Long-term memory motion-compensated prediction extends the spatial displacement utilized in block-based hybrid video coding by a frame reference parameter permitting the use of many previously decoded pictures. This extension of the motion search range significantly increases the motion-compensated prediction gain. However, the amount of motion search related computation is significantly increased by the new approach as well. Based on the triangle inequality, we investigate a modified motion search order especially suited for the long-term memory approach. The concept of half-pel refinement is incorporated into the new search ordering method. A hierarchy of triangle inequalities provides an additional speed-up at the cost of memory. It is demonstrated that lossy methods based on the hierarchy of triangle inequalities give additional speed-ups at small losses of prediction gain. At very minor losses in prediction gain, for the sequences Foreman, Mother-Daughter and Stefan a reduction i...