5 ms·

‘15 stops over 159 total miles’ Sorry but I am having hard time grasping math on this, can you explain a little bit more?

jjk166 6y ago
Let's first consider the case of a single line. There are 210 stations spaced a half mile apart each. You are starting out at station 1, you want to get to station 186. The train starts at station 1, then stops at 2, then 3, then 4... eventually you get to station 196 - you've stopped 195 times and travelled 98 miles. Now consider if you had two lines that split the work. The first line goes to every 15th station, and the second line connects each 15th station with the 14 local stations preceding it. So you get on the first train at station 1, it stops at station 15, station 30, station 45... eventually you get to station 210. You have stopped 14 times and travelled 105 miles. Then you get on the local train, it stops at station 209, 208... until you eventually get to station 196. You have stopped an additional 14 times, and travelled an additional 14 miles, for a total of 28 stops and 119 miles travelled. If you add more lines, there are fewer stops on any given line, but you potentially have to travel a greater total distance as you repeatedly "overshoot" your target. Once you get to 5 lines, each line is just adding one stop between those of the earlier line - you've essentially reduced the commute to a binary search, and adding any more lines won't get you home in fewer stops. Since writing my initial comment, I have realized that in reality you could improve on this worst case scenario further. Instead of having a single local line that just connects a single hub to its local stations, you could have two lines connecting two hubs to the local stations between them, so that a traveler would at most travel half the distance between the hubs and never overshoot. This doubles the number of lines, but halves the worst case number of stops and substantially reduces overshoot. Thus you could reach any node in the city within 10 stops and a maximum trip length of 109 miles. Thus with the same parameters you could conceivably commute make the commute in 43 minutes, which is more reasonable though still double what Salman claims, and it would still be faster to bike if the city was circular. These 10 rail lines would run a total of 1050 miles, which at an extremely generous estimate of $300 million per mile of underground rail is about $315 Billion dollars, over 1.5X the maximum expected cost according to the article.

jjk166 6y ago

Let's first consider the case of a single line. There are 210 stations spaced a half mile apart each. You are starting out at station 1, you want to get to station 186. The train starts at station 1, then stops at 2, then 3, then 4... eventually you get to station 196 - you've stopped 195 times and travelled 98 miles. Now consider if you had two lines that split the work. The first line goes to every 15th station, and the second line connects each 15th station with the 14 local stations preceding it. So you get on the first train at station 1, it stops at station 15, station 30, station 45... eventually you get to station 210. You have stopped 14 times and travelled 105 miles. Then you get on the local train, it stops at station 209, 208... until you eventually get to station 196. You have stopped an additional 14 times, and travelled an additional 14 miles, for a total of 28 stops and 119 miles travelled. If you add more lines, there are fewer stops on any given line, but you potentially have to travel a greater total distance as you repeatedly "overshoot" your target. Once you get to 5 lines, each line is just adding one stop between those of the earlier line - you've essentially reduced the commute to a binary search, and adding any more lines won't get you home in fewer stops. Since writing my initial comment, I have realized that in reality you could improve on this worst case scenario further. Instead of having a single local line that just connects a single hub to its local stations, you could have two lines connecting two hubs to the local stations between them, so that a traveler would at most travel half the distance between the hubs and never overshoot. This doubles the number of lines, but halves the worst case number of stops and substantially reduces overshoot. Thus you could reach any node in the city within 10 stops and a maximum trip length of 109 miles. Thus with the same parameters you could conceivably commute make the commute in 43 minutes, which is more reasonable though still double what Salman claims, and it would still be faster to bike if the city was circular. These 10 rail lines would run a total of 1050 miles, which at an extremely generous estimate of $300 million per mile of underground rail is about $315 Billion dollars, over 1.5X the maximum expected cost according to the article.