[ACM-ICPC] 4713 - Elias Omega Coding
Elias code can be used to efficiently encode positive integers when there is no prior information about the cardinality of the integers to be encoded and it is known that the probability of getting a large integer is smaller than or equal to the probability of getting a small integer. For practical reasons, however, in this contest we do pose a limit on the size of the input integers. For the same reason we restrict the input integer to be greater than 1.
Elias developed three variants of the code: the Elias gamma, Elias delta, and Elias omega coding methods. This problem presents the Elias gamma and Elias omega methods and calls for implementing the Elias omega code. The following is a background, definition, and illustration of the problem.
Background
Suppose that Alice wants to transmit a positive integer
n to Bob through a binary channel and let 
(n) stand for the binary
representation of n. If Bob knows
|(n)| (the number of bits required for the
binary representation of n) in advance, then Alice should use (n) for the
transmission. On the other hand, if Bob does not have this information, then Alice can
first send
|(n)|, using efficient and distinguishable encoding, then she can send
the actual beta code
F1 = (n). The end result is a two field code
< F1, F2 >.
Elias code and its variants differ in the way they encode these two pieces of information (F1 and F2). The main difference between variants lies in the representation of F1. This may imply modifications in the representation of F1. In addition, some of the variants apply repetition or recursion to the representation of F1. We use a specific variant specified below.
Definition
Formally, in Elias gamma coding, a positive integer n is encoded using two concatenated bit
fields. The first field, the prefix, contains
log2n
bits of 0 (
x is the floor of x).
The second field, the postfix, is the actual binary representation of n using
log2n + 1
bits. For example, the binary representation of the decimal number 9 is 1001. Under Elias
coding 9 is encoded as 0001001. The first three leading zeros denote that four bits are
required for the binary representation of 9. The next four bits
contain the binary representation of 9. Elias delta code applies the gamma code to the
prefix(
log2n) of the gamma code and Elias omega code applies a recursion over the
prefix Elias gamma representation of
log2n.
Illustration (detailed example)
To further illustrate the Elias omega code, consider the integer 536870907. The binary representation of this integer is 1 1111 1111 1111 1111 1111 1111 1011 which is required 29 bits. Hence, in the first step it is encoded as follows:
To emphasize, for this contest the recursion stops when the first field of a recursive
stage contains one 0.
Looking at all the bits generated by all the steps and using (n) to denote the binary
representation of n, we have:
- a.
- <28 0's,
(536870907) > =
<0000 0000 0000 0000 0000 0000 0000, 1 1111 1111 1111 1111 1111 1111 1011>
- b.
- < <4 0's,
(28) > ,(536870907) > =
< <0000, 1 1100>, 1 1111 1111 1111 1111 1111 1111 1011>
- c.
- < < <2 0's, (4) >,
(28) > ,(536870907) > =
< < <00, 100>, 1 1100>, 1 1111 1111 1111 1111 1111 1111 1011>
- d.
- < < < <one 0, (2) >, (4) >,
(28) >,
(536870907) > =
< < < <0, 10>, 100>, 1 1100>, 1 1111 1111 1111 1111 1111 1111 1011>
Taking away all the blanks, dots, commas, and brackets we get that the Elias omega code of
536870907 is: 0101001110011111111111111111111111111011. This code is distinguishable
(uniquely decodable). It can be decoded in a unique way and yield back the number 536870907.
Problem statement
Given some positive integers in the range of 2...2 x 109, you are to write a program to produce the Elias omega codes for these integers.
Input
The input contains positive integers each in a separate line. Each integer is in between 2 and 2,000,000,000, inclusive. The end of input is indicated by a zero. The input can contain up to one hundred lines.
Output
The output consists of lines each corresponding to an input integer except the last zero. Each line contains the Elias omega code of each input integer. The output should not contain any blanks or blank lines.
Sample Input
2 510 7 120000 536870905 49 5 0
Sample Output
010 0111000111111110 010111 0101001000011101010011000000 0101001110011111111111111111111111111001 010101110001 010101
看懂就好做了，沒有什麼難度。
#include <stdio.h>
#include <math.h>
void bNum(int n) {
if(n) {
bNum(n/2);
printf("%d", n%2);
}
}
void transform(int n) {
if(n == 1) {
printf("0");
return;
}
transform((int)log2(n));
bNum(n);
}
int main() {
int n;
while(scanf("%d", &n) == 1 && n) {
transform(n);
puts("");
}
return 0;
}