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

## Description
<section id='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.
</section>

## Instructions
<section id='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.
</section>

## Tests
<section id='tests'>

``` yml
tests:
  - text: <code>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)</code> should return <code>405</code>.
    testString: 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, "<code>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)</code> should return <code>405</code>.");
  - text: <code>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)</code> should return <code>510</code>.
    testString: 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, "<code>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)</code> should return <code>510</code>.");
  - text: <code>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)</code> should return <code>145</code>.
    testString: 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, "<code>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)</code> should return <code>145</code>.");
  - text: <code>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)</code> should return <code>185</code>.
    testString: 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, "<code>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)</code> should return <code>185</code>.");
  - text: <code>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)</code> should return <code>237</code>.
    testString: 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, "<code>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)</code> should return <code>237</code>.");
  - text: <code>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)</code> should return <code>317</code>.'
    testString: 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, "<code>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)</code> should return <code>317</code>.");
```

</section>

## Challenge Seed
<section id='challengeSeed'>
<div id='js-seed'>

```js
function knapsack(items, maxweight) {
  // Good luck!
}
```

</div>
</section>

## Solution
<section id='solution'>

```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;
}
```

</section>