I have a friend with a greyhound and [originally] two cats, one grey and one tortie. When she first had the dog, he was kept on a lead in the workshop until the cats no longer spat when they saw him, and he no longer looked at them if they came in. He was then allowed free. Tortie soon put him in his place, Grey was much shyer but eventually ignored him. What was really comic what that on walks, he is not particularly cat-safe - unless the cat is grey or tortie. He then firmly looks the other way.
All animals slept upstairs, but if a cat was sitting on a step, dog would not go up but stood at the bottom and cried for help.