[chore] add markdown versions of DP notebooks

Author: gitgik

dynamic-programming/disk_stacking.md

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+### Disk stacking
+An array's element represents a disk. Each element has `[length, width, height]`
+Write a function that will stack the disks and build a tower with the greatest height, such that each disk has strictly lower dimensions than those below it.
+
+Sample input
+```
+array = [[2, 1, 2], [3, 2, 3], [2, 2, 8], [2, 3, 4], [1, 3, 1], [4, 4, 5]]
+```
+
+Sample output:
+```
+[[4, 4, 5], [3, 2, 3], [2, 1, 2]]
+
+The tower looks something like this
+
+        [2, 1, 2]
+       [ 3, 2, 3 ]
+      [4,   4,   5]
+```
+
+
+
+```python
+def diskStacking(disks):
+    """O(n) space, O(n^2) time"""
+    # sort based on height
+    disks.sort(key=lambda disk: disk[2])
+    diskHeights = [disk[2] for disk in disks]
+    sequences = [None for disk in disks]
+    
+    # maxHeightIndex
+    maxHeightIndex = 0
+    
+    for i in range(1, len(disks)):
+        currentDisk = disks[i]
+        for j in range(0, i):
+            otherDisk = disks[j]
+            # validate dimensions
+            if areValidDimensions(otherDisk, currentDisk):
+                if diskHeights[i] <= currentDisk[2] + diskHeights[j]:
+                    diskHeights[i] = currentDisk[2] + diskHeights[j]
+                    sequences[i] = j
+        # update maxHeightIndex: which will help us backtrack to find the disks involved.
+        if diskHeights[i] >= diskHeights[maxHeightIndex]:
+            maxHeightIndex = i
+    return buildSequence(disks, sequences, maxHeightIndex)
+    
+def buildSequence(array, sequences, currentIdx):
+    results = []
+    while currentIdx is not None:
+        results.append(disks[currentIdx])
+        currentIdx = sequences[currentIdx]
+    return results
+        
+def areValidDimensions(o, c):
+    return o[0] < c[0] and o[1] < c[1] and o[2] < c[2]
+
+```
+
+
+```python
+disks = [[2, 1, 2], [3, 2, 3], [2, 2, 8], [2, 3, 4], [1, 3, 1], [4, 4, 5]]
+diskStacking(disks)
+```
+
+
+
+
+    [[4, 4, 5], [3, 2, 3], [2, 1, 2]]
+
+
+
+
+```python
+
+```

dynamic-programming/fibonacci.md

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+```python
+def fibonacci(n):
+    dp = [0, 1]
+    for i in range(2, n + 1):
+        dp.append(dp[i - 1] + dp[i - 2])
+    return dp[n]
+```
+
+
+```python
+def fibo(n, memoize={1:0,2:1}):
+    if n < 2:
+        return n
+    
+    memoize[n] = fibo(n - 1, memoize) + fibo(n - 2, memoize)
+    return memoize[n]
+```
+
+
+```python
+fibo(5)
+```
+
+
+
+
+    5
+
+
+
+
+```python
+fibonacci(6)
+```
+
+
+
+
+    8
+
+
+
+
+```python
+
+```

dynamic-programming/find_denominations.md

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+### Problem
+You are given an array of length N, where each element i represents the number of ways we can produce i units of change. For example, [1, 0, 1, 1, 2] shows that there is only one way to make [0, 2, or 3] units, and 2 ways of making 4 units.
+
+Given such an array, determine the denominations that must be in use. In the case above, there must be coins with value 2, 3, and 4.
+
+### Approach
+
+There are two situation that will make the value at that index i be incremented:
+1. Coin `i` is one of the denominations
+2. Another coin `j` is one of the denominations and the value of array[i - j] is also non-zero
+    
+For each index:
+- Check the lower denominations j that are known to be part of the solution.
+- Each time we find `i - j` to be non-zero, we have encountered one of the ways of getting to element `i`, so we decrement the value at that index by 1.
+- If after going through all coins and the value at that index is still positive, we know that we must include `i` as part of the solution
+
+We must take note not to double count. For example, when `i == 7` and [3, 4] are in our solution set, that is one way of producing 7.
+When two coins sum up to an index, only the lower one cause us to decrement the value at that index.
+
+    
+
+
+```python
+def find_denominations(array):
+    coins = set()
+    # start from index 1, and start counting from 1 (not zero-based index)
+    for i, val in enumerate(array[1:], 1):
+        if val > 0:
+            for j in coins:
+                if array[i - j] > 0 and (i - j not in coins or i - j <= j):
+                    val -= 1
+            if val > 0:
+                coins.add(i)
+    return coins
+```
+
+
+```python
+find_denominations([1, 0, 1, 1, 2])
+```
+
+
+
+
+    {2, 3, 4}
+
+
+
+The time complexity of this algorithm is O(M * N), where M is the number of coins in our solution and N is the length of our input. 
+
+The space complexity will be O(M), since we require extra space to store the set of solution coins.
+
+
+```python
+
+```

dynamic-programming/kadanes_algorithm.md

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+### Problem
+Given an array of integers, find the largest possible sum of contiguous subarray within the given array.
+
+
+```python
+## solution
+def kadanes_algorithm(array):
+    """Complexity: O(n) time | O(1) space"""
+    
+    local_max = global_max = array[0]
+    for i in range(1, len(array)):
+        # the local maximum for an index is always => max(array[i], local_max of array[i - 1])
+        local_max = max(array[i], array[i] + local_max)
+        global_max = max(global_max, local_max)
+    return global_max
+```
+
+
+```python
+array = [-3, 4, -1, 2, 1, -5, 4, -12, 3, -7]
+kadanes_algorithm(array)
+```
+
+
+
+
+    6
+
+
+
+
+```python
+## unit tests to validate solution
+import unittest
+
+class KadenesTestCase(unittest.TestCase):
+    def test_case_1(self):
+        array = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
+        expected_sum = 55
+        self.assertEqual(kadanes_algorithm(array), expected_sum)
+        
+    def test_case_2(self):
+        array = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
+        expected_sum = 6
+        self.assertEqual(kadanes_algorithm(array), expected_sum)
+
+if __name__ == "__main__":
+    unittest.main(argv=['first-arg-is-ignored'], exit=False)
+```
+
+
+```python
+kadanes_algorithm([1, -1, 2, 3, -6])
+```
+
+
+
+
+    5
+
+
+
+
+```python
+
+```

dynamic-programming/knapsack.md

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+## Knapsack Problem
+You are given an array of arrays. Each subarray in this array contains two integer elements: the first represents an item value, the second an item's weight.
+
+You are also given an integer representing the maximum capacity of the knapsack that you have.
+
+Your goal is to fit items that will collectively fit within the knapsack's capacity while maximizing on their combined value.
+
+Write a function that returns the maximum combined value of the items you'd pick, as well as an array of the item indices picked.
+
+Sample Input: [[1,2],[4,3],[5,6],[6,7]], 10
+Sample Output: 10, [1, 3]
+
+
+
+
+```python
+def knapsack(items, capacity):
+    """O(nc) time | O(nc) space, where n == number of items, c == capacity"""
+    knapsack_values = [[0 for col in range(capacity + 1)] for row in range(len(items) + 1)]
+    
+    for row in range(1, len(items) + 1):
+        value = items[row - 1][0]
+        weight = items[row - 1][1]
+        for col in range(capacity + 1):
+            if weight > col:  # col here represents a given capacity
+                knapsack_values[row][col] = knapsack_values[row - 1][col]
+            else:
+                knapsack_values[row][col] = max(
+                    knapsack_values[row - 1][col],
+                    knapsack_values[row - 1][col - weight] + value
+                )
+    
+    return [knapsack_values[-1][-1], backtrack_items(knapsack_values, items)]
+
+def backtrack_items(values, items):
+    """
+    Backtracks from the bottom right of the 2d array, finding all the items that were put in the knapsack.
+    """
+    results = []
+    row = len(values) - 1
+    col = len(values[0]) - 1
+    
+    while row > 0:
+        if values[row][col] == values[row - 1][col]:
+            row -= 1
+        else:
+            results.append(row - 1)
+            col -= items[row - 1][1]
+            row -= 1
+    return list(reversed(results))
+```
+
+
+```python
+# test it out
+capacity = 10
+items = [[1, 3], [4, 3], [5, 6], [6, 7]]
+
+knapsack(items, capacity)
+```
+
+
+
+
+    [10, [1, 3]]
+
+
+
+### Unit Tests
+
+
+```python
+# unit tests
+import unittest
+
+class KnapsackTestCase(unittest.TestCase):
+    def test_case_1(self):
+        capacity = 10
+        items = [[1, 3], [4, 3], [5, 6], [6, 7]]
+        expected = [10, [1, 3]]
+        self.assertEqual(knapsack(items, capacity), expected)
+        
+    def test_case_when_capacity_is_zero(self):
+        capacity = 0
+        items = [[1, 3], [4, 3], [5, 6], [6, 7]]
+        expected = [0, []]
+        self.assertEqual(knapsack(items, capacity), expected)
+        
+        
+if __name__ == "__main__":
+    unittest.main(argv=[""], exit=False)
+```