Implement Game Physics and Block Mining

You are given two independent coding tasks from the same interview category. Solve both. ### Task A: Flappy-Bird-Style Jump Boundary Function Implement a function for a simple side-scrolling game that updates a bird's vertical position for one simulation tick and reports whether the bird is still inside the playable area. The game uses the following coordinate system: - `y = 0` is the ceiling. - `y = height` is the floor. - Positive velocity moves the bird downward. Function signature: `update_bird(y, velocity, jump_pressed, height, gravity, jump_velocity) -> (new_y, new_velocity, alive)` Rules: 1. If `jump_pressed` is true, set the bird's velocity to `jump_velocity`. `jump_velocity` may be negative. 2. Otherwise, keep the current velocity. 3. Apply gravity to the velocity for this tick: `new_velocity = velocity_after_jump + gravity`. 4. Update position: `new_y = y + new_velocity`. 5. The bird is alive only if `0 <= new_y <= height`. 6. Return the new position, new velocity, and alive status. Handle edge cases such as jumps near the ceiling, falling below the floor, and zero gravity. ### Task B: Build a Cryptocurrency Block with Transaction Dependencies You are building a simplified block-mining simulator. Each pending transaction has: - `id`: unique string - `fee`: integer fee paid by the transaction - `size`: integer size in bytes - `parents`: list of transaction IDs that must appear earlier in the same block if this transaction is included Given a maximum block size `max_size`, choose an ordered list of transaction IDs to include in the block. Rules: 1. A transaction can be included only if all of its parent transactions are also included earlier in the returned order. 2. The total size of all included transactions must be at most `max_size`. 3. Use the following greedy strategy rather than solving the globally optimal knapsack problem: - For each not-yet-included transaction, compute the package consisting of that transaction plus all missing ancestors required to include it. - The package fee is the sum of fees of missing transactions in the package. - The package size is the sum of sizes of missing transactions in the package. - Repeatedly choose the valid package with the highest `package_fee / package_size` ratio that fits in the remaining block size. - Add the package transactions in dependency order. 4. If the dependency graph contains a cycle, report an error. 5. Return the ordered transaction IDs, total fee, and total size. Discuss the time complexity of your implementation and how you would cache ancestor-package computations for large mempools.