Problem Description

You are tasked with implementing and optimizing vector-vector multiplication. This problem has three main parts:

Part 1: Dense Vector Multiplication

Implement the vector-vector multiplication (e.g., dot product) for dense vectors.

Part 2: Optimization for Parallel Architectures

Describe how you would optimize your implementation from Part 1 for a parallel architecture. Consider techniques such as:

• SIMD (Single Instruction, Multiple Data)
• Multithreading
• Memory alignment
• Cache behavior

Part 3: Sparse Vector Multiplication

Explain how you would perform the same vector-vector multiplication operation when the input vectors are sparse. This explanation should include:

• Suitable data structures for sparse vectors.
• How sparsity affects computational complexity.
• How sparsity impacts parallelization strategies.