213 lines
10 KiB
Markdown
213 lines
10 KiB
Markdown
---
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id: 5a23c84252665b21eecc7ed2
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title: Knapsack problem/Bounded
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challengeType: 5
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forumTopicId: 323652
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dashedName: knapsack-problembounded
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---
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# --description--
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The bounded knapsack problem is defined as follows:
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You are given an array of objects representing items to be put in a knapsack. The objects have 4 attributes: name, pieces (the number of the particular item), 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. Keep in mind that each item can appear between 0 and `pieces` times.
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# --instructions--
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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.
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# --hints--
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`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 300)` should return `755`.
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```js
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assert.equal(
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findBestPack(
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[
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{ name: 'map', weight: 9, value: 150, pieces: 1 },
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{ name: 'compass', weight: 13, value: 35, pieces: 1 },
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{ name: 'water', weight: 153, value: 200, pieces: 2 },
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{ name: 'sandwich', weight: 50, value: 60, pieces: 2 },
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{ name: 'glucose', weight: 15, value: 60, pieces: 2 },
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{ name: 'tin', weight: 68, value: 45, pieces: 3 },
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{ name: 'banana', weight: 27, value: 60, pieces: 3 },
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{ name: 'apple', weight: 39, value: 40, pieces: 3 },
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{ name: 'cheese', weight: 23, value: 30, pieces: 1 },
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{ name: 'beer', weight: 52, value: 10, pieces: 3 },
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{ name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
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{ name: 'camera', weight: 32, value: 30, pieces: 1 },
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{ name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
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],
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300
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),
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755
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);
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```
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`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 400)` should return `875`.
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```js
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assert.equal(
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findBestPack(
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[
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{ name: 'map', weight: 9, value: 150, pieces: 1 },
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{ name: 'compass', weight: 13, value: 35, pieces: 1 },
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{ name: 'water', weight: 153, value: 200, pieces: 2 },
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{ name: 'sandwich', weight: 50, value: 60, pieces: 2 },
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{ name: 'glucose', weight: 15, value: 60, pieces: 2 },
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{ name: 'tin', weight: 68, value: 45, pieces: 3 },
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{ name: 'banana', weight: 27, value: 60, pieces: 3 },
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{ name: 'apple', weight: 39, value: 40, pieces: 3 },
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{ name: 'cheese', weight: 23, value: 30, pieces: 1 },
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{ name: 'beer', weight: 52, value: 10, pieces: 3 },
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{ name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
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{ name: 'camera', weight: 32, value: 30, pieces: 1 },
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{ name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
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],
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400
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),
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875
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);
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```
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`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 500)` should return `1015`.
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```js
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assert.equal(
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findBestPack(
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[
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{ name: 'map', weight: 9, value: 150, pieces: 1 },
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{ name: 'compass', weight: 13, value: 35, pieces: 1 },
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{ name: 'water', weight: 153, value: 200, pieces: 2 },
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{ name: 'sandwich', weight: 50, value: 60, pieces: 2 },
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{ name: 'glucose', weight: 15, value: 60, pieces: 2 },
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{ name: 'tin', weight: 68, value: 45, pieces: 3 },
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{ name: 'banana', weight: 27, value: 60, pieces: 3 },
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{ name: 'apple', weight: 39, value: 40, pieces: 3 },
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{ name: 'cheese', weight: 23, value: 30, pieces: 1 },
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{ name: 'beer', weight: 52, value: 10, pieces: 3 },
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{ name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
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{ name: 'camera', weight: 32, value: 30, pieces: 1 },
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{ name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
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],
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500
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),
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1015
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);
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```
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`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 600)` should return `1120`.
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```js
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assert.equal(
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findBestPack(
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[
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{ name: 'map', weight: 9, value: 150, pieces: 1 },
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{ name: 'compass', weight: 13, value: 35, pieces: 1 },
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{ name: 'water', weight: 153, value: 200, pieces: 2 },
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{ name: 'sandwich', weight: 50, value: 60, pieces: 2 },
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{ name: 'glucose', weight: 15, value: 60, pieces: 2 },
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{ name: 'tin', weight: 68, value: 45, pieces: 3 },
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{ name: 'banana', weight: 27, value: 60, pieces: 3 },
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{ name: 'apple', weight: 39, value: 40, pieces: 3 },
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{ name: 'cheese', weight: 23, value: 30, pieces: 1 },
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{ name: 'beer', weight: 52, value: 10, pieces: 3 },
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{ name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
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{ name: 'camera', weight: 32, value: 30, pieces: 1 },
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{ name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
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],
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600
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),
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1120
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);
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```
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`findBestPack([{ name:'map', weight:9, value:150, pieces:1 }, { name:'compass', weight:13, value:35, pieces:1 }, { name:'water', weight:153, value:200, pieces:2 }, { name:'sandwich', weight:50, value:60, pieces:2 }, { name:'glucose', weight:15, value:60, pieces:2 }, { name:'tin', weight:68, value:45, pieces:3 }, { name:'banana', weight:27, value:60, pieces:3 }, { name:'apple', weight:39, value:40, pieces:3 }, { name:'cheese', weight:23, value:30, pieces:1 }, { name:'beer', weight:52, value:10, pieces:3 }, { name:'suntan, cream', weight:11, value:70, pieces:1 }, { name:'camera', weight:32, value:30, pieces:1 }, { name:'T-shirt', weight:24, value:15, pieces:2 }], 700)` should return `1225`.
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```js
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assert.equal(
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findBestPack(
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[
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{ name: 'map', weight: 9, value: 150, pieces: 1 },
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{ name: 'compass', weight: 13, value: 35, pieces: 1 },
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{ name: 'water', weight: 153, value: 200, pieces: 2 },
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{ name: 'sandwich', weight: 50, value: 60, pieces: 2 },
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{ name: 'glucose', weight: 15, value: 60, pieces: 2 },
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{ name: 'tin', weight: 68, value: 45, pieces: 3 },
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{ name: 'banana', weight: 27, value: 60, pieces: 3 },
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{ name: 'apple', weight: 39, value: 40, pieces: 3 },
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{ name: 'cheese', weight: 23, value: 30, pieces: 1 },
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{ name: 'beer', weight: 52, value: 10, pieces: 3 },
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{ name: 'suntan, cream', weight: 11, value: 70, pieces: 1 },
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{ name: 'camera', weight: 32, value: 30, pieces: 1 },
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{ name: 'T-shirt', weight: 24, value: 15, pieces: 2 }
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],
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700
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),
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1225
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);
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```
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# --seed--
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## --seed-contents--
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```js
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function findBestPack(data, maxweight) {
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}
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```
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# --solutions--
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```js
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function findBestPack(data, maxweight) {
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var m = [[0]]; // maximum pack value found so far
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var b = [[0]]; // best combination found so far
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var opts = [0]; // item index for 0 of item 0
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var P = [1]; // item encoding for 0 of item 0
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var choose = 0;
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for (var j = 0; j < data.length; j++) {
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opts[j + 1] = opts[j] + data[j].pieces; // item index for 0 of item j+1
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P[j + 1] = P[j] * (1 + data[j].pieces); // item encoding for 0 of item j+1
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}
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for (var j = 0; j < opts[data.length]; j++) {
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m[0][j + 1] = b[0][j + 1] = 0; // best values and combos for empty pack: nothing
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}
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for (var w = 1; w <= maxweight; w++) {
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m[w] = [0];
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b[w] = [0];
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for (var j = 0; j < data.length; j++) {
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var N = data[j].pieces; // how many of these can we have?
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var base = opts[j]; // what is the item index for 0 of these?
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for (var n = 1; n <= N; n++) {
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var W = n * data[j].weight; // how much do these items weigh?
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var s = w >= W ? 1 : 0; // can we carry this many?
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var v = s * n * data[j].value; // how much are they worth?
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var I = base + n; // what is the item number for this many?
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var wN = w - s * W; // how much other stuff can we be carrying?
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var C = n * P[j] + b[wN][base]; // encoded combination
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m[w][I] = Math.max(m[w][I - 1], v + m[wN][base]); // best value
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choose = b[w][I] = m[w][I] > m[w][I - 1] ? C : b[w][I - 1];
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}
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}
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}
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var best = [];
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for (var j = data.length - 1; j >= 0; j--) {
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best[j] = Math.floor(choose / P[j]);
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choose -= best[j] * P[j];
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}
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var wgt = 0;
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var val = 0;
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for (var i = 0; i < best.length; i++) {
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if (0 == best[i]) continue;
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wgt += best[i] * data[i].weight;
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val += best[i] * data[i].value;
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}
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return val;
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}
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```
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