Add dynamic programming solution for Egg Dropping Problem (#12066)

* Add dynamic programming solution for Egg Dropping Problem

* Revert changes to .pre-commit-config.yaml

Co-authored-by: Christian Clauss <cclauss@me.com>

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---------

Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

Author: 3 people

dynamic_programming/egg_dropping.py

@@ -0,0 +1,70 @@
+"""
+Calculate the minimum number of attempts needed in the worst case to find the
+critical floor from which eggs start breaking when dropped.
+"""
+
+# The Egg Dropping Problem is a classic dynamic programming problem.
+# - You are given `k` eggs and a building with `n` floors. Your goal is to determine
+#   the minimum number of attempts required to find the highest floor `F` from which
+#   if an egg is dropped, it will break. If an egg breaks from floor `F`, it will
+#   also break from any floor above `F`. The challenge is to minimize the worst-case
+#   number of attempts.
+
+
+def egg_dropping(eggs: int, floors: int) -> int:
+    """
+    Calculate the minimum number of attempts needed in the worst case for `eggs`
+    and `floors` using dynamic programming.
+
+    >>> egg_dropping(1, 5)
+    5
+    >>> egg_dropping(2, 6)
+    3
+    >>> egg_dropping(2, 10)
+    4
+    """
+
+    # Base case: No floors require 0 trials, one floor requires 1 trial.
+    if floors in (0, 1):
+        return floors
+    if eggs == 1:
+        return floors
+
+    # Create a DP table to store the results of subproblems.
+    dp = [[0 for _ in range(floors + 1)] for _ in range(eggs + 1)]
+
+    # Fill the base cases for one egg (i.e., we need `i` attempts for `i` floors).
+    for i in range(1, floors + 1):
+        dp[1][i] = i
+
+    # Compute the minimum number of trials in the worst case for each combination.
+    for e in range(2, eggs + 1):
+        for f in range(1, floors + 1):
+            dp[e][f] = 10**9  # Initialize to infinity.
+            for x in range(1, f + 1):
+                res = 1 + max(dp[e - 1][x - 1], dp[e][f - x])
+                dp[e][f] = min(dp[e][f], res)
+
+    return dp[eggs][floors]
+
+
+if __name__ == "__main__":
+    print("\n********* Egg Dropping Problem Using Dynamic Programming ************\n")
+    print("\n*** Enter -1 at any time to quit ***")
+    print("\nEnter the number of eggs and floors separated by a space: ", end="")
+    try:
+        while True:
+            input_data = input().strip()
+            if input_data == "-1":
+                print("\n********* Goodbye!! ************")
+                break
+            else:
+                eggs, floors = map(int, input_data.split())
+                print(
+                    f"The minimum number of attempts required with {eggs} eggs and "
+                    f"{floors} floors is:"
+                )
+                print(egg_dropping(eggs, floors))
+                print("Try another combination of eggs and floors: ", end="")
+    except NameError, ValueError:
+        print("\n********* Invalid input, goodbye! ************\n")