IOI '16 P5 - Unscrambling a Messy Bug

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Points: 25 (partial)
Time limit: 1.0s
Memory limit: 256M

Problem type
Allowed languages
C++

Ilshat is a software engineer working on efficient data structures. One day he invented a new data structure. This data structure can store a set of non-negative n- bit integers, where n is a power of two. That is, n = 2^b for some non-negative integer b.

The data structure is initially empty. A program using the data structure has to follow the following rules:

  • The program can add elements that are n-bit integers into the data structure, one at a time, by using the function add_element(x). If the program tries to add an element that is already present in the data structure, nothing happens.
  • After adding the last element the program should call the function compile_set() exactly once.
  • Finally, the program may call the function check_element(x) to check whether the element x is present in the data structure. This function may be used multiple times.

When Ilshat first implemented this data structure, he made a bug in the function compile_set(). The bug reorders the binary digits of each element in the set in the same manner. Ilshat wants you to find the exact reordering of digits caused by the bug.

Formally, consider a sequence p = [p_0 , \dots , p_{n-1}] in which every number 0 from to n - 1 appears exactly once. We call such a sequence a permutation. Consider an element of the set, whose digits in binary are a_0 , \dots , a_{n-1} (with a_0 being the most significant bit). When the function compile_set() is called, this element is replaced by the element a_{p_0} , a_{p_1} , \dots , a_{p_{n-1}}.

The same permutation is used to reorder the digits of every element. Any permutation is possible, including the possibility that p_i = i for each 0 \le i \le n - 1.

For example, suppose that n=4, p=[2,1,3,0], and you have inserted into the set integers whose binary representations are 0000, 1100 and 0111. Calling the function compile_set changes these elements to 0000, 0101 and 1110, respectively.

Your task is to write a program that finds the permutation by interacting with the data structure. It should (in the following order):

  1. choose a set of n-bit integers,
  2. insert those integers into the data structure,
  3. call the function compile_set to trigger the bug,
  4. check the presence of some elements in the modified set,
  5. use that information to determine and return the permutation.

Note that your program may call the function compile_set only once.

In addition, there is a limit on the number of times your program calls the library functions. Namely, it may

  • call add_element at most w times (w is for "writes"),
  • call check_element at most r times (r is for "reads").

Implementation details

You should implement one function (method):

int[] restore_permutation(int n, int w, int r)
  • n: the number of bits in the binary representation of each element of the set (and also the length of p).
  • w: the maximum number of add_element operations your program can perform.
  • r: the maximum number of check_element operations your program can perform.
  • the function should return the restored permutation p.

Library functions

In order to interact with the data structure, your program should use the following three functions (methods):

  • void add_element(string x)
    This function adds the element described by x to the set.

    • x: a string of 0 and 1 characters giving the binary representation of an integer that should be added to the set. The length of x must be n.
  • void compile_set()
    This function must be called exactly once. Your program cannot call add_element() after this call. Your program cannot call check_element() before this call.

  • bool check_element(string x)
    This function checks whether the element x is in the modified set.

  • x: a string of 0 and 1 characters giving the binary representation of the element that should be checked. The length of x must be n.
  • returns true if element x is in the modified set, and false otherwise.

Note that if your program violates any of the above restrictions, its grading outcome will be Wrong Answer.

For all the strings, the first character gives the most significant bit of the corresponding integer.

The grader fixes the permutation p before the function restore_permutation is called.

Example

The grader makes the following function call:

  • restore_permutation(4, 16, 16). We have n=4 and the program can do at most 16 "writes" and 16 "reads".

The program makes the following function calls:

  • add_element("0001")
  • add_element("0011")
  • add_element("0100")
  • compile_set()
  • check_element("0001") returns false
  • check_element("0010") returns true
  • check_element("0100") returns true
  • check_element("1000") returns false
  • check_element("0011") returns false
  • check_element("0101") returns false
  • check_element("1001") returns false
  • check_element("0110") returns false
  • check_element("1010") returns true
  • check_element("1100") returns false

Only one permutation is consistent with these values returned by check_element(): the permutation p = [2, 1, 3, 0]. Thus, restore_permutation should return [2, 1, 3, 0].

Subtasks

  1. (20 points) n=8, w=256, r=256, p_i \neq i for at most 2 indices i (0 \le i \le n - 1),
  2. (18 points) n=32, w=320, r=1024,
  3. (11 points) n=32, w=1024, r=320,
  4. (21 points) n=128, w=1792, r=1792,
  5. (30 points) n=128, w=896, r=896.

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