In　the　high　efficiency　video　coding　(HEVC)　standard,　motion　estimation　(ME)　adopts　a　quadtree　coding　structure　and　a　larger　search　range　to　improve　the　coding　performance.　These　advanced　coding　tools,　however,　dramatically　increase　the　computational　complexity.　To　accelerate　ME,　fast　methods　have　been　proposed　that　reduce　ME　complexity　at　the　expense　of　rate-distortion　(R-D)　performance.　However,　none　of　these　methods　can　claim　their　tradeoff　to　be　optimal.　In　this　paper,　we　propose　an　optimal　fast　motion　estimation　(FME)　algorithm　based　on　an　analytical　model　of　rate-complexity-distortion　(R-C-D).　We　extend　the　traditional　R-D　model　by　introducing　the　ME　complexity　into　it,　which　enables　us　to　explicitly　express　the　R-D　performance　under　different　complexity　budgets.　Based　on　the　R-C-D　model,　the　proposed　FME　finds　the　R-C-D　optimized　search　ranges　for　some　representative　prediction　units　(PUs),　which　are　then　extended　or　refined　dynamically　according　to　motion　characteristics　for　neighboring　PUs.　The　proposed　FME　enables　an　R-D　performance　close　to　that　of　full　search.　When　compared　with　the　default　FME　method　in　the　reference　software　of　HEVC,　the　proposed　fast　algorithm　can　reduce　the　complexity　by　over　80%　while　improving　the　R-D　performance.　Furthermore,　our　proposed　FME　algorithm　is　hardware　friendly,　as　regular　data　flow　enables　high　data　reuse　efficiency.