Merge pull request #126 from BrianLusina/feat/divide-chocolate

feat(algorithms, binary-search): divide chocolate

Author: BrianLusina

DIRECTORY.md

@@ -120,6 +120,8 @@
   * [Petethebaker](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/petethebaker.py)
   * Search
     * Binary Search
+      * Divide Chocolate
+        * [Test Divide Chocolate](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/search/binary_search/divide_chocolate/test_divide_chocolate.py)
       * Maxruntime N Computers
         * [Test Max Runtime](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/search/binary_search/maxruntime_n_computers/test_max_runtime.py)
       * [Test Binary Search](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/search/binary_search/test_binary_search.py)

algorithms/intervals/remove_intervals/__init__.py

@@ -30,7 +30,7 @@ def remove_covered_intervals(intervals: List[List[int]]) -> int:
     intervals.sort(key=lambda x: (x[0], -x[1]))
 
     # keep track of the last max end seen so far, we use a large negative infinity to cover all possible numbers
-    max_end_seen = float('-inf')
+    max_end_seen = float("-inf")
     count = 0
 
     # We then iterate through the given intervals

algorithms/search/binary_search/divide_chocolate/README.md

@@ -0,0 +1,73 @@
+# Divide Chocolate
+
+You have a chocolate bar made up of several chunks, and each chunk has a certain sweetness, given in an array called 
+sweetness. You want to share the chocolate with k friends. To do this, you’ll make k cuts to divide the bar into k + 1 
+parts. Each part will consist of consecutive chunks.
+
+Being a kind person, you’ll take the piece with the minimum total sweetness and give the other pieces to your friends.
+
+Your task is to find the maximum total sweetness of the piece you will receive if you cut the chocolate bar optimally.
+
+## Constraints
+
+- 0 <= k < sweetness.length <= 10^3
+- 1 <= sweetness[i] <= 10^3
+
+## Solution
+
+As we have a chocolate bar made of several chunks, each with its own sweetness value, dividing the chocolate evenly 
+won’t ensure that we get the chocolate part with the maximum possible minimum sweetness. The next natural thought for 
+solving this problem is to try to cut the chocolate in different ways and find out the one that gives the desired result.
+But it becomes very inefficient when the chocolate bar is long. There are just too many possible combinations of where 
+to cut, and checking each one would take too much time.
+
+So, instead of guessing exactly where to cut, we flip the problem and ask: What is the largest minimum sweetness we can 
+guarantee for ourselves if we divide the chocolate bar into k + 1 pieces?
+
+The idea is to guess a sweetness value, called S , and check if it’s possible to cut the chocolate into k + 1 parts, 
+where each part has at least S sweetness. Go through the chocolate chunks one by one, adding their sweetness. When the 
+total reaches S or more, count it as one piece and start a new one. If it’s possible to make at least k + 1 such pieces, 
+then S is a good guess. If not, the guess is too high and should be lowered. For example, if S=5 , keep adding chunks 
+until the total is 5 or more. Count that as one piece and reset the total to 0 . Repeat this through the whole chocolate 
+bar.
+
+The sweetness value is guessed using the binary search. The lower bound is 1, and the upper bound is the total sweetness 
+divided by (k + 1), as no single piece can have more than that if the chocolate is divided evenly. Start by guessing the
+middle value. If it allows at least k + 1 valid parts, try a higher value. Otherwise, try a lower one. This quickly 
+narrows down the largest sweetness that can be guaranteed.
+
+Now, let’s look at the algorithm steps below:
+
+1. Initialize the binary search range:
+   - Set low (the smallest possible minimum sweetness for a piece) as 1, because sweetness values are at least 1.
+   - Set high (the largest possible minimum sweetness) as the total sweetness divided evenly among k + 1 people, using 
+     the formula sum(sweetness) // (k + 1). This is the best possible sweetness each person could get if the chocolate 
+     were divided perfectly.
+
+2. Start the binary search, and while low is less than or equal to high, do the following:
+   - Calculate the mid-point value, mid = (low + high) // 2. This represents the current guess for the maximum minimum 
+     sweetness we might be able to give to each person.
+   - Use the helper function canDivide(sweetness, k, mid) to check if dividing the chocolate into at least k + 1 pieces 
+     is possible, where each piece has a total sweetness of at least mid. The helper works by summing chunks and counting
+     how many full pieces it can form with at least mid sweetness. 
+   - If it’s possible to divide into k + 1 or more such pieces:
+     - That means we can afford to aim for even higher sweetness per person. So we move the lower bound up: low = mid + 1.
+     - We also store mid as a potential result, result.
+
+3. If it’s not possible to make enough pieces:
+   - Then our guess mid was too high, so we reduce the upper bound: high = mid - 1.
+
+4. After the binary search ends, return the last valid result, which is the maximum possible minimum sweetness we can 
+   guarantee for each person.
+
+### Time Complexity
+
+Let n be the total number of chunks in the chocolate bar, and S be the upper bound for our binary‐search range, i.e., 
+the sum of values in the sweetness list divided by k + 1. Each call to the helper function scans the entire sweetness 
+array once, taking O(n) time. The binary search is performed over the integer range [1…S], which takes O(log S) iterations. 
+Therefore, the overall time complexity of this solution is O(n×logS).
+
+### Space Complexity
+
+We only use a fixed number of extra variables, and no additional arrays or data structures. Therefore the space complexity
+is O(1).

algorithms/search/binary_search/divide_chocolate/__init__.py

@@ -0,0 +1,64 @@
+from typing import List
+
+
+def maximize_sweetness(sweetness: List[int], k: int) -> int:
+    """
+    Finds the maximum possible sweetness of the piece you will receive if you cut the chocolate bar optimally.
+    Args:
+        sweetness (List[int]): A list of sweetness values for each chunk of the chocolate bar.
+        k (int): The number of friends to share the chocolate bar with.
+    Returns:
+        int: The maximum possible sweetness of the piece you will receive if you cut the chocolate bar optimally.
+    """
+    lower_bound = 1
+    upper_bound = sum(sweetness) // (k + 1)
+
+    while lower_bound < upper_bound:
+        current_sweetness = 0
+        pieces_count = 0
+        mid = (lower_bound + upper_bound + 1) // 2
+
+        for s in sweetness:
+            current_sweetness += s
+            if current_sweetness >= mid:
+                pieces_count += 1
+                current_sweetness = 0
+
+        # check if pieces_count is at least k+1
+        if pieces_count >= k + 1:
+            lower_bound = mid
+        else:
+            upper_bound = mid - 1
+
+    return lower_bound
+
+
+def maximize_sweetness_2(sweetness: List[int], k: int) -> int:
+    """
+    Finds the maximum possible sweetness of the piece you will receive if you cut the chocolate bar optimally.
+    Args:
+        sweetness (List[int]): A list of sweetness values for each chunk of the chocolate bar.
+        k (int): The number of friends to share the chocolate bar with.
+    Returns:
+        int: The maximum possible sweetness of the piece you will receive if you cut the chocolate bar optimally.
+    """
+    low, high = 1, sum(sweetness) // (k + 1)
+    result = low
+    while low <= high:
+        mid = (low + high) // 2
+        if can_divide(sweetness, k, mid):
+            result = mid
+            low = mid + 1
+        else:
+            high = mid - 1
+    return result
+
+
+def can_divide(sweetness: List[int], k: int, min_sweetness: int) -> bool:
+    total_sweetness, pieces = 0, 0
+    for sweet in sweetness:
+        total_sweetness += sweet
+        if total_sweetness >= min_sweetness:
+            pieces += 1
+            total_sweetness = 0
+    return pieces >= k + 1

algorithms/search/binary_search/divide_chocolate/test_divide_chocolate.py

@@ -0,0 +1,28 @@
+import unittest
+from typing import List
+from parameterized import parameterized
+from algorithms.search.binary_search.divide_chocolate import maximize_sweetness, maximize_sweetness_2
+
+TEST_CASES = [
+    ([1, 2, 3, 4, 5, 6, 7, 8, 9], 5, 6),
+    ([5], 0, 5),
+    ([1, 2, 2, 1, 2, 2, 1], 3, 2),
+    ([1, 1, 1, 1, 1, 1, 1], 6, 1),
+    ([7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7], 20, 7),
+]
+
+
+class MaximizeSweetnessTestCase(unittest.TestCase):
+    @parameterized.expand(TEST_CASES)
+    def test_maximize_sweetness(self, sweetness: List[int], k: int, expected: int):
+        actual = maximize_sweetness(sweetness, k)
+        self.assertEqual(expected, actual)
+
+    @parameterized.expand(TEST_CASES)
+    def test_maximize_sweetness_2(self, sweetness: List[int], k: int, expected: int):
+        actual = maximize_sweetness_2(sweetness, k)
+        self.assertEqual(expected, actual)
+
+
+if __name__ == "__main__":
+    unittest.main()