You are asked to solve two coding problems. ## Problem 1: Sparse vector dot product Given two vectors **A** and **B** of the same length **n** (potentially large, e.g., up to 100,000), most entries are zero. Implement a function (or a small class API) to compute the dot product: \[ A \cdot B = \sum_{i=0}^{n-1} A_i \times B_i \] **Input representation (sparse):** Each vector is provided as a list of non-zero entries, e.g. a list of pairs `(index, value)` sorted by index (or equivalently a map from index to value). **Output:** Return the integer dot product. **Constraints/notes:** - Indices are in `[0, n-1]`. - Values can be negative. - Aim for time proportional to the number of non-zero entries. ## Problem 2: Count connected islands in a grid Given an `m x n` 2D grid of characters (or integers) where `'1'` represents land and `'0'` represents water, count the number of **islands**. An island is a maximal set of land cells connected **4-directionally** (up, down, left, right). **Input:** `grid[m][n]` containing `'0'`/`'1'`. **Output:** The number of islands. **Constraints/notes:** - You may modify the grid in-place or use auxiliary memory. - Consider edge cases like empty grid, all water, all land, and thin grids (1 row/1 column).