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In this paper we present a more efficient approach to the problem of finding the connected components in binary images. In conventional connected components algorithms, the main data structure to compute and store the connected components is the region label image. We replace the region label image with a singly-linked list of line segments (or runs) for each region. This enables us to design a very fast and memory efficient connected components algorithm. Most conventional algorithms require (at least) two raster scans. Those that only need one raster scan, require irregular and unbounded image access. The proposed algorithm is a single pass regular access algorithm and only requires access to the three most recently processed image lines at any given time. Experimental results demonstrate that our algorithm is considerably faster than the fastest conventional algorithm. Additionally, our novel region coding data structure uses much less memory in typical cases than the traditional region label image. Even in worst case situations the processing time of our algorithm is linear with the number of pixels in an image.