Develop Reference

How to find figures with even areas

We have shape with n angles and its coordinates. I need to divide that figure into 2 figures with the same area. Division is performed in parallel to the axis y. The task is to find such x-coordinate, which would comply with the terms.
Test,E.g
4 (n angles)
0 0
0 2
2 2
2 0
The output must be 1
I wrote
n = int(input())
coordinates = []
for i in range(n):
x,y = map(int,input().split())
coordinates.append([x,y])
s = 0
for i in range(len(coordinates)-1):
s += coordinates[i][0]*coordinates[i+1][1]
s -= coordinates[i][1]*coordinates[i+1][0]
s = s/2
x = 0
print(s)
And after that I don't have any ideas what to do next..

changing the boolean values of an array according to a formula for the indices

I want to create a 64 components array showing all the squares in which the two rooks of an empty chessboard could move from their current position. So far I am doing it with for and while loops.
I first create a function just to better visualize the board:
import numpy as np
def from_array_to_matrix(v):
m=np.zeros((8,8)).astype('int')
for row in range(8):
for column in range(8):
m[row,column]=v[row*8+column]
return m
and here I show how I actually build the array:
# positions of the two rooks
a=np.zeros(64).astype('int')
a[15] = 1
a[25] = 1
print from_array_to_matrix(a)
# attack_a will be all the squares where they could move in the empty board
attack_a=np.zeros(64).astype('int')
for piece in np.where(a)[0]:
j=0
square=piece+j*8
while square<64:
attack_a[square]=1
j+=1
square=piece+j*8
j=0
square=piece-j*8
while square>=0:
attack_a[square]=1
j+=1
square=piece-j*8
j=0
square=piece+j
while square<8*(1+piece//8):
attack_a[square]=1
j+=1
square=piece+j
j=0
square=piece-j
while square>=8*(piece//8):
attack_a[square]=1
j+=1
square=piece-j
print attack_a
print from_array_to_matrix(attack_a)
I have been advised to avoid for and while loops whenever it is possible to use other ways, because they tend to be time consuming. Is there any way to achieve the same result without iterating the process with for and while loops ?
Perhaps using the fact that the indices to which I want to assign the value 1 can be determined by a function.

There are a couple of different ways to do this. The simplest thing is of course to work with matrices.
But you can vectorize operations on the raveled array as well. For example, say you had a rook at position 0 <= n < 64 in the linear array. To set the row to one, use integer division:
array[8 * (n // 8):8 * (n // 8 + 1)] = True
To set the column, use modulo:
array[n % 8::8] = True
You can convert to a matrix using reshape:
matrix = array.reshape(8, 8)
And back using ravel:
array = martix.ravel()
Or reshape:
array = matrix.reshape(-1)
Setting ones in a matrix is even simpler, given a specific row 0 <= m < 8 and column 0 <= n < 8:
matrix[m, :] = matrix[:, n] = True
Now the only question is how to vectorize multiple indices simultaneously. As it happens, you can use a fancy index in one axis. I.e, the expression above can be used with an m and n containing multiple elements:
m, n = np.nonzero(matrix)
matrix[m, :] = matrix[:, n] = True
You could even play games and do this with the array, also using fancy indexing:
n = np.nonzero(array)[0]
r = np.linspace(8 * (n // 8), 8 * (n // 8 + 1), 8, False).T.ravel()
c = np.linspace(n % 8, n % 8 + 64, 8, False)
array[r] = array[c] = True
Using linspace allows you to generate multiple sequences of the same size simultaneously. Each sequence is a column, so we transpose before raveling, although this is not required.

Use reshaping to convert 1-D array to 8x8 2-D matrix and then numpy advance indexing to select rows and columns to set to 1:
import numpy as np
def from_array_to_matrix(v):
return v.reshape(8,8)
# positions of the two rooks
a=np.zeros(64).astype('int')
a[15] = 1
a[25] = 1
a = from_array_to_matrix(a)
# attack_a will be all the squares where they could move in the empty board
attack_a=np.zeros(64).astype('int')
attack_a = from_array_to_matrix(attack_a)
#these two lines replace your for and while loops
attack_a[np.where(a)[0],:] = 1
attack_a[:,np.where(a)[1]] = 1
output:
a:
[[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]]
attack_a:
[[0 1 0 0 0 0 0 1]
[1 1 1 1 1 1 1 1]
[0 1 0 0 0 0 0 1]
[1 1 1 1 1 1 1 1]
[0 1 0 0 0 0 0 1]
[0 1 0 0 0 0 0 1]
[0 1 0 0 0 0 0 1]
[0 1 0 0 0 0 0 1]]

How to assign ones and zeros to specific indices of an array using numpy?

I wanted to construct a 6 x 9 matrix with entries zeros and ones in a specific way as follows. In the zeroth row column, 0 to 2 should be 1 and in the first-row column,3 to 5 should be one and in the second-row column, 6 to 8 should be one, with all the other entries to be zeros. In the third row, element 0,3,6 should be one and the other should be zeros. In the fourth row, element 1,4,7 should be one and the other elements should be zeros. In the fifth row,2,5,8 should be one and the remaining should be zeros. Half of the rows follow one way enter the value 1 and the other half of the row follows different procedures to enter the value one. How do extend this some 20 x 100 case where the first 10 rows follow one procedure as mentioned above and the second half follows different procedures
The 6x9 by matrix looks as follows
[[1,1,1,0,0,0,0,0,0],
[0,0,0,1,1,1,0,0,0],
[0,0,0,0,0,0,1,1,1],
[1,0,0,1,0,0,1,0,0],
[0,1,0,0,1,0,0,1,0],
[0,0,1,0,0,1,0,0,1]]
EDIT: Code I used to create this matrix:
import numpy as np
m=int(input("Enter the value of m, no. of points = "))
pimatrix=np.zeros((2*m +1)*(m**2)).reshape((2*m+1),(m**2))
for i in range(2*m + 1):
for j in range(m**2):
if((i<m) and ((j<((i+1)*m) and j>=(i*m)))):
pimatrix[i][j]=1
if (i>(m-1)):
for k in range(-1,m-1,1):
if(j == i+(k*m)):
pimatrix[i][j]=1
if i==2*m:
pimatrix[i][j]=1
print(pimatrix)

Try to use numpy.put function numpy.put

The best approach depends on the rules you plan to follow, but an easy approach would be to initialise the array as an array of zeroes:
import numpy as np
a = np.zeros([3, 4], dtype = int)
You can then write the logic to loop over the appropriate rows and set 1's as needed. You can simply access any element of the array by its coordinates:
a[2,1] = 1
print(a)
Result:
[[0 0 0 0]
[0 0 0 0]
[0 1 0 0]]
Without a general rule, it's hard to say what your intended logic is exactly, but assuming these rules: the top half of the array has runs of three ones on each consecutive row, starting in the upper left and moving down a row at the end of every run, until it reaches the bottom of the top half, where it wraps around to the top; the bottom half has runs of single ones, following the same pattern.
Implementing that, with your given example:
import numpy as np
a = np.zeros([6, 9], dtype=int)
def set_ones(a, run_length, start, end):
for n in range(a.shape[1]):
a[start + ((n // run_length) % (end - start)), n] = 1
set_ones(a, 3, 0, a.shape[0] // 2)
set_ones(a, 1, a.shape[0] // 2, a.shape[0])
print(a)
Result:
[[1 1 1 0 0 0 0 0 0]
[0 0 0 1 1 1 0 0 0]
[0 0 0 0 0 0 1 1 1]
[1 0 0 1 0 0 1 0 0]
[0 1 0 0 1 0 0 1 0]
[0 0 1 0 0 1 0 0 1]]

Find all paths of length 2 in a graph

I've tried to create an algorithm for finding all paths of length 2, but it doesn't seem to work properly:
input_split = input().split(' ')
node_count = int(input_split[0])
input_count = int(input_split[1])
items = np.zeros((node_count, node_count), dtype=np.int32) # matrix of adjacency
for j in range(input_count):
split = input().split(' ')
x = int(split[0]) - 1 # convert 1 based coordinates to 0 based
y = int(split[1]) - 1
items[x][y] = 1
items[y][x] = 1
result = np.linalg.matrix_power(items, 2)
result_sum = int(np.sum(result) / 2) # reverse paths are counted only once
print(result_sum)
Sample input:
6 7
1 2
2 3
3 1
2 4
4 5
5 6
6 2
The result should be 11, but it prints 18.

You're on the right track when calculating the square of the adjacency matrix. After the exponentiation you would get result matrix that looks like this:
[[2 1 1 1 0 1]
[1 4 1 0 2 0]
[1 1 2 1 0 1]
[1 0 1 2 0 2]
[0 2 0 0 2 0]
[1 0 1 2 0 2]]
First you need to exclude all diagonal entries from this matrix, because those denote walks that are not paths, as their starting and ending node is the same. Note that for length 2 that is the only way how nodes can be repeating.
The other entries need to be counted only once, because of symmetry. So only look at the upper right triangle of the matrix.
One way to do it is:
result_sum = 0
for i in range(input_count - 1):
for j in range(i + 1, input_count - 1):
result_sum += result[i][j]
print(result_sum) # prints 11
More Pythonic way, one-liner using numpy.trace():
result_sum = (np.sum(result) - np.trace(result)) // 2

You are calculating walks, which would include walks 6-7-6 (which is not a P2)
this discussion might help:
https://math.stackexchange.com/questions/1890620/finding-path-lengths-by-the-power-of-adjacency-matrix-of-an-undirected-graph

Gernerate all the possible undirected graphs

What is an efficient solution to generate all the possible graphs using an incidence matrix?
The problems is equivalent of generating all the possible binary triangular matrix.
My first idea was to use python with itertools. For instance, for generating all the possibile 4x4 matrix
for b in itertools.combinations_with_replacement((0,1), n-3):
b_1=[i for i in b]
for c in itertools.combinations_with_replacement((0,1), n-2):
c_1=[i for i in c]
for d in itertools.combinations_with_replacement((0,1), n-1):
d_1=[i for i in d]
and then you create the matrix adding the respective number of zeroes..
But this is not correct since we skip some graphs...
So, any ideas?
Perhaps i can use the isomorphism between R^n matrix and R^(n*n) vector, and generate all the possibile vector of 0 and 1, and then cut it into my matrix, but i think there's a more efficient solutions.
Thank you
I add the matlab tab because it's a problem you can have in numerical analysis and matlab.

I assume you want lower triangular matrices, and that the diagonal needs not be zero. The code can be easily modified if that's not the case.
n = 4; %// matrix size
vals = dec2bin(0:2^(n*(n+1)/2)-1)-'0'; %// each row of `vals` codes a matrix
mask = tril(reshape(1:n^2, n, n))>0; %// decoding mask
for v = vals.' %'// `for` picks one column each time
matrix = zeros(n); %// initiallize to zeros
matrix(mask) = v; %// decode into matrix
disp(matrix) %// Do something with `matrix`
end
Each iteration gives one possible matrix. For example, the first matrices for n=4 are
matrix =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
matrix =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 1
matrix =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 1 0
matrix =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 1 1

Here is an example solution using numpy that generates all simple graphs:
It first generates the indices of the upper triangular part iu. The loop converts the number k to it's binary representation and then assigns it to the upper triangular part G[iu].
import numpy as np
n = 4
iu = np.triu_indices(n,1) # Start at first minor diagonal
G = np.zeros([n,n])
def dec2bin(k, bitlength=0):
return [1 if digit=='1' else 0 for digit in bin(k)[2:].zfill(bitlength)]
for k in range(0,2**(iu[0].size)):
G[iu] = dec2bin(k, iu[0].size)
print(G)