Dot Product of Two Sparse Vectors

Dot Product of Two Sparse Vectors

What is this problem about?

The Dot Product of Two Sparse Vectors coding problem involves calculating the dot product of two vectors that contain a large number of zeros. A sparse vector is one where most elements are zero. Computing the dot product using a standard array approach would be inefficient because most multiplications would involve zero. The goal is to design a data structure that stores only the non-zero elements and computes the dot product efficiently.

Why is this asked in interviews?

This question is a favorite at companies like Meta and Google because it tests both data structure design and algorithmic optimization. It simulates real-world scenarios in machine learning and data science where datasets are often sparse. The Dot Product of Two Sparse Vectors interview question evaluates how a candidate handles memory constraints and whether they can optimize a common mathematical operation for specialized data.

Algorithmic pattern used

There are two primary design interview patterns for this problem:

1. Hash Table: Store the index and value of each non-zero element in a hash map. To compute the dot product, iterate through the keys of the smaller map and check if the index exists in the larger map.
2. Two Pointers: Store non-zero elements as a list of pairs (index, value) sorted by index. Use a two-pointer approach to find common indices and multiply their values. This is often preferred if the indices are sorted.

Example explanation

Suppose we have two vectors:

• v1 = [1, 0, 0, 2, 3]
• v2 = [0, 3, 0, 4, 0]

1. Non-zero representation for v1: {(0, 1), (3, 2), (4, 3)} (as pairs or hash map).
2. Non-zero representation for v2: {(1, 3), (3, 4)}.
3. Calculating Dot Product:
   • Check index 0 from v1: Not in v2.
   • Check index 3 from v1: In v2 with value 4. Product = 2 * 4 = 8.
   • Check index 4 from v1: Not in v2. Total Dot Product = 8.

Common mistakes candidates make

• Iterating over the full array: Using a standard O(N) loop that processes all elements, including zeros, which defeats the purpose of the optimization.
• Ignoring the smaller vector: Not optimizing the iteration to always loop through the vector with fewer non-zero elements.
• Inefficient storage: Using a storage method that doesn't allow for quick lookup or traversal of non-zero indices.

Interview preparation tip

Be ready to discuss the trade-offs between the Hash Table and Two Pointers approaches. Hash tables offer O(1) average lookup but may have higher memory overhead, while two pointers on sorted lists are very memory-efficient and cache-friendly.