Competitive Programming

Elevator Problem

A building has floors numbered 0 through n≤10^18. There is a single elevator with four buttons: "go to floor 0", "+a floors", "+b floors", and "+c floors". We have a,b,c≤ 10^6. Compute the number of unreachable floors.


Solution:

This feels like a number theory problem, but trying to solve it by GCDs and casework will not lead to success. Instead, treat it as a graph theory problem. First, note that if you can reach floor x, you can reach all floors of the form x+k*a. Hence, for each remainder modulo a all we need is the smallest reachable x with this remainder. These can be found by using Dijkstra’s shortest path algorithm on a graph with a nodes. The nodes are the remainder classes, and from each node there are two edges, corresponding to +b and +c.

The stunning thing about this problem is the asymmetry of the solution: you are treating one button differently from the other two.

Credits : Michal Forišek on Quora, scientist, competitive programmer

www.tg-me.com/us/Competitive Programming/com.Competitive_Programming_Cpp/53

18.9K viewsNov 8, 2019 at 15:06

Elevator Problem

A building has floors numbered 0 through n≤10^18. There is a single elevator with four buttons: "go to floor 0", "+a floors", "+b floors", and "+c floors". We have a,b,c≤ 10^6. Compute the number of unreachable floors.


Solution:

This feels like a number theory problem, but trying to solve it by GCDs and casework will not lead to success. Instead, treat it as a graph theory problem. First, note that if you can reach floor x, you can reach all floors of the form x+k*a. Hence, for each remainder modulo a all we need is the smallest reachable x with this remainder. These can be found by using Dijkstra’s shortest path algorithm on a graph with a nodes. The nodes are the remainder classes, and from each node there are two edges, corresponding to +b and +c.

The stunning thing about this problem is the asymmetry of the solution: you are treating one button differently from the other two.

Credits : Michal Forišek on Quora, scientist, competitive programmer

BY Competitive Programming