One substring, grown from each end

One substring, grown from each end

The same 7-character string built two ways: leftwards, one character at a time from its last, and rightwards from its first. At every step both the forward interval and the interval in the index of the reversed text are held, and the two are always the same width — they are the same occurrences counted twice, and the assertion that they are is what catches every off-by-one this structure can have. The widths fall from 1,047 to 1 either way, and the final interval is the same four numbers by either route. What the second index buys is not space: it is the freedom to add the next character at whichever end the search wants.

Drawn at 700 × 430, wide on the page. Everything above is what bidirectional-search returns with no arguments; the caption is the generator's own, computed from the numbers in the drawing rather than written beside it.

7 essays call bidirectional-search. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about — which is what optcheck and figfill exist to catch.