You are given two separate coding tasks from an interview. ## Task 1: Optimize storage for vectors with repeated values, then compute dot product You are given integer vectors `A` and `B` of the same length `n` (potentially very large). The vectors often contain long *runs* of the same value (i.e., many **consecutive duplicates**). 1. Design a compressed representation for a vector using run-length encoding (RLE), e.g. represent `[-2, -2, -2, 5, 5, 0, 0, 0, 0]` as `[(-2,3), (5,2), (0,4)]`. 2. Using only the compressed representations of `A` and `B` (without fully expanding them), compute the dot product: \[ A \cdot B = \sum_{i=0}^{n-1} A[i] \times B[i] \] ### Input/Output - **Input:** two arrays `A` and `B` (or their compressed forms) with the same length. - **Output:** the dot product as an integer (or 64-bit integer if needed). ### Constraints (typical) - `1 <= n <= 10^7` - Values fit in 32-bit signed integer; the dot product may require 64-bit. - Compression should be significantly smaller than `n` when there are long runs. ### Example - `A = [1,1,1,2,2]` → `[(1,3),(2,2)]` - `B = [3,3,4,4,4]` → `[(3,2),(4,3)]` - Dot product = `1*3 + 1*3 + 1*4 + 2*4 + 2*4 = 22` ## Task 2: Compute exclusive execution time from nested logs You are given `n` functions labeled `0..n-1` and a list of execution logs. Each log entry is a string: - `"<id>:start:<timestamp>"` or `"<id>:end:<timestamp>"` Rules: - Function calls can be nested (single-threaded CPU). - When a function is running, its time should be counted **exclusively**, excluding time spent in functions it calls. - Timestamps are integers and are inclusive at `end` (i.e., an `end` at time `t` includes the unit time `t`). ### Input/Output - **Input:** integer `n`, array `logs` of strings. - **Output:** array `ans` of length `n` where `ans[i]` is the exclusive time of function `i`. ### Example - `n = 2` - `logs = ["0:start:0","1:start:2","1:end:5","0:end:6"]` - Output: `[3,4]` (Explanation: function 0 runs at times 0–1 and 6 → 3 units; function 1 runs at times 2–5 → 4 units.)