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IOI 2010 Day 2 – Traffic Congestion

Although Canada is a large country, many areas are uninhabited, and most of the population lives near the southern border. The Trans-Canada Highway, completed in 1962, connects the people living in this strip of land, from St. John’s in the East to Victoria in the West, a distance of 7 821 km.

Canadians like hockey. After a hockey game, thousands of fans get in their cars and drive home from the game, causing heavy congestion on the roads. A wealthy entrepreneur wants to buy a hockey team and build a new hockey arena. Your task is to help him select a location for the arena to minimize the traffic congestion after a hockey game.

 

The country is organized into cities connected by a network of roads. All roads are bidirectional, and there is exactly one route connecting any pair of cities. A route connecting the cities c0 and ck is a sequence of distinct cities c0, …, ck such that there is a road from ci-1 to ci for each i. The new arena must be built in one of the cities, which we will call the arena city. After a hockey game, all of the hockey fans travel from the arena city to their home city, except those who already live in the arena city. The amount of congestion on each road is proportional to the number of hockey fans that travel along the road. You must locate the arena city such that the amount of congestion on the most congested road is as small as possible. If there are several equally good locations, you may choose any one.

You are to implement a procedure LocateCentre(N,P,S,D). N is a positive integer, the number of cities. The cities are numbered from 0 to N-1. P is an array of N positive integers; for each i, P[i] is the number of hockey fans living in the city numbered i. The total number of hockey fans in all the cities will be at most 2 000 000 000. S and D are arrays of N-1 integers each, specifying the locations of roads. For each i, there is a road connecting the two cities whose numbers are S[i] and D[i]. The procedure must return an integer, the number of the city that should be the arena city.

Example

As an example, consider the network of five cities in the top diagram on the right, where cities 0, 1 and 2 contain 10 hockey fans each, and cities 3 and 4 contain 20 hockey fans each. The middle diagram shows the congestions when the new arena is in city 2, the worst congestion being 40 on the thicker arrow. The bottom diagram shows the congestions when the new arena is in city 3, the worst congestion being 30 on the thicker arrow. Therefore, city 3 would be a better location for the arena than city 2. The data for this example are in grader.in.3a.

Note

We remind contestants that with the given constraints, it is possible to submit a solution that passes Subtask 3 and fails Subtask 2. However, remember that your final score for the entire task is determined by only one of your submissions.

Subtask 1 [25 points]

Assume that all the cities lie in a straight line from East to West, and that the roads all follow this straight line with no branches. More specifically, assume that for all i with 0 ≤ i ≤ N-2, S[i] = i and D[i] = i+1.

There are at most 1000 cities.

Subtask 2 [25 points]

Make the same assumptions as in Subtask 1, but there are at most 1 000 000 cities.

Subtask 3 [25 points]

The assumptions from Subtask 1 may no longer be true.

There are at most 1000 cities.

Subtask 4 [25 points]

The assumptions from Subtask 1 may no longer be true.

There are at most 1 000 000 cities.

Implementation Details

  • Use the RunC programming and test environment
  • Implementation folder: /home/ioi2010-contestant/traffic/ (prototype: traffic.zip)
  • To be implemented by contestant: traffic.c or traffic.cpp or traffic.pas
  • Contestant interface: traffic.h or traffic.pas
  • Grader interface: none
  • Sample grader: grader.c or grader.cpp or grader.pas
  • Sample grader input: grader.in.1grader.in.2
    Note: The first line of the input file contains N. The following N lines contain P[i] for i between 0 and N-1. The following N-1 lines contain pairs S[i] D[i] for i between 0 and N-2.
  • Expected output for sample grader input: grader.expect.1grader.expect.2 etc.
  • Compile and run (command line): runc grader.c or runc grader.cpp or runc grader.pas
  • Compile and run (gedit plugin): Control-R, while editing any implementation file.
  • Submit (command line): submit grader.c or submit grader.cpp or submit grader.pas
  • Submit (gedit plugin): Control-J, while editing any implementation or grader file.

Spoiler for: Solution Show