Linear time isotonic and unimodal regression in the L1L1 and L∞L∞ norms

We consider L1L1-isotonic regression and L∞L∞ isotonic and unimodal regression. For L1L1-isotonic regression, we present a linear time algorithm when the number of outputs are bounded. We extend the algorithm to construct an approximate isotonic regression in linear time when the output range is bounded. We present linear time algorithms for L∞L∞ isotonic and unimodal regression.