Implement a small set of Python functions to detect likely pirated Shopify themes by comparing their extracted feature sets using the **Jaccard similarity coefficient**. The Jaccard similarity between two sets $A$ and $B$ is defined as: $$J(A, B) = \frac{|A \cap B|}{|A \cup B|}$$ This problem has two parts: Part 1 builds the core similarity primitive, and Part 2 reuses it to scan a catalog of custom themes against a set of known pirated themes. ### Constraints & Assumptions - Inputs are plain Python lists of strings (Part 1) and lists of dictionaries (Part 2). No external libraries beyond the standard library are required. - `features` strings are treated as opaque tokens (extracted theme attributes, asset names, file hashes, CSS classes, etc.); equality is exact string equality. - Lists may contain duplicate strings; duplicates must not affect the score (the inputs are conceptually sets). - A single comparison runs over modest feature lists (tens to a few hundred tokens). The full pirated-vs-custom scan is $O(P \times C)$ pairwise comparisons, where $P$ and $C$ are the number of pirated and custom themes respectively — assume both fit comfortably in memory. - Functions must not mutate their input objects, and output ordering must be deterministic across runs. ### Clarifying Questions to Ask - Are `features` strings case-sensitive, and should they be normalized (trimmed/lowercased) before comparison, or compared exactly as given? - Is the threshold comparison inclusive (`>= threshold`) or strict (`> threshold`)? Should `threshold` be validated to lie in `[0, 1]`? - Can the same custom theme appear more than once in the input, and if so should it be flagged once or per occurrence? - For a flagged theme, do we need only the single best pirated match, or all pirated matches at or above the threshold? - Are `theme_id` values guaranteed unique and non-null within each list? ### Part 1: Basic Jaccard similarity Given two lists of strings, implement a function that returns their Jaccard similarity. Requirements: - Return a `float` in the closed interval `[0, 1]`. - Treat the input lists as sets, so duplicate strings must not change the score. - If **both** lists are empty, return `1.0` (two empty sets are identical). - If **exactly one** list is empty, return `0.0`. Example: ```python list_1 = ["a", "b", "c", "d"] list_2 = ["b", "c", "d", "e"] # intersection = {b, c, d} (size 3), union = {a, b, c, d, e} (size 5) # expected score = 3 / 5 = 0.6 ``` ```hint Data structure Converting each list to a `set` gives you O(1) membership and built-in `&` (intersection) and `|` (union) operators, and it deduplicates for free. ``` ```hint Edge cases Decide the empty-input behavior *before* dividing — when the union is empty (both inputs empty) the formula is $0/0$, so you must short-circuit it. Map "both empty → 1.0" and "one empty → 0.0" explicitly rather than relying on the division. ``` #### What This Part Should Cover - A correct, deduplicating implementation that maps the formula directly onto `set` operations. - All empty-input edge cases handled explicitly, with the division-by-zero case ($0/0$) short-circuited before any arithmetic. - A guaranteed `float` result in `[0, 1]`, plus awareness that comparing floats with `>=` near a boundary can invite rounding surprises. ### Part 2: Identify likely pirated custom themes You are given two lists of theme 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 (string). - `features`: a list of strings representing extracted theme signals. This key may be missing or its value may be an empty list. Implement: ```python def find_likely_pirated_themes(pirated_themes, custom_themes, threshold=0.8): ... ``` For each custom theme: 1. Compare it against **every** known pirated theme using the Part 1 Jaccard similarity over `features`. 2. Find the single best-matching pirated theme (highest similarity). 3. Flag the custom theme if its best similarity score is **greater than or equal to** `threshold`. For each flagged custom theme, the returned result must include: - `custom_theme_id` - `matched_pirated_theme_id` - `similarity_score` Additional requirements: - Handle missing or empty `features` gracefully (a missing key must not raise). - Do not mutate any input object. - Make the output deterministic (a stable, well-defined ordering and a defined tie-break rule when two pirated themes score equally). ```hint Where to start Reuse your Part 1 function as the inner kernel. For each custom theme, loop over all pirated themes and keep a running "best" (score and the pirated `theme_id` that produced it). ``` ```hint Robustness Use `theme.get("features", [])` so a missing key yields an empty list instead of a `KeyError`. An empty custom feature set should never falsely match — your Part 1 empty-set rules already guarantee it scores `0.0` against any *non-empty* pirated theme. ``` ```hint Determinism & tie-breaks If the input order of `pirated_themes` is not guaranteed stable, two themes can tie for "best". Pick an explicit tie-break — e.g. iterate in a fixed order and only replace the best on a *strictly greater* score, or break ties by `theme_id`. Document whichever rule you choose so the output is reproducible. ``` #### Clarifying Questions for this Part - When a pirated theme's own `features` is empty, should it be skipped as a candidate entirely? (If not, Part 1's "both empty → 1.0" rule could make an empty custom theme match it and clear any `threshold <= 1.0`.) - Should a custom theme with missing/empty `features` ever appear in the output, or always be excluded? #### What This Part Should Cover - Clean reuse of the Part 1 function as the inner kernel rather than re-deriving the union/intersection math. - Graceful handling of missing / `None` / empty `features` on both sides (no exceptions) and no mutation of any input. - An explicit, *stated* tie-break rule together with a deterministic output ordering. - A return shape matching the spec exactly (`custom_theme_id`, `matched_pirated_theme_id`, `similarity_score`), plus a guard that avoids emitting a bogus flag when `pirated_themes` is empty. ### What a Strong Answer Covers These dimensions span both parts: - Modular design where Part 2 is a thin scan layered on top of the single Part 1 primitive, with no duplicated similarity math. - Awareness of overall correctness boundaries: scores live in `[0, 1]`, float-comparison caution near the `threshold`, and the $O(P \times C)$ cost of the all-pairs scan. - A few well-chosen test cases that exercise both parts together: identical sets, disjoint sets, partial overlap, empty/missing features, and duplicate tokens. ### Follow-up Questions - The naive scan is $O(P \times C)$ pairwise comparisons. If there were millions of themes, how would you make near-duplicate detection scalable (e.g. MinHash + Locality-Sensitive Hashing, or inverted indexes on features)? - Jaccard weights every feature equally. How would you incorporate the fact that some features (a rare custom file hash) are far more discriminative than common ones (a generic CSS class) — e.g. TF-IDF weighting or weighted Jaccard? - How would you choose and validate the `threshold` in production — what precision/recall trade-off, and how would you measure it against labeled piracy cases? - The `features` here are exact tokens. How would you extend this to detect *near*-identical assets (minified CSS with renamed classes, re-encoded images) rather than only exact-match signals?