Compute Theme Similarity

Implement Python functions to compare theme similarity using the Jaccard coefficient. ### Part 1: Basic Jaccard similarity Given two lists of strings, implement a function that returns their Jaccard similarity: `Jaccard(A, B) = size(intersection(A, B)) / size(union(A, B))` Requirements: - Return a float in `[0, 1]`. - Treat the input lists as sets, so duplicate strings should not affect the score. - If both lists are empty, return `1.0`. - If only one list is empty, return `0.0`. Example: ```python list_1 = ["a", "b", "c", "d"] list_2 = ["b", "c", "d", "e"] ``` The expected score is `3 / 5 = 0.6`. ### Part 2: Identify likely pirated custom themes You are given two lists of dictionaries: ```python pirated_themes = [ {"theme_id": "p1", "features": ["a", "b", "c"]}, {"theme_id": "p2", "features": ["x", "y", "z"]} ] custom_themes = [ {"theme_id": "c1", "features": ["a", "b", "d"]}, {"theme_id": "c2", "features": ["x", "y", "z"]} ] ``` Each theme dictionary contains: - `theme_id`: a unique theme identifier. - `features`: a list of strings representing extracted theme attributes, assets, file hashes, CSS classes, or other comparable signals. Implement a function: ```python def find_likely_pirated_themes(pirated_themes, custom_themes, threshold=0.8): ... ``` For each custom theme: 1. Compare it with every known pirated theme using Jaccard similarity over `features`. 2. Find the best matching pirated theme. 3. Return custom themes whose best similarity score is greater than or equal to `threshold`. The returned result should include, for each flagged custom theme: - `custom_theme_id` - `matched_pirated_theme_id` - `similarity_score` Handle missing or empty `features` lists gracefully, do not mutate the input objects, and make the output deterministic.