The distances are a function of throughput and speed at which you want it to take place.
You asked about the Norfolk and Southern , which was spliced together largely out of a series of logging railroad rights of way. It was a manual block, station to station operation except for interlockers . The latter were at :terminal areas involving NP Beltline mostly, in the Norfolk area ; the bridges across the wide mouths of rivers in the western parts of the sounds; the Raleigh station area's Boylan Tower; etc. In other words where they crossed railroads at grade. Boylan tower dealt with crossing of Seaboard Air Line mainline from Richmond to Florida , and the Southern Railway's leased North Carolina Railroad mainline from Wilmington, N.C. to Charlotte via the northern Piedmont. THe latter went via Durham,Burlington,and Greensboro where it connected to the Washington , Charlotte, Spartanburg,Atlanta , Birmingham , et al to NO mainline.
The Norfolk and Southern was acquired and merged into Southern in 1970s prior to merger of Norfolk and Western and Southern that lead to revival of names for combined road. Nevertheless, a complex terminal area crossing such as the one controlled by Boylan Tower would have had a mixture of advanced and absolute signalling controlled by line of sight requirements of all trackage . Furthermore, to make it work safely , a single dispatcher from one carrier (originally Seaboard Air Line , at Seaboard Division Office Building , located a few blocks from Governor's Mansion in Raleigh through 1970s) needs to be in charge of the plant . The need for advanced signals , or combination signals is related both to speed and visibility and complexity of the route options (i.e. junctions?,crossings?, passing sidings and whether they are remote or manual, etc.). The absence or presence of cab signals , and before them ASSD's ( i.e.Blasters) had a bearing on the matter as fog,smoke, and cold weather precipitation severely restricted visibility.
If you have a set of variables defined by the trainspeed and types of trains, and the number of running tracks and length and location of sidings, given radio dispatching , and cab signals you can work out formulas for minimum number of signals to accomplish your train moving goal. Specific route condition deviations from the ideal , such as the junctions , and crossings unique to the line have to be factored into your model.
O.S. Nock's ,SINGLE LINE RAILWAYS; David and Charles for UK Railway Advisory Services;, 1966; is a book written by one of the world's most distinguished raifans , who also happened to be a professional engineer. So his descriptions of all forms of train working, by diverse forms of control are very helpful for a layman or professional as they explain what was the technological history of signalling. There are other railway engineering texts, (modern ones from Wiley in U.S.) , but if your math is tolerable , and you are really interested in this matter you should get the manuals and occassional papers put out by the train control system manufacturers. Bennet Feely , is a bookdealer in the Pittsburgh suburbs that specializes in the sale of signal company brochures and technical papers.
If your interest doesn't extend that far, buy ETTs and track charts for the railroad lines you want to know about for the epoch you want to know about. The railway historical societies of the various carriers almost always have copies of these for sale.
Good-Luck, PJB