A telephone company wants to build a new telephone network in a city.
The company has the goal that each person in the city should be able
to call each other person. Of course, it is not possible to build direct
connections between every pair of persons. Instead, the company uses a
network made up of several layers.
We denote a network switch in layer j by S(j). A switch S(0)
consists of one input, one output and a cable connecting the input to the output. A
switch S(j) with j > 0 consists of 2j inputs, 2j outputs and
two switches S(j−1). Input i of S(j) (0 ≤ i < 2j) is connected via a cable
to the inputs i mod 2j−1 of each of the two
switches S(j−1). Similarly, output i of S(j) is connected to the
outputs i mod 2j−1 of each of the two switches S(j−1).
Figure 1: The connections between a switch S(j) and the two switches
S(j−1) it consists of.
We are considering a network with one switch S(n) in the outermost layer.
It can be shown that any input and output of switch S(n) has a unique connection path to any
of the S(0) switches. Therefore, any input of S(n) can be connected to
any of its outputs, and the connection path is uniquely determined by specifying through
which switch S(0) the connection is established.
We number the switches S(0) belonging to the switch S(n) from 0
to 2n−1. The i-th switch S(0) is defined as follows. Write the
number i in binary as bn−1bn−2…b0. This defines a path
from an input of S(n) to the i-th switch S(0) via the following
procedure: for each j, bj=0 indicates that the path extends from
S(j+1) to the first S(j) switch to which it is directly connected, and bj=1 indicates that the
path extends to the second S(j) switch. Note that regardless of
which input of S(n) is selected, this path arrives at the same
S(0) switch, which is given the number i. See also the figure
below the sample data for details of how the numbering works.
Sometimes multiple connections
are needed at the same time. In order to avoid interference, each of the inputs and outputs
of all switches S(j) (0 ≤ j ≤ n)
can be used by at most one connection. Given a set of connection requests, can you find
connection paths for each request such that the connection paths are disjoint?
Input
On the first line a positive integer: the number of test cases, at
most 100. After that per test case:
One line with two integers n (1 ≤ n ≤ 16) and m (1 ≤ m ≤ 2n): the layer of the outermost switch and the number of connection
requests.
m lines, each with two integers ai and bi (0 ≤ ai, bi < 2n). Each such line represents a connection request from input number ai of S(n) to
output number bi. You may assume that the integers ai are
pairwise distinct, and the integers bi are pairwise distinct as
well.
Output
Per test case:
One line with m integers s1, …, sm, where si is the number of the S(0) switch through which the connection of input ai to output bi is established.
The connection paths should be disjoint. You may print any valid
solution, and you may assume that there is at least one valid solution.
Sample in- and output
Input
Output
2
1 1
0 1
3 5
0 3
1 4
2 5
3 6
4 7
0
3 0 1 2 4
Figure 2: A possible connection scheme for the second sample case,
with used cables in bold.
