Let's solve this step-by-step as requested. --- ### **STEP 1: Introduction to the Concept** **Concept Introduction:** This question deals with the **lattice of queries** in data warehousing, where each node represents a possible query and the associated number is the cost to answer that query. The task is to **materialize (store)** 4 views (nodes), other than the top view, such that the overall query cost is minimized. The **greedy algorithm** is used here, which at each step selects the view that gives the maximum reduction in total cost. --- ### **STEP 2: Understanding the Lattice** Here is the lattice structure and costs: - **Top node:** `a (200)` - Down to `b (150)` and `c (170)` - `b (150)` down to `d (80)` and `e (100)` - `c (170)` down to `e (100)` and `f (60)` - `d (80)` down to `g (50)` - `e (100)` down to `g (50)` and `h (20)` - `f (60)` down to `h (20)` - `g (50)` and `h (20)` down to `none (1)` (not to be considered) --- ### **STEP 3: Apply the Greedy Algorithm** #### **How the Greedy Algorithm Works:** - In each step, pick the view that gives the **maximum cost saving** if materialized. - We need to pick **4 views** (other than the top view `a`). #### **Step-by-step selection:** 1. **First Selection:** - Materializing the **lowest-cost views** (leaf nodes) usually gives the highest cost savings. - Candidate leaf nodes: `g (50)` and `h (20)` - **Materialize `h (20)`**: - Any query that can use `h` will now cost 20 instead of higher ancestors. - Queries affected: `h, f, e, c, a` 2. **Second Selection:** - Next best saving: **Materialize `g (50)`** - Queries affected: `g, d, e, b, a` 3. **Third Selection:** - Next, consider nodes like `d (80)`, `e (100)`, or `f (60)`. - **Materialize `f (60)`** (as it covers a chain not covered by `g` or `h`). - Queries affected: `f, c, a` (but `h` already covers `f` partially) 4. **Fourth Selection:** - Consider `d (80)` or `e (100)`. - **Materialize `d (80)`** (since `e` is already partially covered by `g` and `h`). - Queries affected: `d, b, a` **Selected Views:** `h (20)`, `g (50)`, `f (60)`, `d (80)` --- ### **STEP 4: Final Answer in a Few Sentences** **Final Answer:** By applying the greedy algorithm, the 4 views to be materialized are: **h (20), g (50), f (60), and d (80)**. These selections minimize the total cost of answering all queries by always choosing the view that provides the greatest immediate cost reduction. --- ### **STEP 5: Summary of Steps** - **Step 1:** Introduced the problem and greedy algorithm concept. - **Step 2:** Analyzed the query lattice and costs. - **Step 3:** Applied the greedy algorithm to select views step-by-step. - **Step 4:** Summarized the resulting optimal materialized views. - **Step 5:** Briefly recapped what was done in each step.