Implement weighted random choice

Implement a simplified version of np.random.choice without calling the built-in function. You are given an array items of length n and an optional probability array p of length n, where p[i] is the probability of returning items[i] and the probabilities sum to 1 within numerical tolerance.

Requirements:

• If p is omitted, sample uniformly.
• Support returning a single sample.
• Extend the function to support size = k , returning k independent samples with replacement.
• Explain how you would validate invalid inputs such as negative probabilities, length mismatches, or probabilities that do not sum correctly.
• Discuss the time and space complexity of your approach.

Bonus: describe an efficient approach for repeated sampling or for sampling without replacement.