In the footsteps of “Our won: TopCoder Open 2019” I publish tasks from the Algorithm track (classic sports programming. In an hour and a half, you need to solve three problems in Java, C #, C ++ or Python.)
1. Pie for six
Formulation of the problem
The time limit is 4 seconds.
You have a pie. Seen from above, the pie has the shape of a (strictly) convex polygon. You are given the coordinates of the vertices in integers X and Y.
You have five friends. You want to divide the cake into six pieces of equal area (but not necessarily the same shape). Of course, anyone can do it in five cuts, but only the pros can do it in three cuts.
Find three cuts in straight lines through one point that will divide the cake into six equal-sized pieces. Print {x, y, d1, d2, d3}, where (x, y) is the common point of all three cuts, and d1, d2, d3 are the cut angles of the cuts in radians.
Definition
Class: CakeForSix
Method: cut
Parameters: int [], int []
Returns: double []
Method signature: double [] cut (int [] x, int [] y)
(be sure your method is public)
Method: cut
Parameters: int [], int []
Returns: double []
Method signature: double [] cut (int [] x, int [] y)
(be sure your method is public)
Notes
- The positive direction along the x axis is 0 (radian), the positive direction along the y axis is pi / 2 (radian).
- The cut in the direction d is similar to the cut in the direction pi * k + d for any integer k.
- You can output any directions, they do not have to be from [0, pi).
- The grader will calculate the area of your six pieces of cake in doubles. The answer will be accepted if the relative or absolute difference between them is less than 10 ^ (- 4).
- More precisely, let X and Y be the smallest and largest of your six areas calculated by the grader. Then your answer will be accepted if Y <max (X + 10 ^ (- 4), X * 1 + 10 ^ (- 4))).
- (In the initial version of the problem, accuracy 1e-7 was used instead of 1e-4. To solve this problem in the archive, the accuracy limit was reduced due to the presence of call cases that most likely make the task unsolvable with accuracy 1e-7. In an ideal world, restrictions do not allow such cases and still require high accuracy, so solving the problem with some general numerical optimization is not easy.)
Limitations
- x contains from 3 to 50 elements inclusive.
- y contains as many elements as x.
- all coordinates between 0 and 10,000 inclusive
- x and y define a convex polygon in the counterclockwise direction.
Original in English
Time limit is 4 seconds.
You have a cake. Seen from above, the cake is a (strictly) convex polygon. You are given the coordinates of its vertices in the int [] sx and y.
You have five friends. You now want to cut the cake into six pieces of equal area (but not necessarily equal shape). Of course, anyone can do that in five cuts - but only a true pro can do it in three!
Find three straight-line cuts passing through the same point that cut the cake into six equally large parts. Return {x, y, d1, d2, d3}, where (x, y) is the common point of the three cuts, and d1, d2, d3 are their directions in radians.
Definition
Class: CakeForSix
Method: cut
Parameters: int [], int []
Returns: double []
Method signature: double [] cut (int [] x, int [] y)
(be sure your method is public)
Notes
- The positive direction along the x axis is 0 (radians), the positive direction along the y axis is pi / 2 (radians).
- A cut in direction d is the same as a cut in direction pi * k + d for any integer k.
- You may return any directions, they do not have to be from [0, pi).
- The grader will compute the areas of your six cake pieces in doubles. The answer will be accepted if the relative or absolute difference between them is less than 10 ^ (- 4).
- More precisely, let X and Y be the smallest and the largest of your six areas, as computed by the grader. Then, your answer will be accepted if Y <max (X + 10 ^ (- 4), X * (1 + 10 ^ (- 4))).
- (The original version of the problem used 1e-7 precision instead of 1e-4. For upsolving this problem in the archive the precision limit was lowered due to the existence of challenge cases that most likely make the task unsolvable with 1e-7 precision . In an ideal world the constraints would not allow such cases and still require high precision, so that it isn't easy to solve the problem via some general numeric optimization.)
Constraints
- x will have between 3 and 50 elements, inclusive.
- y will have the same number of elements as x.
- All coordinates will be between 0 and 10,000, inclusive.
- x and y will describe a convex polygon in counterclockwise order.
Problem Statement
Time limit is 4 seconds.
You have a cake. Seen from above, the cake is a (strictly) convex polygon. You are given the coordinates of its vertices in the int [] sx and y.
You have five friends. You now want to cut the cake into six pieces of equal area (but not necessarily equal shape). Of course, anyone can do that in five cuts - but only a true pro can do it in three!
Find three straight-line cuts passing through the same point that cut the cake into six equally large parts. Return {x, y, d1, d2, d3}, where (x, y) is the common point of the three cuts, and d1, d2, d3 are their directions in radians.
Definition
Class: CakeForSix
Method: cut
Parameters: int [], int []
Returns: double []
Method signature: double [] cut (int [] x, int [] y)
(be sure your method is public)
Notes
- The positive direction along the x axis is 0 (radians), the positive direction along the y axis is pi / 2 (radians).
- A cut in direction d is the same as a cut in direction pi * k + d for any integer k.
- You may return any directions, they do not have to be from [0, pi).
- The grader will compute the areas of your six cake pieces in doubles. The answer will be accepted if the relative or absolute difference between them is less than 10 ^ (- 4).
- More precisely, let X and Y be the smallest and the largest of your six areas, as computed by the grader. Then, your answer will be accepted if Y <max (X + 10 ^ (- 4), X * (1 + 10 ^ (- 4))).
- (The original version of the problem used 1e-7 precision instead of 1e-4. For upsolving this problem in the archive the precision limit was lowered due to the existence of challenge cases that most likely make the task unsolvable with 1e-7 precision . In an ideal world the constraints would not allow such cases and still require high precision, so that it isn't easy to solve the problem via some general numeric optimization.)
Constraints
- x will have between 3 and 50 elements, inclusive.
- y will have the same number of elements as x.
- All coordinates will be between 0 and 10,000, inclusive.
- x and y will describe a convex polygon in counterclockwise order.
Examples
0)
{0, 20, 30, 50, 30, 20}
{10, 0, 0, 10, 20, 20}
Returns:
{24.999999999437453, 9.999999999500002, 0.0, 0.7266423406817211, 2.4149503129080787}
Symmetric but not regular hexagon. An example of an answer corresponds to cutting it in half horizontally and making two other cuts in the center, which divide each part into three parts.
one)
{0, 1000, 0}
{0, 0, 1000}
Returns:
{333.3333333331763, 333.3333333332546, 0.7853981633986264, 2.0344439357948154, 2.6779450445891753}
Right triangle. Again, we can start with one of three cuts along the axis of symmetry.
2)
{40, 70, 90, 90, 50}
{30, 20, 40, 100, 60}
Returns:
{69.79517771922892, 52.77575974637605, 2.0616329654335885, 3.637826104091601, 4.32123485812475}
Wrong pentagon.
3)
{300, 400, 300, 200}
{500, 600, 700, 600}
Returns: {299.99999999974995, 599.9999999995, 0.0, 1.107148717794088, 2.034443935795705}
A square rotated 45 degrees.
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