Explain the algorithm for computing the dot product of two compressed (RLE-encoded) vectors in a single pass without unpacking them. What is the asymptotic time complexity?

The dot product of two RLE-compressed vectors can be computed in a single pass without fully unpacking them, using the idea of parallel block traversal.

Each vector is represented as a sequence of pairs (value, length), for example: [(v1, l1), (v2, l2), ...].

Algorithm:

1. Initialize two pointers to the first blocks of each vector.
2. At each step, take the current blocks from both vectors.
3. Calculate the minimum block length minLen = min(l1, l2).
4. Add to the result the product of the block values multiplied by minLen: result += v1 * v2 * minLen.
5. Decrease the block lengths by minLen.
6. If the block length in one of the vectors becomes 0, move to the next block of that vector.
7. Repeat until all blocks of both vectors are processed.

Asymptotic complexity:

• Time proportional to the total number of blocks in both RLE representations, i.e., O(n + m), where n and m are the number of blocks in the first and second vectors respectively.

Thus, we efficiently compute the dot product without unpacking the vectors.