<p><it>Abstract—</it>This short note presents constant-time algorithms for labeling the connected components of an image on a network of processors with a wide reconfigurable bus. The algorithms are based on a processor indexing scheme which employs constant-weight codes. The use of such codes enables identifying a single representative processor for each component in a constant number of steps. The proposed algorithms can label an $<tmath>N\times N</tmath>$ image in $<tmath>O</tmath>$(1) time using $<tmath>N^2</tmath>$ processors, which is optimal. Furthermore, the proposed techniques lead to an $<tmath>O(\hbox{\bf log}\, N /\hbox{\bf log}\,\hbox{\bf log}\, N)</tmath>$-time image labeling algorithm on a network of $<tmath>N^2</tmath>$ processors with a reconfigurable bus of width $<tmath>\hbox{\bf log}\, N</tmath>$ bits. It is shown that these techniques can be applied to labeling an undirected $<tmath>N</tmath>$-vertex graph represented by an adjacency matrix.</p><p><it>Index Terms—</it>Reconfigurable networks of processors, parallel processing, constant-time algorithms, constant-weight codes, labeling connected components, image computations, graphs, array processing.</p>