IOI '16 P1 - Detecting Molecules

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Points: 15 (partial)
Time limit: 0.6s
Memory limit: 256M

Problem type
Allowed languages
C, C++

Petr is working for a company that has built a machine for detecting molecules. Each molecule has a positive integer weight. The machine has a detection range [l, u], where l and u are positive integers. The machine can detect a set of molecules if and only if this set contains a subset of the molecules with total weight belonging to the machine's detection range.

Formally, consider n molecules with weights w_0, \dots, w_{n-1}. The detection is successful if there is a set of distinct indices I = \{i_0, \dots, i_{m-1}\} such that l \le w_{i_0} + \dots + w_{i_{m-1}} \le u.

Due to specifics of the machine, the gap between l and u is guaranteed to be greater than or equal to the weight gap between the heaviest and the lightest molecule. Formally, u - l \ge w_{max} - w_{min}, where w_{max} = \max(w_0, \dots, w_{n-1}) and w_{min} = \min(w_0, \dots, w_{n-1}).

Your task is to write a program which either finds any one subset of molecules with total weight within the detection range, or determines that there is no such subset.

Implementation details

You should implement one function (method):

int find_subset(int l, int u, int[] w, int n, int[] result)
  • l and u: the endpoints of the detection range,
  • w: weights of the molecules,
  • n: the number of elements in w (i.e., the number of molecules),
  • result: an array of integers. The method should write the of indices of molecules that form any one such subset to the first m cells of array result. If there are several correct answers, write any of them.
  • if the required subset does not exist, the function should not write anything to the result array and it should return 0.

Your program may write the indices into the result array in any order.

Sample 1

find_subset(15, 17, [6, 8, 8, 7], 4, result)

In this example we have four molecules with weights 6, 8, 8 and 7. The machine can detect subsets of molecules with total weight between 15 and 17, inclusive. Note, that 17 -15 \ge 8 - 6. The total weight of molecules 1 and 3 is w_1 + w_3 = 8 + 7 = 15, so the result array can contain [1, 3] and the function should return 2. Other possible correct answers are [1, 2] (w_1 + w_2 = 8 + 8 = 16) and [2, 3] (w_2 + w_3 = 8 + 7 = 15).

Sample 2

find_subset(14, 15, [5, 5, 6, 6], 4, result)

In this example we have four molecules with weights 5, 5, 6 and 6, and we are looking for a subset of them with total weight between 14 and 15, inclusive. Again, note that 15 - 14 \ge 6 -5. There is no subset of molecules with total weight between 14 and 15 so the function should return 0.

Sample 3

find_subset(10, 20, [15, 17, 16, 18], 4, result)

In this example we have four molecules with weights 15, 17, 16 and 18, and we are looking for a subset of them with total weight between 10 and 20, inclusive. Again, note that 20 - 10 \ge 18 - 15. Any subset consisting of exactly one element has total weight between 10 and 20, so the possible correct answers are: [0], [1], [2] and [3].

Subtasks

  1. (9 points): 1 \le n \le 100, 1 \le w_i \le 100, 1 \le u, l \le 1\,000, all w_i are equal.
  2. (10 points): 1 \le n \le 100, 1 \le w_i, u, l \le 1\,000 and \max(w_0, \dots, w_{n-1})-\min(w_0, \dots, w_{n-1}) \le 1.
  3. (12 points): 1 \le n \le 100 and 1 \le w_i, u, l \le 1\,000.
  4. (15 points): 1 \le n \le 10\,000 and 1 \le w_i, u, l \le 10\,000.
  5. (23 points): 1 \le n \le 10\,000 and 1 \le w_i, u, l \le 500\,000.
  6. (31 points): 1 \le n \le 200\,000 and 1 \le w_i, u, l < 2^{31}.

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