---
id: 5a23c84252665b21eecc7ed1
title: Knapsack problem/0-1
challengeType: 5
forumTopicId: 323649
dashedName: knapsack-problem0-1
---

# --description--

The 0-1 knapsack problem is defined as follows:

You are given an array of objects representing items to be put in a knapsack. The objects have 3 attributes: name, weight, and value. The items need to be selected so that the total weight does not exceed the maximum weight and the value is maximized.

# --instructions--

Write a function to solve the knapsack problem. The function is given the array of objects and the maximum weight as parameters. It should return the maximum total value possible.

# --hints--

`knapsack([{ name:'map', weight:9, value:150 }, { name:'compass', weight:13, value:35 }, { name:'water', weight:153, value:200 }, { name:'sandwich', weight:50, value:160 }, { name:'glucose', weight:15, value:60 }, { name:'tin', weight:68, value:45 }, { name:'banana', weight:27, value:60 }, { name:'apple', weight:39, value:40 }], 100)` should return `405`.

```js
assert.equal(
  knapsack(
    [
      { name: 'map', weight: 9, value: 150 },
      { name: 'compass', weight: 13, value: 35 },
      { name: 'water', weight: 153, value: 200 },
      { name: 'sandwich', weight: 50, value: 160 },
      { name: 'glucose', weight: 15, value: 60 },
      { name: 'tin', weight: 68, value: 45 },
      { name: 'banana', weight: 27, value: 60 },
      { name: 'apple', weight: 39, value: 40 }
    ],
    100
  ),
  405
);
```

`knapsack([{ name:'map', weight:9, value:150 }, { name:'compass', weight:13, value:35 }, { name:'water', weight:153, value:200 }, { name:'sandwich', weight:50, value:160 }, { name:'glucose', weight:15, value:60 }, { name:'tin', weight:68, value:45 }, { name:'banana', weight:27, value:60 }, { name:'apple', weight:39, value:40 }], 200)` should return `510`.

```js
assert.equal(
  knapsack(
    [
      { name: 'map', weight: 9, value: 150 },
      { name: 'compass', weight: 13, value: 35 },
      { name: 'water', weight: 153, value: 200 },
      { name: 'sandwich', weight: 50, value: 160 },
      { name: 'glucose', weight: 15, value: 60 },
      { name: 'tin', weight: 68, value: 45 },
      { name: 'banana', weight: 27, value: 60 },
      { name: 'apple', weight: 39, value: 40 }
    ],
    200
  ),
  510
);
```

`knapsack([{ name:'cheese', weight:23, value:30 }, { name:'beer', weight:52, value:10 }, { name:'suntan cream', weight:11, value:70 }, { name:'camera', weight:32, value:30 }, { name:'T-shirt', weight:24, value:15 }, { name:'trousers', weight:48, value:10 }, { name:'umbrella', weight:73, value:40 }], 100)` should return `145`.

```js
assert.equal(
  knapsack(
    [
      { name: 'cheese', weight: 23, value: 30 },
      { name: 'beer', weight: 52, value: 10 },
      { name: 'suntan cream', weight: 11, value: 70 },
      { name: 'camera', weight: 32, value: 30 },
      { name: 'T-shirt', weight: 24, value: 15 },
      { name: 'trousers', weight: 48, value: 10 },
      { name: 'umbrella', weight: 73, value: 40 }
    ],
    100
  ),
  145
);
```

`knapsack([{ name:'cheese', weight:23, value:30 }, { name:'beer', weight:52, value:10 }, { name:'suntan cream', weight:11, value:70 }, { name:'camera', weight:32, value:30 }, { name:'T-shirt', weight:24, value:15 }, { name:'trousers', weight:48, value:10 }, { name:'umbrella', weight:73, value:40 }], 200)` should return `185`.

```js
assert.equal(
  knapsack(
    [
      { name: 'cheese', weight: 23, value: 30 },
      { name: 'beer', weight: 52, value: 10 },
      { name: 'suntan cream', weight: 11, value: 70 },
      { name: 'camera', weight: 32, value: 30 },
      { name: 'T-shirt', weight: 24, value: 15 },
      { name: 'trousers', weight: 48, value: 10 },
      { name: 'umbrella', weight: 73, value: 40 }
    ],
    200
  ),
  185
);
```

`knapsack([{ name:'waterproof trousers', weight:42, value:70 }, { name:'waterproof overclothes', weight:43, value:75 }, { name:'note-case', weight:22, value:80 }, { name:'sunglasses', weight:7, value:20 }, { name:'towel', weight:18, value:12 }, { name:'socks', weight:4, value:50 }, { name:'book', weight:30, value:10 }], 100)` should return `237`.

```js
assert.equal(
  knapsack(
    [
      { name: 'waterproof trousers', weight: 42, value: 70 },
      { name: 'waterproof overclothes', weight: 43, value: 75 },
      { name: 'note-case', weight: 22, value: 80 },
      { name: 'sunglasses', weight: 7, value: 20 },
      { name: 'towel', weight: 18, value: 12 },
      { name: 'socks', weight: 4, value: 50 },
      { name: 'book', weight: 30, value: 10 }
    ],
    100
  ),
  237
);
```

`knapsack([{ name:'waterproof trousers', weight:42, value:70 }, { name:'waterproof overclothes', weight:43, value:75 }, { name:'note-case', weight:22, value:80 }, { name:'sunglasses', weight:7, value:20 }, { name:'towel', weight:18, value:12 }, { name:'socks', weight:4, value:50 }, { name:'book', weight:30, value:10 }], 200)` should return `317`.'

```js
assert.equal(
  knapsack(
    [
      { name: 'waterproof trousers', weight: 42, value: 70 },
      { name: 'waterproof overclothes', weight: 43, value: 75 },
      { name: 'note-case', weight: 22, value: 80 },
      { name: 'sunglasses', weight: 7, value: 20 },
      { name: 'towel', weight: 18, value: 12 },
      { name: 'socks', weight: 4, value: 50 },
      { name: 'book', weight: 30, value: 10 }
    ],
    200
  ),
  317
);
```

# --seed--

## --seed-contents--

```js
function knapsack(items, maxweight) {

}
```

# --solutions--

```js
function knapsack(items, maxweight) {
  var _ = {
    max: function(e) {
      var mx = e[0];
      e.forEach(function(f) {
        if (mx < f) mx = f;
      });
      return mx;
    },
    map: function(array, func) {
      return array.map(func);
    },
    isUndefined: function(a) {
      if (a) {
        return false;
      }
      return true;
    },
    range: function(start, end, step) {
      var a = [];
      var f = (f = (i, end) => i < end);
      if (start > end) f = (i, end) => i > end;

      for (var i = start; f(i, end); i += step) a.push(i);
      return a;
    }
  };

  var valuefn = e => e.value;
  var weightfn = e => e.weight;
  var _epsilon = 0.01;
  var _p = _.max(_.map(items, valuefn));
  var _k = (_epsilon * _p) / items.length;

  var _memo = (function() {
    var _mem = {};
    var _key = function(i, w) {
      return i + '::' + w;
    };
    return {
      get: function(i, w) {
        return _mem[_key(i, w)];
      },
      put: function(i, w, r) {
        _mem[_key(i, w)] = r;
        return r;
      }
    };
  })();

  var _m = function(i, w) {
    i = Math.round(i);
    w = Math.round(w);

    if (i < 0 || w === 0) {
      // empty base case
      return { items: [], totalWeight: 0, totalValue: 0 };
    }

    var mm = _memo.get(i, w);
    if (!_.isUndefined(mm)) {
      return mm;
    }

    var item = items[i];
    if (weightfn(item) > w) {
      //item does not fit, try the next item
      return _memo.put(i, w, _m(i - 1, w));
    }
    // this item could fit.
    // are we better off excluding it?
    var excluded = _m(i - 1, w);
    // or including it?
    var included = _m(i - 1, w - weightfn(item));
    if (
      included.totalValue + Math.floor(valuefn(item) / _k) >
      excluded.totalValue
    ) {
      // better off including it
      // make a copy of the list
      var i1 = included.items.slice();
      i1.push(item);
      return _memo.put(i, w, {
        items: i1,
        totalWeight: included.totalWeight + weightfn(item),
        totalValue: included.totalValue + Math.floor(valuefn(item) / _k)
      });
    }
    //better off excluding it
    return _memo.put(i, w, excluded);
  };
  var scaled = _m(items.length - 1, maxweight);

  var val = 0;
  scaled.items.forEach(function(e) {
    val += e.value;
  });
  return val;
}
```
