Added randomized algorithms

randomized_algorithms/miller_rabin_primality test.py

@@ -0,0 +1,54 @@
+import random
+
+
+def miller_rabin_primality_test(n: int, k: int = 5) -> bool:
+    """
+    Probabilistic primality test using Miller-Rabin algorithm.
+
+    Args:
+        n (int): Number to test for primality.
+        k (int): Number of iterations for accuracy.
+
+    Returns:
+        bool: True if n is probably prime, False if composite.
+
+    Examples:
+    >>> miller_rabin_primality_test(2)
+    True
+    >>> miller_rabin_primality_test(15)
+    False
+    >>> miller_rabin_primality_test(17)
+    True
+    """
+    if n <= 1:
+        return False
+    if n <= 3:
+        return True
+    if n % 2 == 0:
+        return False
+
+    # Write n-1 as 2^r * d
+    d = n - 1
+    r = 0
+    while d % 2 == 0:
+        d //= 2
+        r += 1
+
+    for _ in range(k):
+        a = random.randint(2, n - 2)
+        x = pow(a, d, n)
+        if x == 1 or x == n - 1:
+            continue
+        for _ in range(r - 1):
+            x = pow(x, 2, n)
+            if x == n - 1:
+                break
+        else:
+            return False
+    return True
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()

randomized_algorithms/randomized_quicksort.py

@@ -0,0 +1,33 @@
+import random
+from typing import List
+
+
+def randomized_quicksort(arr: List[int]) -> List[int]:
+    """
+    Sorts a list of integers using randomized QuickSort algorithm.
+
+    Args:
+        arr (List[int]): List of integers to sort.
+
+    Returns:
+        List[int]: Sorted list of integers.
+
+    Examples:
+    >>> randomized_quicksort([3, 6, 1, 8, 4])
+    [1, 3, 4, 6, 8]
+    >>> randomized_quicksort([])
+    []
+    """
+    if len(arr) <= 1:
+        return arr
+    pivot = random.choice(arr)
+    less = [x for x in arr if x < pivot]
+    equal = [x for x in arr if x == pivot]
+    greater = [x for x in arr if x > pivot]
+    return randomized_quicksort(less) + equal + randomized_quicksort(greater)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()

randomized_algorithms/reservoir_sampling.py

@@ -0,0 +1,37 @@
+import random
+from typing import List, Iterator, TypeVar
+
+T = TypeVar("T")
+
+
+def reservoir_sampling(stream: Iterator[T], k: int) -> List[T]:
+    """
+    Randomly select k items from a stream of unknown length using reservoir sampling.
+
+    Args:
+        stream (Iterator[T]): Input data stream.
+        k (int): Number of items to sample.
+
+    Returns:
+        List[T]: List of k randomly sampled items.
+
+    Examples:
+    >>> stream = iter(range(1, 11))
+    >>> len(reservoir_sampling(stream, 5))
+    5
+    """
+    reservoir: List[T] = []
+    for i, item in enumerate(stream):
+        if i < k:
+            reservoir.append(item)
+        else:
+            j = random.randint(0, i)
+            if j < k:
+                reservoir[j] = item
+    return reservoir
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()