largest divisible subset (#9825)

* largest divisible subset

* minor tweaks

* adding more test cases

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

* improving code for better readability

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

* update

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

* update

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

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* suggested changes done, and further modfications

* final update

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* Update largest_divisible_subset.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Update largest_divisible_subset.py

---------

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

dynamic_programming/largest_divisible_subset.py

@@ -0,0 +1,74 @@
+from __future__ import annotations
+
+
+def largest_divisible_subset(items: list[int]) -> list[int]:
+    """
+    Algorithm to find the biggest subset in the given array such that for any 2 elements
+    x and y in the subset, either x divides y or y divides x.
+    >>> largest_divisible_subset([1, 16, 7, 8, 4])
+    [16, 8, 4, 1]
+    >>> largest_divisible_subset([1, 2, 3])
+    [2, 1]
+    >>> largest_divisible_subset([-1, -2, -3])
+    [-3]
+    >>> largest_divisible_subset([1, 2, 4, 8])
+    [8, 4, 2, 1]
+    >>> largest_divisible_subset((1, 2, 4, 8))
+    [8, 4, 2, 1]
+    >>> largest_divisible_subset([1, 1, 1])
+    [1, 1, 1]
+    >>> largest_divisible_subset([0, 0, 0])
+    [0, 0, 0]
+    >>> largest_divisible_subset([-1, -1, -1])
+    [-1, -1, -1]
+    >>> largest_divisible_subset([])
+    []
+    """
+    # Sort the array in ascending order as the sequence does not matter we only have to
+    # pick up a subset.
+    items = sorted(items)
+
+    number_of_items = len(items)
+
+    # Initialize memo with 1s and hash with increasing numbers
+    memo = [1] * number_of_items
+    hash_array = list(range(number_of_items))
+
+    # Iterate through the array
+    for i, item in enumerate(items):
+        for prev_index in range(i):
+            if ((items[prev_index] != 0 and item % items[prev_index]) == 0) and (
+                (1 + memo[prev_index]) > memo[i]
+            ):
+                memo[i] = 1 + memo[prev_index]
+                hash_array[i] = prev_index
+
+    ans = -1
+    last_index = -1
+
+    # Find the maximum length and its corresponding index
+    for i, memo_item in enumerate(memo):
+        if memo_item > ans:
+            ans = memo_item
+            last_index = i
+
+    # Reconstruct the divisible subset
+    if last_index == -1:
+        return []
+    result = [items[last_index]]
+    while hash_array[last_index] != last_index:
+        last_index = hash_array[last_index]
+        result.append(items[last_index])
+
+    return result
+
+
+if __name__ == "__main__":
+    from doctest import testmod
+
+    testmod()
+
+    items = [1, 16, 7, 8, 4]
+    print(
+        f"The longest divisible subset of {items} is {largest_divisible_subset(items)}."
+    )