The run-length encoding part came pretty naturally.
Propose a run-length encoding (RLE) scheme that stores each run as a (value, count) pair, achieving O(R) space where R is the number of runs. For the dot product, use two pointers to traverse both compressed vectors simultaneously, aligning runs by their logical positions and accumulating the product of overlapping segments. Emphasize that no decompression is needed and the time complexity is O(R1 + R2).
Pro tip: Mention that this approach is essentially a merge of two sorted lists of intervals, and highlight that it handles cases where runs partially overlap by splitting the overlap. Also, note that the same technique generalizes to other operations like addition or equality checks.
Propose storing the vector as a list of (value, count) pairs, where each pair represents a consecutive run of identical values. Explain that this uses space proportional to the number of runs, not the total length.
Initialize pointers i and j to the start of each compressed vector, and maintain remaining counts for the current runs. Also track the current logical position to know when runs end.
While both pointers are within bounds, determine the length of the overlapping segment between the current runs: min(remaining count in run i, remaining count in run j). Multiply the values of the two runs by this overlap length and add to the dot product.
After processing the overlap, decrement the remaining counts of both runs by the overlap length. If a run's remaining count becomes zero, advance its pointer to the next run. Continue until all runs are processed.
State that the algorithm runs in O(R1 + R2) time and uses O(1) extra space. Discuss edge cases such as different numbers of runs, zero-length vectors, and runs with zero counts (if allowed).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.