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Joseph Born Kadane is a statistician whose work connects Bayesian statistics, decision theory and applications of probability to practical evidence. In his own research statement, he describes work on prior elicitation: constructing probability distributions from an investigator's background knowledge, as well as on the foundations and computation of Bayesian inference.^[ref]

His institutional biography records that he joined Carnegie Mellon in 1971 and headed its statistics department from 1972 to 1981.^[ref] In computing, his name is also attached to Kadane's algorithm for the maximum subarray problem: the best segment ending at a new position either extends the preceding best-ending segment or starts at that position. This gives a linear-time dynamic programming solution.

Given a finite sequence of real numbers, find a nonempty contiguous block with maximum sum. Its restriction to contiguous blocks distinguishes it from choosing an arbitrary subset. The problem admits a linear-time solution using Kadane's algorithm.

Let $e_i$ be the best sum of a segment ending at position $i$, allowing the empty segment with sum zero. The maximum subarray problem recurrence isAn optimal nonempty segment ending at $i$ extends the best suffix ending at $i-1$, or starts fresh; a negative suffix cannot help. Keep its start index and record endpoints whenever a new overall best is found:

suffix = 0; best = 0
start = 0; bestLeft = 0; bestRight = 0
for i = 0,...,n-1:
    suffix = suffix + A[i]
    if suffix < 0:
        suffix = 0; start = i+1
    if suffix > best:
        best = suffix
        bestLeft = start; bestRight = i+1
return A[bestLeft:bestRight]

The half-open endpoint interval is empty initially, so empty input and all-negative input correctly return an empty segment. The invariant identifies the best nonnegative suffix and the best segment in the scanned prefix, proving correctness. This is Kadane's algorithm, using $O(n)$ time and $O(1)$ auxiliary space, excluding the returned slice. The displayed example yields sum $55$, from the segment $10,-5,40,10$.

This version of Kadane's algorithm, named for Joseph Born Kadane, permits the empty subsequence. For $j\ge 1$, let $E_j$ be the maximum of zero and the weight of a contiguous subsequence ending exactly at position $j$. Let $B_j$ be the best weight anywhere in the first $j$ positions. UseA segment ending at $j$ either extends one ending at $j-1$ or starts at $j$; the zero candidate permits discarding a negative predecessor and the empty answer. Any best segment of the first $j$ positions either ends at $j$ or already occurs in the first $j-1$. These two decompositions prove the dynamic programming equations by induction.

Here is an implementation returning endpoints, using zero-based indexing:There is exactly one addition per input element, hence $n$ additions and $O(n)$ comparisons. Auxiliary storage is $O(1)$. The all-negative case retains interval $[0,0)$ and weight zero. The example returns $[3,6)$ and weight 8, hence the desired contiguous subsequence.