UT Invitational Programming Contest Fall 2019


2019-11-23 10:00 AKST

UT Invitational Programming Contest Fall 2019


2019-11-23 15:00 AKST
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Problem J
Red Light, Green Light

A recent study has shown that long commutes can have negative effects on one’s happiness. Upon reading this study, Jay had suddenly become quite unhappy with his long commute to work; it seems the study is true after all. Jay now has frequent dreams in which he has a short daily commute, and these pleasant dreams often cause Jay to sleep through his alarms in the morning.

On the day of his Very Important Meeting, Jay once again oversleeps, to his dismay. As he leaves his house at time $0$, he realizes he only has $T$ minutes to reach his workplace. The city Jay works in contains $N$ intersections (labeled $1$ to $N$) and $M$ one-way roads between intersections that vary in length. His house is located at intersection $1$, and his workplace is located at intersection $N$.

The real culprits responsible for Jay’s long commute are the traffic lights that are located on every road. Jay has driven around the city so much that he’s realized that each light operates on a cycle: for the $i$th road, the light is green for $g_ i$ minutes, then red for $r_ i$ minutes, then green for $g_ i$ minutes, and so on. Jay has also figured out the specific values of $g_ i$ and $r_ i$ for every road in the city, and he has calculated the first time $t_ i$ after time $0$ (when Jay leaves his house) that each road’s traffic light is green. Here, $t_ i$ will fall somewhere in the inclusive range $[0, r_ i]$. Jay can only take a road if the light is green; otherwise, he must wait at the intersection or choose to take a different road. Once he enters a road, he can continue to the end even if the light turns red before he reaches the next intersection. If Jay arrives at an intersection right as an outgoing road’s light switches, he must obey the new color of the light.

While Jay wants to make it to his meeting on time, he also wants to avoid driving too fast. Knowing all this information about the city’s traffic lights, Jay wants to know the minimum speed he needs to drive in order to make it to his meeting in time.


The first line of input contains three space-separated integers, $N$ ($2 \leq N \leq 50\, 000$), $M$ ($1 \leq M \leq 50\, 000$), and $T$ ($1 \leq T \leq 10^6$). Then follows $M$ lines, the $i$th of which describes the $i$th road in the city. Each of the $M$ lines will contain six space-separated integers, $u_ i$, $v_ i$ ($1 \leq u_ i, v_ i \leq N$), $l_ i$ ($1 \leq l_ i \leq 10^6$), $g_ i$, $r_ i$ ($1 \leq g_ i, r_ i \leq 10^4$), $t_ i$ ($0 \leq t_ i \leq r_ i$), denoting that the $i$th road is from intersection $u_ i$ to intersection $v_ i$, has a length of $l_ i$ feet, and the period of the road is described by $g_ i$, $r_ i$, and $t_ i$ (as explained in the statement above). There will be at most one road from city $u$ to city $v$ for all $u$, $v$, and there will be no roads from a city to itself (note that there may be a road from $u$ to $v$ AND $v$ to $u$).

It is guaranteed that Jay will be able to reach his workplace from home.


Output the minimum speed Jay needs to drive in feet per minute in order to reach his Very Important Meeting on time. Answers within a relative or absolute error of $10^{-6}$ will be accepted.

Sample Input 1 Sample Output 1
4 4 12
1 2 4 1 1 0
1 3 6 2 2 1
2 4 8 3 4 2
3 4 4 4 6 3