Add approaches for Knapsack (#2791)

* Add approaches for Knapsack

* Fix formatting in introduction.md

Author: kahgoh

exercises/practice/knapsack/.approaches/config.json

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+{
+  "introduction": {
+    "authors": [
+      "kahgoh"
+    ]
+  },
+  "approaches": [
+    {
+      "uuid": "bf439128-7610-4959-b9f6-bd033c979d8e",
+      "slug": "recursive",
+      "title": "Recursive",
+      "blurb": "Use recursion to work out whether each item should be added.",
+      "authors": [
+        "kahgoh"
+      ]
+    },
+    {
+      "uuid": "c51300cb-94f8-4a33-8f64-56deebaa1a22",
+      "slug": "dynamic-programming",
+      "title": "Dynamic Programming",
+      "blurb": "Build up a table of maximum values.",
+      "authors": [
+        "kahgoh"
+      ]
+    }
+  ]
+}

exercises/practice/knapsack/.approaches/dynamic-programming/content.md

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+# Dynamic Programming
+
+```java
+import java.util.List;
+
+class Knapsack {
+
+    int maximumValue(int maxWeight, List<Item> items) {
+        int [][]maxValues = new int[maxWeight + 1][items.size() + 1];
+
+        for (int nItems = 0; nItems <= items.size(); nItems++) {
+            maxValues[0][nItems] = 0;
+        }
+
+        for (int itemIndex = 0; itemIndex < items.size(); itemIndex++) {
+            Item item = items.get(itemIndex);
+
+            for (int capacity = 0; capacity <= maxWeight; capacity++) {
+                if (capacity < item.weight) {
+                    maxValues[capacity][itemIndex + 1] = maxValues[capacity][itemIndex];
+                } else {
+                    int valueWith = maxValues[capacity - item.weight][itemIndex] + item.value;
+                    int valueWithout = maxValues[capacity][itemIndex];
+                    maxValues[capacity][itemIndex + 1] = Math.max(valueWith, valueWithout);
+                }
+            }
+        }
+
+        return maxValues[maxWeight][items.size()];
+    }
+}
+```
+
+This approach works by building a table of maximum total values.
+The table is represented by the 2D [array][array] `maxValues`.
+The table's axes are the capacity (starting from 0 and going up to the `maxWeight`) and the number of items (starting from 0 and going up to length of the `items` list).
+The number of items always count from the first item.
+1 is added to the `maxWeight` and the number of items to allow space for 0 weight and 0 items.
+
+The first [for loop][for-loop] fills table for when there are no items available.
+In this case, the maximum value must be 0 because there are no items.
+
+The next for loops fills the rest of the table.
+The outer for loop iterates over each item.
+The inner loop fills calculates and stores the maximum value for the item and capacity.
+When storing, 1 is added to the `itemIndex` on the left hand side because the first item in the `maxValues` array represents no items.
+If the knapsack doesn't have enough capacity for the item, then maximum value is same as the maximum value _without_ the item.
+The maximum value without the item is obtained simply by looking up the value for the previous item and capacity in the table (i.e `maxValues[capacity][itemIndex]`).
+
+Otherwise, the maximum value is the greater of the value with and without the item.
+The maximum value _with_ (`valueWith`) the item is obtained by, first, looking up the maximum value _without_ the item and enough capacity for the item (i.e `capacity - item.weight`).
+The item's value (`item.value`) is then added to get the maximum value for the get the maximum value _with_ the item.
+
+After the table is completely filled, the maximum value is obtained from the table.
+
+[array]: https://docs.oracle.com/javase/tutorial/java/nutsandbolts/arrays.html
+[for-loop]: https://docs.oracle.com/javase/tutorial/java/nutsandbolts/for.html

exercises/practice/knapsack/.approaches/dynamic-programming/snippet.txt

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+for (int itemIndex = 0; itemIndex < items.size(); itemIndex++) {
+    for (int capacity = 0; capacity <= maxWeight; capacity++) {
+        int valueWith = maxValues[capacity - item.weight][itemIndex] + item.value;
+        int valueWithout = maxValues[capacity][itemIndex];
+        maxValues[capacity][itemIndex + 1] = Math.max(valueWith, valueWithout);
+    }
+}

exercises/practice/knapsack/.approaches/introduction.md

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+# Introduction
+
+There are a couple of approaches to solve Knapsack.
+You can recursively determine whether each item should be added to the knapsack.
+Or, you can solve it iteratively using a dynamic programming approach.
+
+## General guidance
+
+The key to solving Knapsack is to determine whether each item should be added to the knapsack or not.
+An item should be added only if:
+
+1. There is enough capacity to add the item and:
+2. It increases the total value.
+
+## Approach: Recursive
+
+```java
+import java.util.List;
+
+class Knapsack {
+
+    int maximumValue(int maxWeight, List<Item> items) {
+        if (items.isEmpty()) {
+            return 0;
+        }
+
+        List<Item> remainingItems = items.subList(1, items.size());
+        int valueWithout = maximumValue(maxWeight, remainingItems);
+
+        Item item = items.get(0);
+        if (item.weight > maxWeight) {
+            return valueWithout;
+        }
+
+        int valueWith = maximumValue(maxWeight - item.weight, remainingItems) + item.value;
+        return Math.max(valueWithout, valueWith);
+    }
+}
+```
+
+For more information, check the [Recursive approach][approach-recursive].
+
+## Approach: Dynamic programming
+
+```java
+import java.util.List;
+
+class Knapsack {
+
+    int maximumValue(int maxWeight, List<Item> items) {
+        int [][]maxValues = new int[maxWeight + 1][items.size() + 1];
+
+        for (int nItems = 0; nItems <= items.size(); nItems++) {
+            maxValues[0][nItems] = 0;
+        }
+
+        for (int itemIndex = 0; itemIndex < items.size(); itemIndex++) {
+            Item item = items.get(itemIndex);
+
+            for (int capacity = 0; capacity <= maxWeight; capacity++) {
+                if (capacity < item.weight) {
+                    maxValues[capacity][itemIndex + 1] = maxValues[capacity][itemIndex];
+                } else {
+                    int valueWith = maxValues[capacity - item.weight][itemIndex] + item.value;
+                    int valueWithout = maxValues[capacity][itemIndex];
+                    maxValues[capacity][itemIndex + 1] = Math.max(valueWith, valueWithout);
+                }
+            }
+        }
+
+        return maxValues[maxWeight][items.size()];
+    }
+}
+```
+
+For more information, check the [dynamic programming approach][approach-dynamic].
+
+## Which approach to use?
+
+The recursive approach is inefficient because it recalculates the maximum value for some item combinations a number of times.
+The dynamic programming approach avoids this by storing them in an [array][array].
+In addition, the dynamic programming approach is also an iterative approach and avoids overhead of making method calls.
+
+[approach-recursive]: https://exercism.org/tracks/java/exercises/knapsack/approaches/recursive
+[approach-dynamic]: https://exercism.org/tracks/java/exercises/knapsack/approaches/dynamic-programming
+[array]: https://docs.oracle.com/javase/tutorial/java/nutsandbolts/arrays.html

exercises/practice/knapsack/.approaches/recursive/content.md

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+# Recursive
+
+```java
+import java.util.List;
+
+class Knapsack {
+
+    int maximumValue(int maxWeight, List<Item> items) {
+        if (items.isEmpty()) {
+            return 0;
+        }
+
+        List<Item> remainingItems = items.subList(1, items.size());
+        int valueWithout = maximumValue(maxWeight, remainingItems);
+
+        Item item = items.get(0);
+        if (item.weight > maxWeight) {
+            return valueWithout;
+        }
+
+        int valueWith = maximumValue(maxWeight - item.weight, remainingItems) + item.value;
+        return Math.max(valueWithout, valueWith);
+    }
+}
+```
+
+The approach uses recursion to find the maximum value with and without each item.
+Each iteration works on the first item in the list (`items.get(0)`).
+If the list is empty, the maximum value must be `0`.
+Otherwise, a recursive call is made to work out the maximum value _without_ the first item (`maximumValue(maxWeight, remainingItems)`).
+
+If the there is not enough capacity to add the current item, the maximum value is the same as the maximum value without the first item.
+Otherwise, another recursive call is made to calculate the maximum value _with_ the first item.
+Since the item is included, the recursive call is done with the item's weight subtracted from the maximum weight (`maxWeight - item.weight`) and its value (`item.value`) is added.
+The maximum value is greater of these two values.

exercises/practice/knapsack/.approaches/recursive/snippet.txt

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+int maximumValue(int maxWeight, List<Item> items) {
+    int valueWithout = maximumValue(maxWeight, remainingItems);
+    int valueWith = maximumValue(maxWeight - item.weight, remainingItems) + item.value;
+    return Math.max(valueWithout, valueWith);
+}