Okay, start by making a contigs file:

$ sort-bed inputs.unsorted.bed > inputs.bed
$ bedops --merge inputs.bed > contigs.bed

Now use bedmap with both --echo-map and --echo-map-size to report two pieces of information: a list of mapped transcripts (which overlap the contig) and their associated lengths:

$ bedmap --echo-map --echo-map-size contigs.bed inputs.bed > perContigTranscriptsWithSizes.txt

For each contig, this format is as follows:

transcript-1 ; transcript-2 ; ... ; transcript-N | length-1 ; length-2 ; ... ; length-N

Note that the transcripts and lengths are separated from each other by the pipe character (|), while the lists of individual transcripts and lengths are separated by semi-colons (;).

We feed this result to awk to split the transcripts and lengths into separate arrays. We find the index of the longest length and report that transcript. We pass those transcripts to sort-bed to report a sorted BED file containing the longest-length transcripts in sorted order:

$ awk '{ \
    split($0, components, "|"); \
    transcriptsStr = components[1]; \
    lengthsStr = components[2]; \
    max = 0; \
    maxIdx = -1; \
    numLengths = split(lengthsStr, lengths, ";"); \
    for (idx = 1; idx <= numLengths; idx++) { \
        if (lengths[idx] > max) { \
            maxIdx = idx; \
            max = lengths[idx]; \
        } \
    } \
    split(transcriptsStr, transcripts, ";"); \
    maxTranscript = transcripts[maxIdx]; \
    print maxTranscript; \
}' perContigTranscriptsWithSizes.txt \
| sort-bed - > answer.bed

Putting this all together into one pipeline:

$ sort-bed inputs.unsorted.bed > inputs.bed
$ bedops --merge inputs.bed \
    | bedmap --echo-map --echo-map-size - inputs.bed \
    | awk '{ \
        split($0, components, "|"); \
        transcriptsStr = components[1]; \
        lengthsStr = components[2]; \
        max = 0; \
        maxIdx = -1; \
        numLengths = split(lengthsStr, lengths, ";"); \
        for (idx = 1; idx <= numLengths; idx++) { \
            if (lengths[idx] > max) { \
                maxIdx = idx; \
                max = lengths[idx]; \
            } \
        } \
        split(transcriptsStr, transcripts, ";"); \
        maxTranscript = transcripts[maxIdx]; \
        print maxTranscript; \
    }' - \
    | sort-bed - > answer.bed

Here is an alternative to mergeBed that preserves ids:

$ sort-bed inputs.unsorted.bed > inputs.bed
$ bedops --merge inputs.bed | bedmap --echo --echo-map-id-uniq --delim '\t' - inputs.bed > answer.bed

The file answer.bed has the same structure as your demo output.bed file.

If you want more attributes of mapped elements than just the ids (such as ranges, etc.) there are other --echo-map-* operands. See the documentation link below for examples.

Links for docs:

• sort-bed: http://bedops.readthedocs.org/en/latest/content/reference/file-management/sorting/sort-bed.html
• bedops --merge: http://bedops.readthedocs.org/en/latest/content/reference/set-operations/bedops.html#merge-m-merge
• bedmap --echo-map-id-uniq: http://bedops.readthedocs.org/en/latest/content/reference/statistics/bedmap.html#echo