Mindstorms EV3: Patrolling a number of rooms - what are my options?
For moving on defined path you can achieve amazing accuracy with LEGO MINDSTORMS EV3 odometry combined with good gyroscope but this still will result in a collision sooner or later.

Examples of such systems can be found here and here

Both of these systems use MICROINFINITY CruizCore XG1300L gyro/accelerometer but there are other good gyroscopes.

The only way to limit the growth of positioning error (and eventual collision) are absolute positioning methods such as beacons, artificial/natural landmarks recognition or model matching.

The state of the art in robotics is SLAM which could be implented on top of system with LIDAR, odometry and possibly gyroscope, preferably with ready-made ROS SLAM algorithms and computation on PC (EV3 is too slow). Making such system work is quite involved and may take months of work.

I'm not sure if this would be any faster than following a line though.

You can also consider following the wall, with Maze Solving Algorithms to move around the rooms.