The genome can be modeled as a set of strings (chromosomes) of distinguished elements called genes. Genome duplication is an important source of new gene functions and novel physiological pathways. Originally (ancestrally), a duplicated genome contains two identical copies of each chromosome, but through the genomic rearrangement mutational processes of reciprocal translocation (prefix and/or suffix exchanges between chromosomes) and substring reversals, this simple doubled structure is disrupted. At the time of observation, each of the chromosomes resulting from the accumulation of rearrangements can be decomposed into a succession of conserved segments, such that each segment appears exactly twice in the genome. We present exact algorithms for reconstructing the ancestral doubled genome in linear time, minimizing the number of rearrangement mutations required to derive the observed order of genes along the present-day chromosomes. Somewhat different techniques are required for a translocations-only model, a translocations/reversals one, both of these in the multichromosomal context (eukaryotic nuclear genomes), and a reversals-only model, for single chromosome prokaryotic and organellar genomes. We apply these methods to the yeast genome which is thought to have doubled, and to the liverwort mitochondrial genome, whose duplicate genes are unlikely to have arisen by genome doubling.