in Figure 15-6, if the item weight and unit value are viewed as the horizontal and vertical axes of a two-dimensional chart respectively, the fractional knapsack problem can be transformed into "seeking the largest area enclosed within a limited horizontal axis range". This analogy can help us understand the effectiveness of the greedy strategy from a geometric perspective.

Figure 15-6   Geometric representation of the fractional knapsack problem

Feel free to drop your insights, questions or suggestions