feat(maths): add repunit theorem helpers

dynamic_programming/knapsack.py

@@ -39,6 +39,72 @@ def knapsack(w, wt, val, n):
     return dp[n][w_], dp
 
 
+def knapsack_space_optimized(
+    capacity: int, weights: list[int], values: list[int], num_items: int
+) -> int:
+    """
+    Solve the 0/1 knapsack problem with O(capacity) extra space.
+
+    It uses a 1D dynamic programming array and iterates capacities in reverse
+    for each item to avoid reusing the same item more than once.
+
+    >>> knapsack_space_optimized(50, [10, 20, 30], [60, 100, 120], 3)
+    220
+    >>> knapsack_space_optimized(0, [10, 20, 30], [60, 100, 120], 3)
+    0
+    >>> knapsack_space_optimized(6, [4, 3, 2, 3], [3, 2, 4, 4], 4)
+    8
+    >>> knapsack_space_optimized(-1, [1], [1], 1)
+    Traceback (most recent call last):
+        ...
+    ValueError: The knapsack capacity cannot be negative.
+    >>> knapsack_space_optimized(1, [1], [1], -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: The number of items cannot be negative.
+    >>> knapsack_space_optimized(1, [1], [1], 2)
+    Traceback (most recent call last):
+        ...
+    ValueError: The number of items exceeds the provided input lengths.
+    >>> knapsack_space_optimized(1, [-1], [1], 1)
+    Traceback (most recent call last):
+        ...
+    ValueError: Weight at index 0 cannot be negative.
+    >>> knapsack_space_optimized(1, [1], [1.5], 1)
+    Traceback (most recent call last):
+        ...
+    TypeError: Value at index 0 must be an integer.
+    """
+    if num_items < 0:
+        raise ValueError("The number of items cannot be negative.")
+    if capacity < 0:
+        raise ValueError("The knapsack capacity cannot be negative.")
+    if num_items > len(weights) or num_items > len(values):
+        raise ValueError("The number of items exceeds the provided input lengths.")
+    for item_index in range(num_items):
+        item_weight = weights[item_index]
+        item_value = values[item_index]
+        if not isinstance(item_weight, int):
+            msg = f"Weight at index {item_index} must be an integer."
+            raise TypeError(msg)
+        if item_weight < 0:
+            msg = f"Weight at index {item_index} cannot be negative."
+            raise ValueError(msg)
+        if not isinstance(item_value, int):
+            msg = f"Value at index {item_index} must be an integer."
+            raise TypeError(msg)
+
+    dp = [0] * (capacity + 1)
+    for item_index in range(num_items):
+        item_weight = weights[item_index]
+        item_value = values[item_index]
+        for current_capacity in range(capacity, item_weight - 1, -1):
+            dp[current_capacity] = max(
+                dp[current_capacity], item_value + dp[current_capacity - item_weight]
+            )
+    return dp[capacity]
+
+
 def knapsack_with_example_solution(w: int, wt: list, val: list):
     """
     Solves the integer weights knapsack problem returns one of

maths/repunit_theorem.py

@@ -0,0 +1,63 @@
+"""
+Utilities related to repunits and a classical repunit divisibility theorem.
+
+A repunit of length ``k`` is the number made of ``k`` ones:
+``R_k = 11...1``.
+
+For every positive integer ``n`` with ``gcd(n, 10) = 1``,
+there exists a repunit ``R_k`` divisible by ``n``.
+"""
+
+from math import gcd
+
+
+def has_repunit_multiple(divisor: int) -> bool:
+    """
+    Check whether a divisor admits a repunit multiple.
+
+    >>> has_repunit_multiple(7)
+    True
+    >>> has_repunit_multiple(13)
+    True
+    >>> has_repunit_multiple(2)
+    False
+    >>> has_repunit_multiple(25)
+    False
+    """
+    if divisor <= 0:
+        raise ValueError("divisor must be a positive integer")
+    return gcd(divisor, 10) == 1
+
+
+def least_repunit_length(divisor: int) -> int:
+    """
+    Return the smallest length ``k`` such that repunit ``R_k`` is divisible by divisor.
+
+    Uses modular arithmetic to avoid constructing huge integers.
+
+    >>> least_repunit_length(3)
+    3
+    >>> least_repunit_length(7)
+    6
+    >>> least_repunit_length(41)
+    5
+    """
+    if divisor <= 0:
+        raise ValueError("divisor must be a positive integer")
+    if not has_repunit_multiple(divisor):
+        raise ValueError("divisor must be coprime to 10")
+
+    remainder = 0
+    for length in range(1, divisor + 1):
+        remainder = (remainder * 10 + 1) % divisor
+        if remainder == 0:
+            return length
+
+    # Unreachable when gcd(divisor, 10) == 1 (pigeonhole principle theorem).
+    raise ArithmeticError("no repunit length found for divisor")
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()