Dot Product of Two Sparse Vectors

Background Knowledge

The problem deals with sparse vectors, which are vectors that contain mostly zero elements. In such cases, storing all the elements can be inefficient, especially when dealing with high-dimensional vectors. To optimize storage and computation, we only store the non-zero elements of the sparse vectors. This is a common technique used in various applications, including linear algebra and machine learning.

In the context of this problem, we are given two sparse vectors represented as arrays, and we need to compute their dot product. The dot product of two vectors is the sum of the products of their corresponding elements. For sparse vectors, we can take advantage of the fact that most elements are zero to reduce the number of computations required. This is where hash maps come into play, as they can be used to efficiently store and retrieve the non-zero elements of the sparse vectors.

To understand the problem better, let's consider an example. Suppose we have two sparse vectors vec1 and vec2, each with a few non-zero elements. We can represent these vectors using hash maps, where the keys are the indices of the non-zero elements and the values are the corresponding element values. For instance, vec1 might be represented as {0: 1, 3: 2, 5: 3}, indicating that the non-zero elements are at indices 0, 3, and 5 with values 1, 2, and 3, respectively.