Median of Two Sorted Streaming Order Feeds

A consolidated order feed aggregator at Citadel and HRT receives two independent sorted streams of executed transaction prices, nums1 (size $M$) and nums2 (size $N$).

Both feeds are sorted in non-decreasing numerical order. To establish the benchmark mid-market price across both venues, calculate the overall median price of the combined dataset.

Your Task

Implement find_median_sorted_arrays(nums1: list[float], nums2: list[float]) -> float that returns the median price in strictly $\mathcal{O}(\log(\min(M, N)))$ time without merging the two arrays into memory.

Constraints

• $0 \le M, N \le 10^5$ and $M + N \ge 1$
• $-10^6 \le nums1[i], nums2[j] \le 10^6$
• Execution must run in logarithmic time $\mathcal{O}(\log(\min(M, N)))$.

Examples

nums1 = [1.0, 3.0]
nums2 = [2.0]
# Combined sorted: [1.0, 2.0, 3.0]. Median = 2.0
# Output: 2.0

nums1 = [1.0, 2.0]
nums2 = [3.0, 4.0]
# Combined sorted: [1.0, 2.0, 3.0, 4.0]. Median = (2.0 + 3.0) / 2 = 2.5
# Output: 2.5