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A description is provided of the software algorithms developed for the CMS tracker both for reconstructing charged-particle trajectories in proton-proton interactions and for using the resulting tracks to estimate the positions of the LHC luminous region and individual primary-interaction vertices. Despite the very hostile environment at the LHC, the performance obtained with these algorithms is found to be excellent. For t bar t events under typical 2011 pileup conditions, the average track-reconstruction efficiency for promptly-produced charged particles with transverse momenta of p T> 0.9 GeV is 94% for pseudorapidities of| η|< 0.9 and 85% for 0.9<| η|< 2.5. The inefficiency is caused mainly by hadrons that undergo nuclear interactions in the tracker material. For isolated muons, the corresponding efficiencies are essentially 100%. For isolated muons of p T= 100GeV emitted at| η|< 1.4, the resolutions are approximately …