Creating a non-right Pascal's triangle (centered) in python - python-2.7

I need to write a code that inputs a non-right Pascal's triangle given the nth level as an input where the first row is the 0th level. Apart from that, at the end of each row the level must be indicated.
Here's what I've made so far:
level = input('Please input nth level: ')
x = -1
y = 1
while x < level:
x = x+1
d = str(11**x)
while y < level:
y = y+1
print " ",
for m,n in enumerate(d):
print str(n) + " ",
while y < level:
y = y+1
print " ",
print x
When I input 3, it outputs:
1 0
1 1 1
1 2 1 2
1 3 3 1 3
My desired output is:
   1 0
1 1 1
1 2 1 2
1 3 3 1 3

You could use str.format to center the string for you:
level = int(raw_input('Please input nth level: '))
N = level*2 + 5
for x in range(level+1):
d = ' '.join(str(11**x))
print('{d:^{N}} {x:>}'.format(N=N, d=d, x=x))
Please input nth level: 4
1 0
1 1 1
1 2 1 2
1 3 3 1 3
1 4 6 4 1 4
Note that if d = '1331', then you can add a space between each digit using ' '.join(d):
In [29]: d = '1331'
In [30]: ' '.join(d)
Out[30]: '1 3 3 1'
Note that using d = str(11**x) is a problematic way of computing the numbers in Pascal's triangle since it does not give you the correct digits for x >= 5. For example,
Please input nth level: 5
1 0
1 1 1
1 2 1 2
1 3 3 1 3
1 4 6 4 1 4
1 6 1 0 5 1 5 <-- Should be 1 5 10 10 5 1 !
You'll probably want to compute the digits in Pascal's triangle a different way.

Related

Generate graph with vertices equal to elements of std::next_permutation, with edges between pairwise adjacent permutations

Consider the code snippet:
#include <vector>
#include <algorithm>
#include <stdio.h>
int main(){
int szvec = 4;
std::vector<int> vecint(szvec);
for (size_t i = 0, szi = szvec; i < szi; i++){
vecint[i] = i;
}
do{
for (size_t i = 0, szi = vecint.size(); i < szi; i++){
printf("%d\t", vecint[i]);
}
printf("\n");
} while (std::next_permutation(vecint.begin(), vecint.end()));
}
The output of this is the set of 4! different permutations.:
0 1 2 3 (Permutation 1)
0 1 3 2 (Permutation 2)
0 2 1 3
0 2 3 1
0 3 1 2
0 3 2 1
1 0 2 3
1 0 3 2
1 2 0 3
1 2 3 0
1 3 0 2
1 3 2 0
2 0 1 3
2 0 3 1
2 1 0 3 (Permutation 15)
2 1 3 0
2 3 0 1
2 3 1 0
3 0 1 2
3 0 2 1
3 1 0 2
3 1 2 0
3 2 0 1 (Permutation 23)
3 2 1 0 (Permutation 24)
I am interested in constructing an undirected graph with 24 nodes, (each node represents one permutation), with an edge connecting i and j if these permutations differ from each other in exactly one contiguous pair of elements. For instance, Permutation 1 and Permutation 2 will be connected since they differ in the pairwise exchange over contiguous indices. Permutation 1 and Permutation 15 will NOT be connected since even though they differ in pairwise exchange of elements, this exchange is not over contiguous indices. Permutation 23 and Permutation 24 will be connected.
The only way I can think of now of how to do this is to select each pair of permutations and evaluate explicitly whether they should be connected. Is there a more efficient way of doing this analytically? What I mean by analytical here is that given a particular node i, can we efficiently enumerate all other permutations it will be connected to? Note that szvec can be an arbitrary integer, not only 4.
ETA: With szvec = 3, we have six permutations:
0 1 2 (Permutation 1)
0 2 1 (Permutation 2)
1 0 2 (Permutation 3)
1 2 0 (Permutation 4)
2 0 1 (Permutation 5)
2 1 0 (Permutation 6)
and the graph looks so:
___________________
| |
5 - 6 - 4 - 3 - 1 - 2

Create a dummy variable for the last rows based on on another variable

I would like to create a dummy variable that will look at the variable "count" and label the rows as 1 starting from the last row of each id. As an example ID 1 has count of 3 and the last three rows of this id will have such pattern: 0,0,1,1,1 Similarly, ID 4 which has a count of 1 will have 0,0,0,1. The IDs have different number of rows. The variable "wish" shows what I want to obtain as a final output.
input byte id count wish str9 date
1 3 0 22sep2006
1 3 0 23sep2006
1 3 1 24sep2006
1 3 1 25sep2006
1 3 1 26sep2006
2 4 1 22mar2004
2 4 1 23mar2004
2 4 1 24mar2004
2 4 1 25mar2004
3 2 0 28jan2003
3 2 0 29jan2003
3 2 1 30jan2003
3 2 1 31jan2003
4 1 0 02dec1993
4 1 0 03dec1993
4 1 0 04dec1993
4 1 1 05dec1993
5 1 0 08feb2005
5 1 0 09feb2005
5 1 0 10feb2005
5 1 1 11feb2005
6 3 0 15jan1999
6 3 0 16jan1999
6 3 1 17jan1999
6 3 1 18jan1999
6 3 1 19jan1999
end
For future questions, you should provide your failed attempts. This shows that you have done your part, namely, research your problem.
One way is:
clear
set more off
*----- example data -----
input ///
byte id count wish str9 date
1 3 0 22sep2006
1 3 0 23sep2006
1 3 1 24sep2006
1 3 1 25sep2006
1 3 1 26sep2006
2 4 1 22mar2004
2 4 1 23mar2004
2 4 1 24mar2004
2 4 1 25mar2004
3 2 0 28jan2003
3 2 0 29jan2003
3 2 1 30jan2003
3 2 1 31jan2003
4 1 0 02dec1993
4 1 0 03dec1993
4 1 0 04dec1993
4 1 1 05dec1993
5 1 0 08feb2005
5 1 0 09feb2005
5 1 0 10feb2005
5 1 1 11feb2005
6 3 0 15jan1999
6 3 0 16jan1999
6 3 1 17jan1999
6 3 1 18jan1999
6 3 1 19jan1999
end
list, sepby(id)
*----- what you want -----
bysort id: gen wish2 = _n > (_N - count)
list, sepby(id)
I assume you already sorted your date variable within ids.
One way to accomplish this would be to use within-group row numbers using 'bysort'-type logic:
***Create variable of within-group row numbers.
bysort id: gen obsnum = _n
***Calculate total number of rows within each group.
by id: egen max_obsnum = max(obsnum)
***Subtract the count variable from the group row count.
***This is the number of rows where we want the dummy to equal zero.
gen max_obsnum_less_count = max_obsnum - count
***Create the dummy to equal one when the row number is
***greater than this last variable.
gen dummy = (obsnum > max_obsnum_less_count)
***Clean up.
drop obsnum max_obsnum max_obsnum_less_count

Regular Expression to match few characters from a string

I am trying to find a string within another string. However, I am trying to match even if one or more character is not matching.
Let me explain with an example :
Let's say I have a string 'abcdefghij'. Now if the string to match is 'abcd',
I could write strfind('abcdefghij', 'abc')
Now, I have a string 'adcf'. Notice that, there is a mismatch in two characters, I would consider it as a match.
Any idea how to do it ?
I know, this is not the most optimal code.
Example :
a='abcdefghijk';
b='xbcx'
c='abxx'
d='axxd'
e='abcx'
f='xabc'
g='axcd'
h='abxd'
i ='abcd'
All these strings should match with a. I hope this example makes it more clear. The idea is, if there is a mismatch of 1 or 2 characters also, it should be considered as a match.
You could do it like this:
A = 'abcdefghij'; % Main string
B = 'adcf'; % String to be found
tolerance = 2; % Maximum number of different characters to tolerate
nA = numel(A);
nB = numel(B);
pos = find(sum(A(mod(cumsum([(1:nA)' ones(nA, nB - 1)], 2) - 1, nA) + 1) == repmat(B, nA, 1), 2) >= nB - tolerance);
In this case it will return pos = [1 3]'; because "adcf" can be matched on the first position (matching "a?c?") and on the third position (matching "?d?f")
Explanation:
First, we take the sizes of A and B
Then, we create the matrix [(1:nA)' ones(nA, nB - 1)], which gives us this:
Output:
1 1 1 1
2 1 1 1
3 1 1 1
4 1 1 1
5 1 1 1
6 1 1 1
7 1 1 1
8 1 1 1
9 1 1 1
10 1 1 1
We perform a cumulative sum to the right, using cumsum, to achieve this:
Output:
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
5 6 7 8
6 7 8 9
7 8 9 10
8 9 10 11
9 10 11 12
10 11 12 13
And use the mod function so each number is between 1 and nA, like this:
Output:
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
5 6 7 8
6 7 8 9
7 8 9 10
8 9 10 1
9 10 1 2
10 1 2 3
We then use that matrix as an index for the A matrix.
Output:
abcd
bcde
cdef
defg
efgh
fghi
ghij
hija
ijab
jabc
Note this matrix has all possible substrings of A with size nB.
Now we use repmat to replicate B down, 'nA rows'.
Output:
adcf
adcf
adcf
adcf
adcf
adcf
adcf
adcf
adcf
adcf
And perform a direct comparison:
Output:
1 0 1 0
0 0 0 0
0 1 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
Summing to the right give us this:
Output:
2
0
2
0
0
0
0
0
0
0
Which are the number of character matches on each possible substring.
To finish, we use find to select the indexes of the matches within our tolerance.
In your code
c=a-b is not valid (Matrix dimensions not same)
If you need at least one match, not in order, (as your example says), you can have something like this :-
>> a='abcdefgh';
>> b='adcf';
>> sum(ismember(a,b)) ~= 0
ans =
1

Formula that uses previous value

In Stata I want to have a variable calculated by a formula, which includes multiplying by the previous value, within blocks defined by a variable ID. I tried using a lag but that did not work for me.
In the formula below the Y-1 is intended to signify the value above (the lag).
gen Y = 0
replace Y = 1 if count == 1
sort ID
by ID: replace Y = (1+X)*Y-1 if count != 1
X Y count ID
. 1 1 1
2 3 2 1
1 6 3 1
3 24 4 1
2 72 5 1
. 1 1 2
1 2 2 2
7 16 3 2
Your code can be made a little more concise. Here's how:
input X count ID
. 1 1
2 2 1
1 3 1
3 4 1
2 5 1
. 1 2
1 2 2
7 3 2
end
gen Y = count == 1
bysort ID (count) : replace Y = (1 + X) * Y[_n-1] if count > 1
The creation of a dummy (indicator) variable can exploit the fact that true or false expressions are evaluated as 1 or 0.
Sorting before by and the subsequent by command can be condensed into one. Note that I spelled out that within blocks of ID, count should remain sorted.
This is really a comment, not another answer, but it would be less clear if presented as such.
Y-1, the lag in the formula would be translated as seen in the below.
gen Y = 0
replace Y = 1 if count == 1
sort ID
by ID: replace Y = (1+X)*Y[_n-1] if count != 1

Find all puddles on the square (algorithm)

The problem is defined as follows:
You're given a square. The square is lined with flat flagstones size 1m x 1m. Grass surround the square. Flagstones may be at different height. It starts raining. Determine where puddles will be created and compute how much water will contain. Water doesn't flow through the corners. In any area of ​​grass can soak any volume of water at any time.
Input:
width height
width*height non-negative numbers describing a height of each flagstone over grass level.
Output:
Volume of water from puddles.
width*height signs describing places where puddles will be created and places won't.
. - no puddle
# - puddle
Examples
Input:
8 8
0 0 0 0 0 1 0 0
0 1 1 1 0 1 0 0
0 1 0 2 1 2 4 5
0 1 1 2 0 2 4 5
0 3 3 3 3 3 3 4
0 3 0 1 2 0 3 4
0 3 3 3 3 3 3 0
0 0 0 0 0 0 0 0
Output:
11
........
........
..#.....
....#...
........
..####..
........
........
Input:
16 16
8 0 1 0 0 0 0 2 2 4 3 4 5 0 0 2
6 2 0 5 2 0 0 2 0 1 0 3 1 2 1 2
7 2 5 4 5 2 2 1 3 6 2 0 8 0 3 2
2 5 3 3 0 1 0 3 3 0 2 0 3 0 1 1
1 0 1 4 1 1 2 0 3 1 1 0 1 1 2 0
2 6 2 0 0 3 5 5 4 3 0 4 2 2 2 1
4 2 0 0 0 1 1 2 1 2 1 0 4 0 5 1
2 0 2 0 5 0 1 1 2 0 7 5 1 0 4 3
13 6 6 0 10 8 10 5 17 6 4 0 12 5 7 6
7 3 0 2 5 3 8 0 3 6 1 4 2 3 0 3
8 0 6 1 2 2 6 3 7 6 4 0 1 4 2 1
3 5 3 0 0 4 4 1 4 0 3 2 0 0 1 0
13 3 6 0 7 5 3 2 21 8 13 3 5 0 13 7
3 5 6 2 2 2 0 2 5 0 7 0 1 3 7 5
7 4 5 3 4 5 2 0 23 9 10 5 9 7 9 8
11 5 7 7 9 7 1 0 17 13 7 10 6 5 8 10
Output:
103
................
..#.....###.#...
.......#...#.#..
....###..#.#.#..
.#..##.#...#....
...##.....#.....
..#####.#..#.#..
.#.#.###.#..##..
...#.......#....
..#....#..#...#.
.#.#.......#....
...##..#.#..##..
.#.#.........#..
......#..#.##...
.#..............
................
I tried different ways. Floodfill from max value, then from min value, but it's not working for every input or require code complication. Any ideas?
I'm interesting algorithm with complexity O(n^2) or o(n^3).
Summary
I would be tempted to try and solve this using a disjoint-set data structure.
The algorithm would be to iterate over all heights in the map performing a floodfill operation at each height.
Details
For each height x (starting at 0)
Connect all flagstones of height x to their neighbours if the neighbour height is <= x (storing connected sets of flagstones in the disjoint set data structure)
Remove any sets that connected to the grass
Mark all flagstones of height x in still remaining sets as being puddles
Add the total count of flagstones in remaining sets to a total t
At the end t gives the total volume of water.
Worked Example
0 0 0 0 0 1 0 0
0 1 1 1 0 1 0 0
0 1 0 2 1 2 4 5
0 1 1 2 0 2 4 5
0 3 3 3 3 3 3 4
0 3 0 1 2 0 3 4
0 3 3 3 3 3 3 0
0 0 0 0 0 0 0 0
Connect all flagstones of height 0 into sets A,B,C,D,E,F
A A A A A 1 B B
A 1 1 1 A 1 B B
A 1 C 2 1 2 4 5
A 1 1 2 D 2 4 5
A 3 3 3 3 3 3 4
A 3 E 1 2 F 3 4
A 3 3 3 3 3 3 A
A A A A A A A A
Remove flagstones connecting to the grass, and mark remaining as puddles
1
1 1 1 1
1 C 2 1 2 4 5 #
1 1 2 D 2 4 5 #
3 3 3 3 3 3 4
3 E 1 2 F 3 4 # #
3 3 3 3 3 3
Count remaining set size t=4
Connect all of height 1
G
C C C G
C C 2 D 2 4 5 #
C C 2 D 2 4 5 #
3 3 3 3 3 3 4
3 E E 2 F 3 4 # #
3 3 3 3 3 3
Remove flagstones connecting to the grass, and mark remaining as puddles
2 2 4 5 #
2 2 4 5 #
3 3 3 3 3 3 4
3 E E 2 F 3 4 # # #
3 3 3 3 3 3
t=4+3=7
Connect all of height 2
A B 4 5 #
A B 4 5 #
3 3 3 3 3 3 4
3 E E E E 3 4 # # #
3 3 3 3 3 3
Remove flagstones connecting to the grass, and mark remaining as puddles
4 5 #
4 5 #
3 3 3 3 3 3 4
3 E E E E 3 4 # # # #
3 3 3 3 3 3
t=7+4=11
Connect all of height 3
4 5 #
4 5 #
E E E E E E 4
E E E E E E 4 # # # #
E E E E E E
Remove flagstones connecting to the grass, and mark remaining as puddles
4 5 #
4 5 #
4
4 # # # #
After doing this for heights 4 and 5 nothing will remain.
A preprocessing step to create lists of all locations with each height should mean that the algorithm is close to O(n^2).
This seems to be working nicely. The idea is it is a recursive function, that checks to see if there is an "outward flow" that will allow it to escape to the edge. If the values that do no have such an escape will puddle. I tested it on your two input files and it works quite nicely. I copied the output for these two files for you. Pardon my nasty use of global variables and what not, I figured it was the concept behind the algorithm that mattered, not good style :)
#include <fstream>
#include <iostream>
#include <vector>
using namespace std;
int SIZE_X;
int SIZE_Y;
bool **result;
int **INPUT;
bool flowToEdge(int x, int y, int value, bool* visited) {
if(x < 0 || x == SIZE_X || y < 0 || y == SIZE_Y) return true;
if(visited[(x * SIZE_X) + y]) return false;
if(value < INPUT[x][y]) return false;
visited[(x * SIZE_X) + y] = true;
bool left = false;
bool right = false;
bool up = false;
bool down = false;
left = flowToEdge(x-1, y, value, visited);
right = flowToEdge(x+1, y, value, visited);
up = flowToEdge(x, y+1, value, visited);
down = flowToEdge(x, y-1, value, visited);
return (left || up || down || right);
}
int main() {
ifstream myReadFile;
myReadFile.open("test.txt");
myReadFile >> SIZE_X;
myReadFile >> SIZE_Y;
INPUT = new int*[SIZE_X];
result = new bool*[SIZE_X];
for(int i = 0; i < SIZE_X; i++) {
INPUT[i] = new int[SIZE_Y];
result[i] = new bool[SIZE_Y];
for(int j = 0; j < SIZE_Y; j++) {
int someInt;
myReadFile >> someInt;
INPUT[i][j] = someInt;
result[i][j] = false;
}
}
for(int i = 0; i < SIZE_X; i++) {
for(int j = 0; j < SIZE_Y; j++) {
bool visited[SIZE_X][SIZE_Y];
for(int k = 0; k < SIZE_X; k++)//You can avoid this looping by using maps with pairs of coordinates instead
for(int l = 0; l < SIZE_Y; l++)
visited[k][l] = 0;
result[i][j] = flowToEdge(i,j, INPUT[i][j], &visited[0][0]);
}
}
for(int i = 0; i < SIZE_X; i++) {
cout << endl;
for(int j = 0; j < SIZE_Y; j++)
cout << result[i][j];
}
cout << endl;
}
The 16 by 16 file:
1111111111111111
1101111100010111
1111111011101011
1111000110101011
1011001011101111
1110011111011111
1100000101101011
1010100010110011
1110111111101111
1101101011011101
1010111111101111
1110011010110011
1010111111111011
1111110110100111
1011111111111111
1111111111111111
The 8 by 8 file
11111111
11111111
11011111
11110111
11111111
11000011
11111111
11111111
You could optimize this algorithm easily and considerably by doing several things. A: return true immediately upon finding a route would speed it up considerably. You could also connect it globally to the current set of results so that any given point would only have to find a flow point to an already known flow point, and not all the way to the edge.
The work involved, each n will have to exam each node. However, with optimizations, we should be able to get this much lower than n^2 for most cases, but it still an n^3 algorithm in the worst case... but creating this would be very difficult(with proper optimization logic... dynamic programming for the win!)
EDIT:
The modified code works for the following circumstances:
8 8
1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 1
1 0 1 1 1 1 0 1
1 0 1 0 0 1 0 1
1 0 1 1 0 1 0 1
1 0 1 1 0 1 0 1
1 0 0 0 0 1 0 1
1 1 1 1 1 1 1 1
And these are the results:
11111111
10000001
10111101
10100101
10110101
10110101
10000101
11111111
Now when we remove that 1 at the bottom we want to see no puddling.
8 8
1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 1
1 0 1 1 1 1 0 1
1 0 1 0 0 1 0 1
1 0 1 1 0 1 0 1
1 0 1 1 0 1 0 1
1 0 0 0 0 1 0 1
1 1 1 1 1 1 0 1
And these are the results
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1