*The following is superseded by our later work Functional pearl: finding a densest segment.*

Sharon Curtis and Shin-Cheng Mu. Submitted.

[PDF]

**errata**:

- Page 3: “This input sequence does not have a solution…” what we meant was “This input does not have a prefix that is within bounds.” We used another example where the input does not have a feasible segment at all before changing to example, but I forgot to change the text accordingly.
- Page 4, Proof of Theorem 3.2: the first
`mds`

should be_{M}x ⇑_{d}win (a:x)`mds`

;_{M}x ⇑_{d}wp (trim (a:x))`a : x <`

and_{b}L`a : x ≥`

should respectively be_{b}L`trim (a : x) <`

and_{b}L`trim (a : x) ≥`

._{b}L

Thanks to Josh Ko for pointing out both errors.

The problem of finding a maximally dense segment (MDS) of a list is a generalisation of the well-known maximum segment sum (MSS) problem, but its solution is more challenging. We extend and illuminate some recent work on this problem with a formal development of a linear-time online algorithm, in the form of a sliding window zygomorphism. The development highlights some elegant properties of densities, involving partitions which are decreasing and all right-skew.

Code and supplementary proofs are available online.

**keywords**: program derivation, segment problem, maximum density, sliding window, zygomorphism, right-skew.

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