I am interested in writing a function generate(n,m) which exhaustively generating strings of length n(n-1)/2 consisting solely of +/- characters. These strings will then be transformed into an n × n symmetric (-1,0,1)-matrix in the following way:
toTriangle["+--+-+-++-"]
{{1, -1, -1, 1}, {-1, 1, -1}, {1, 1}, {-1}}
toMatrix[%, 0] // MatrixForm
| 0 1 -1 -1 1 |
| 1 0 -1 1 -1 |
matrixForm = |-1 -1 0 1 1 |
|-1 1 1 0 -1 |
| 1 -1 1 -1 0 |
Thus the given string represents the upper-right triangle of the matrix, which is then reflected to generate the rest of it.
Question: How can I generate all +/- strings such that the resulting matrix has precisely m -1's per row?
For example, generate(5,3) will give all strings of length 5(5-1)/2 = 10 such that each row contains precisely three -1's.
I'd appreciate any help with constructing such an algorithm.
This is the logic to generate every matrix for a given n and m. It's a bit convoluted, so I'm not sure how much faster than brute force an implementation would be; I assume the difference will become more pronounced for larger values.
(The following will generate an output of zeros and ones for convenience, where zero represents a plus and a one represents a minus.)
A square matrix where each row has m ones translates to a triangular matrix where these folded row/columns have m ones:
x 0 1 0 1 x 0 1 0 1 0 1 0 1
0 x 1 1 0 x 1 1 0 1 1 0
1 1 x 0 0 x 0 0 0 0
0 1 0 x 1 x 1 1
1 0 0 1 x x
Each of these groups overlaps with all the other groups; choosing values for the first k groups means that the vertical part of group k+1 is already determined.
We start by putting the number of ones required per row on the diagonal; e.g. for (5,2) that is:
2 . . . .
2 . . .
2 . .
2 .
2
Then we generate every bit pattern with m ones for the first group; there are (n-1 choose m) of these, and they can be efficiently generated, e.g. with Gosper's hack.
(4,2) -> 0011 0101 0110 1001 1010 1100
For each of these, we fill them in in the matrix, and subtract them from the numbers of required ones:
X 0 0 1 1
2 . . .
2 . .
1 .
1
and then recurse with the smaller triangle:
2 . . .
2 . .
1 .
1
If we come to a point where some of the numbers of required ones on the diagonal are zero, e.g.:
2 . . .
1 . .
0 .
1
then we can already put a zero in this column, and generate the possible bit patterns for fewer columns; in the example that would be (2,2) instead of (3,2), so there's only one possible bit pattern: 11. Then we distribute the bit pattern over the columns that have a non-zero required count under them:
2 . 0 . X 1 0 1
1 . . 0 . .
0 . 0 .
1 0
However, not all possible bit patterns will lead to valid solutions; take this example:
2 . . . . X 0 0 1 1
2 . . . 2 . . . 2 . . . X 0 1 1
2 . . 2 . . 2 . . 2 . . 2 . .
2 . 1 . 1 . 0 . 0 .
2 1 1 0 0
where we end up with a row that requires another 2 ones while both columns can no longer take any ones. The way to spot this situation is by looking at the list of required ones per column that is created by each option in the penultimate step:
pattern required
0 1 1 -> 2 0 0
1 0 1 -> 1 1 0
1 1 0 -> 1 0 1
If the first value in the list is x, then there must be at least x non-zero values after it; which is false for the first of the three options.
(There is room for optimization here: in a count list like 1,1,0,6,0,2,1,1 there are only 2 non-zero values before the 6, which means that the 6 will be decremented at most 2 times, so its minimum value when it becomes the first element will be 4; however, there are only 3 non-zero values after it, so at this stage you already know this list will not lead to any valid solutions. Checking this would add to the code complexity, so I'm not sure whether that would lead to an improvement in execution speed.)
So the complete algorithm for (n,m) starts with:
Create an n-sized list with all values set to m (count of ones required per group).
Generate all bit patterns of size n-1 with m ones; for each of these:
Subtract the pattern from a copy of the count list (without the first element).
Recurse with the pattern and the copy of the count list.
and the recursive steps after that are:
Receive the sequence so far, and a count list.
The length of the count list is n, and its first element is m.
Let k be the number of non-zero values in the count list (without the first element).
Generate all bit pattern of size k with m ones; for each of these:
Create a 0-filled list sized n-1.
Distribute the bit pattern over it, skipping the columns with a zero count.
Add the value list to the sequence so far.
Subtract the value list from a copy of the count list (without the first element).
If the first value in the copy of the count list is greater than the number of non-zeros after it, skip this pattern.
At the deepest recursion level, store the sequence, or else:
Recurse with the sequence so far, and the copy of the count list.
Here's a code snippet as a proof of concept; in a serious language, and using integers instead of arrays for the bitmaps, this should be much faster:
function generate(n, m) {
// if ((n % 2) && (m % 2)) return; // to catch (3,1)
var counts = [], pattern = [];
for (var i = 0; i < n - 1; i++) {
counts.push(m);
pattern.push(i < m ? 1 : 0);
}
do {
var c_copy = counts.slice();
for (var i = 0; i < n - 1; i++) c_copy[i] -= pattern[i];
recurse(pattern, c_copy);
}
while (revLexi(pattern));
}
function recurse(sequence, counts) {
var n = counts.length, m = counts.shift(), k = 0;
for (var i = 0; i < n - 1; i++) if (counts[i]) ++k;
var pattern = [];
for (var i = 0; i < k; i++) pattern.push(i < m ? 1 : 0);
do {
var values = [], pos = 0;
for (var i = 0; i < n - 1; i++) {
if (counts[i]) values.push(pattern[pos++]);
else values.push(0);
}
var s_copy = sequence.concat(values);
var c_copy = counts.slice();
var nonzero = 0;
for (var i = 0; i < n - 1; i++) {
c_copy[i] -= values[i];
if (i && c_copy[i]) ++nonzero;
}
if (c_copy[0] > nonzero) continue;
if (n == 2) {
for (var i = 0; i < s_copy.length; i++) {
document.write(["+ ", "− "][s_copy[i]]);
}
document.write("<br>");
}
else recurse(s_copy, c_copy);
}
while (revLexi(pattern));
}
function revLexi(seq) { // reverse lexicographical because I had this lying around
var max = true, pos = seq.length, set = 1;
while (pos-- && (max || !seq[pos])) if (seq[pos]) ++set; else max = false;
if (pos < 0) return false;
seq[pos] = 0;
while (++pos < seq.length) seq[pos] = set-- > 0 ? 1 : 0;
return true;
}
generate(5, 2);
Here are the number of results and the number of recursions for values of n up to 10, so you can compare them to check correctness. When n and m are both odd numbers, there are no valid results; this is calculated correctly, except in the case of (3,1); it is of course easy to catch these cases and return immediately.
(n,m) results number of recursions
(4,0) (4,3) 1 2 2
(4,1) (4,2) 3 6 7
(5,0) (5,4) 1 3 3
(5,1) (5,3) 0 12 20
(5,2) 12 36
(6,0) (6,5) 1 4 4
(6,1) (6,4) 15 48 76
(6,2) (6,3) 70 226 269
(7,0) (7,6) 1 5 5
(7,1) (7,5) 0 99 257
(7,2) (7,4) 465 1,627 2,313
(7,3) 0 3,413
(8,0) (8,7) 1 6 6
(8,1) (8,6) 105 422 1,041
(8,2) (8,5) 3,507 13,180 23,302
(8,3) (8,4) 19,355 77,466 93,441
(9,0) (9,8) 1 7 7
(9,1) (9,7) 0 948 4,192
(9,2) (9,6) 30,016 119,896 270,707
(9,3) (9,5) 0 1,427,457 2,405,396
(9,4) 1,024,380 4,851,650
(10,0) (10,9) 1 8 8
(10,1) (10,8) 945 4440 18930
(10,2) (10,7) 286,884 1,210,612 3,574,257
(10,3) (10,6) 11,180,820 47,559,340 88,725,087
(10,4) (10,5) 66,462,606 313,129,003 383,079,169
I doubt that you really want all variants for large n,m values - number of them is tremendous large.
This problem is equivalent to generation of m-regular graphs (note that if we replace all 1's by zeros and all -1's by 1 - we can see adjacency matrix of graph. Regular graph - degrees of all vertices are equal to m).
Here we can see that number of (18,4) regular graphs is about 10^9 and rises fast with n/m values. Article contains link to program genreg intended for such graphs generation. FTP links to code and executable don't work for me - perhaps too old.
Upd: Here is another link to source (though 1996 year instead of paper's 1999)
Simple approach to generate one instance of regular graph is described here.
For small n/m values you can also try brute-force: fill the first row with m ones (there are C(n,m) variants and for every variants fill free places in the second row and so on)
Written in Wolfram Mathematica.
generate[n_, m_] := Module[{},
x = Table[StringJoin["i", ToString[i], "j", ToString[j]],
{j, 1, n}, {i, 2, n}];
y = Transpose[x];
MapThread[(x[[#, ;; #2]] = y[[#, ;; #2]]) &,
{-Range[n - 1], Reverse#Range[n - 1]}];
Clear ## Names["i*"];
z = ToExpression[x];
Clear[s];
s = Reduce[Join[Total## == m & /# z,
0 <= # <= 1 & /# Union[Flatten#z]],
Union#Flatten[z], Integers];
Clear[t, u, v];
Array[(t[#] =
Partition[Flatten[z] /.
ToRules[s[[#]]], n - 1] /.
{1 -> -1, 0 -> 1}) &, Length[s]];
Array[Function[a,
(u[a] = StringJoin[Flatten[MapThread[
Take[#, 1 - #2] &,
{t[a], Reverse[Range[n]]}]] /.
{1 -> "+", -1 -> "-"}])], Length[s]];
Array[Function[a,
(v[a] = MapThread[Insert[#, 0, #2] &,
{t[a], Range[n]}])], Length[s]]]
Timing[generate[9, 4];]
Length[s]
{202.208, Null}
1024380
The program takes 202 seconds to generate 1,024,380 solutions. E.g. the last one
u[1024380]
----++++---++++-+-+++++-++++--------
v[1024380]
0 -1 -1 -1 -1 1 1 1 1
-1 0 -1 -1 -1 1 1 1 1
-1 -1 0 -1 1 -1 1 1 1
-1 -1 -1 0 1 1 -1 1 1
-1 -1 1 1 0 1 1 -1 -1
1 1 -1 1 1 0 -1 -1 -1
1 1 1 -1 1 -1 0 -1 -1
1 1 1 1 -1 -1 -1 0 -1
1 1 1 1 -1 -1 -1 -1 0
and the first ten strings
u /# Range[10]
++++----+++----+-+-----+----++++++++
++++----+++----+-+------+--+-+++++++
++++----+++----+-+-------+-++-++++++
++++----+++----+--+---+-----++++++++
++++----+++----+---+--+----+-+++++++
++++----+++----+----+-+----++-++++++
++++----+++----+--+-----+-+--+++++++
++++----+++----+--+------++-+-++++++
++++----+++----+---+---+--+--+++++++
This question already has answers here:
Given an XOR and SUM of two numbers, how to find the number of pairs that satisfy them?
(2 answers)
Closed 5 years ago.
You are given a sum S and X , you need to find , if it there exist two numbers a and b such that a+b = S and a^b = X
I used a loop upto S/2 and check if it is possible or not
for(int i=0;i<=s/2;i++)
{
if(i^(s-i)==X)
return true;
}
complexity : O(n)
Need some better approach
Given that a+b = (a XOR b) + (a AND b)*2 (from here) we can calculate (a AND b):
If S < X => not possible, otherwise take S-X. If this is odd => not possible, otherwise (a AND b) = (S-X)/2.
now we can look at the bits of a and b individually. Checking all four combinations we see there is only one result that is impossible namely XOR and AND both 1.
So if (a XOR b) AND (a AND b) != 0 there is no solution. Otherwise one can find a and b that solve the equation.
if (S < X) return false;
Y = S - X;
if (Y is odd) return false;
if ((X & (Y/2)) != 0) return false;
return true;
Without previous knowledge of the equation, a+b = a^b + (a&b)*2, we can think of another solution. This solution is O(logK) where K is the maximum possible value of S and X. That is, if S and X are unsigned int then K is 2^32 - 1.
Start from the MSB of S and X. With the information that whether summing this bit must provide carry or not, we can check for this bit with condition that whether summing the bits to the right should provide carry or not.
Case1 ) summing must not provide carry
S X | need carry from right
------------------------------
0 0 | no (a = 0, b = 0)
0 1 | impossible
1 0 | yes (a = 0, b = 0)
1 1 | no (a = 1, b = 0 or 0,1)
Case2 ) summing must provide carry
S X | need carry from right
------------------------------
0 0 | no (a = 1, b = 1)
0 1 | yes (a = 1, b = 0 or 0,1)
1 0 | yes (a = 1, b = 1)
1 1 | impossible
There is a special case for the MSB where the carry doesn't matter.
Case3 ) don't care
S X | need carry from right
------------------------------
0 0 | no (a = 0, b = 0 or 1,1)
0 1 | yes (a = 1, b = 0 or 0,1)
1 0 | yes (a = 0, b = 0 or 1,1)
1 1 | no (a = 1, b = 0 or 0,1)
Lastly, the LSB must end with 'no need carry from right'.
The implementation and test code is here. It compares the output of the accepted solution and this solution.
I assume that you are assuming that your input sequence is sorted.
If it is, then this problem is as good as finding pair for a given sum and while checking sum check for their (a^b == SUM) this should be enough and this problem can be solved in O(n). URL
And you can't do better than that. In worst case you have to visit each element atleast once.
So I am writing a haskell program to calculate the largest power of a number that divides a factorial.
largestPower :: Int -> Int -> Int
Here largestPower a b has find largest power of b that divides a!.
Now I understand the math behind it, the way to find the answer is to repeatedly divide a (just a) by b, ignore the remainder and finally add all the quotients. So if we have something like
largestPower 10 2
we should get 8 because 10/2=5/2=2/2=1 and we add 5+2+1=8
However, I am unable to figure out how to implement this as a function, do I use arrays or just a simple recursive function.
I am gravitating towards it being just a normal function, though I guess it can be done by storing quotients in an array and adding them.
Recursion without an accumulator
You can simply write a recursive algorithm and sum up the result of each call. Here we have two cases:
a is less than b, in which case the largest power is 0. So:
largestPower a b | a < b = 0
a is greater than or equal to b, in that case we divide a by b, calculate largestPower for that division, and add the division to the result. Like:
| otherwise = d + largestPower d b
where d = (div a b)
Or putting it together:
largestPower a b | a < b = 1
| otherwise = d + largestPower d b
where d = (div a b)
Recursion with an accumuator
You can also use recursion with an accumulator: a variable you pass through the recursion, and update accordingly. At the end, you return that accumulator (or a function called on that accumulator).
Here the accumulator would of course be the running product of divisions, so:
largestPower = largestPower' 0
So we will define a function largestPower' (mind the accent) with an accumulator as first argument that is initialized as 1.
Now in the recursion, there are two cases:
a is less than b, we simply return the accumulator:
largestPower' r a b | a < b = r
otherwise we multiply our accumulator with b, and pass the division to the largestPower' with a recursive call:
| otherwise = largestPower' (d+r) d b
where d = (div a b)
Or the full version:
largestPower = largestPower' 1
largestPower' r a b | a < b = r
| otherwise = largestPower' (d+r) d b
where d = (div a b)
Naive correct algorithm
The algorithm is not correct. A "naive" algorithm would be to simply divide every item and keep decrementing until you reach 1, like:
largestPower 1 _ = 0
largestPower a b = sumPower a + largestPower (a-1) b
where sumPower n | n `mod` b == 0 = 1 + sumPower (div n b)
| otherwise = 0
So this means that for the largestPower 4 2, this can be written as:
largestPower 4 2 = sumPower 4 + sumPower 3 + sumPower 2
and:
sumPower 4 = 1 + sumPower 2
= 1 + 1 + sumPower 1
= 1 + 1 + 0
= 2
sumPower 3 = 0
sumPower 2 = 1 + sumPower 1
= 1 + 0
= 1
So 3.
The algorithm as stated can be implemented quite simply:
largestPower :: Int -> Int -> Int
largestPower 0 b = 0
largestPower a b = d + largestPower d b where d = a `div` b
However, the algorithm is not correct for composite b. For example, largestPower 10 6 with this algorithm yields 1, but in fact the correct answer is 4. The problem is that this algorithm ignores multiples of 2 and 3 that are not multiples of 6. How you fix the algorithm is a completely separate question, though.
Given a set of N numbers x1, x2, ..., xN, how can you find an ordering of them to maximize the minimum absolute difference between adjacent numbers? This is probably an NP hard problem, so any efficient approximate method will do.
Let's say you've defined your data as x_i for i=1, ..., n. We can define binary variables p_{ij} for i=1, ..., n, and j=1, ..., n, which are 1 if number i is in sorted order j and 0 otherwise. Adding a variable e, our optimization model would be something like:
The constraints with the absolute values ensure that e (our minimum gap) does not exceed the gap between each pair of adjacent elements in our sorted sequence. However, absolute values aren't allowed in linear optimization models, and in general you need to add a binary variable to model an absolute value being greater than or equal to some other value. So let's add binary variable r_j, j=2, ..., n, and replace our problematic constraints:
Here M is a large number; 2(max(x) - min(x)) should be sufficiently large. Now, we're ready to actually implement this model. You can use any MIP solver; I'll use the lpSolveAPI in R because it's free and easily accessible. p_{ij} are stored in variables 1 through n^2; r_j are stored in variables n^2+1 through n^2+n-1; and e is stored in variable n^2+n.
x = 1:5
n = length(x)
M = 2*(max(x) - min(x))
library(lpSolveAPI)
mod = make.lp(0, n^2+n)
set.type(mod, 1:(n^2+n-1), "binary")
set.objfn(mod, c(rep(0, n^2+n-1), 1))
lp.control(mod, sense="max")
for (j in 2:n) {
base.cons <- rep(0, n^2+n)
base.cons[seq(j-1, by=n, length.out=n)] = x
base.cons[seq(j, by=n, length.out=n)] = -x
base.cons[n^2+j-1] = M
first.cons = base.cons
first.cons[n^2+n] = -1
add.constraint(mod, first.cons, ">=", 0)
second.cons = -base.cons
second.cons[n^2+n] = -1
add.constraint(mod, second.cons, ">=", -M)
}
for (j in 1:n) {
this.cons = rep(0, n^2+n)
this.cons[seq(j, by=n, length.out=n)] = 1
add.constraint(mod, this.cons, "=", 1)
}
for (i in 1:n) {
this.cons = rep(0, n^2+n)
this.cons[seq((i-1)*n+1, i*n)] = 1
add.constraint(mod, this.cons, "=", 1)
}
Now we're ready to solve the model:
solve(mod)
# [1] 0
get.objective(mod)
# [1] 2
get.variables(mod)
# [1] 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 2
And lastly we can extract the sorted list using the x_i and p_{ij} variables:
sapply(1:n, function(j) sum(get.variables(mod)[seq(j, by=n, length.out=n)]*x))
# [1] 1 3 5 2 4
I'd like to print (or send to a file in a human-readable format like below) arbitrary size square tables where each table cell contains the number of steps required to solve Euclid's algorithm for the two integers in the row/column headings like this (table written by hand, but I think the numbers are all correct):
1 2 3 4 5 6
1 1 1 1 1 1 1
2 1 1 2 1 2 1
3 1 2 1 2 3 1
4 1 1 2 1 2 2
5 1 2 3 2 1 2
6 1 1 1 2 2 1
The script would ideally allow me to choose the start integer (1 as above or 11 as below or something else arbitrary) and end integer (6 as above or 16 as below or something else arbitrary and larger than the start integer), so that I could do this too:
11 12 13 14 15 16
11 1 2 3 4 4 3
12 2 1 2 2 2 2
13 3 2 1 2 3 3
14 4 2 2 1 2 2
15 4 2 3 2 1 2
16 3 2 3 2 2 1
I realize that the table is symmetric about the diagonal and so only half of the table contains unique information, and that the diagonal itself is always a 1-step algorithm.
See this and for a graphical representation of what I'm after, but I'd like to know the actual number of steps for any two integers which the image doesn't show me.
I have the algorithms (there's probably better implementations, but I think these work):
The step counter:
def gcd(a,b):
"""Step counter."""
if b > a:
x = a
a = b
b = x
counter = 0
while b:
c = a % b
a = b
b = c
counter += 1
return counter
The list builder:
def gcd_steps(n):
"""List builder."""
print("Table of size", n - 1, "x", n - 1)
list_of_steps = []
for i in range(1, n):
for j in range(1, n):
list_of_steps.append(gcd(i,j))
print(list_of_steps)
return list_of_steps
but I'm totally hung up on how to write the table. I thought about a double nested for loop with i and j and stuff, but I'm new to Python and haven't a clue about the best way (or any way) to go about writing the table. I don't need special formatting like something to offset the row/column heads from the table cells as I can do that by eye, but just getting everything to line up so that I can read it easily is proving too difficult for me at my current skill level, I'm afraid. I'm thinking that it probably makes sense to print/output within the two nested for loops as I'm calculating the numbers I need which is why the list builder has some print statements as well as returning the list, but I don't know how to work the print magic to do what I'm after.
Try this. The programs computes data row by row and prints each row when it's available,
in order to limit memory usage.
import sys, os
def gcd(a,b):
k = 0
if b > a:
a, b = b, a
while b > 0:
a, b = b, a%b
k += 1
return k
def printgcd(name, a, b):
f = open(name, "wt")
s = ""
for i in range(a, b + 1):
s = "{}\t{}".format(s, i)
f.write("{}\n".format(s))
for i in range(a, b + 1):
s = "{}".format(i)
for j in range (a, b + 1):
s = "{}\t{}".format(s, gcd(i, j))
f.write("{}\n".format(s))
f.close()
printgcd("gcd-1-6.txt", 1, 6)
The preceding won't return a list with all computed values, since they are destroyed on purpose. It's easy to do however. Here is a solution with a hash table
def printgcd2(name, a, b):
f = open(name, "wt")
s = ""
h = { }
for i in range(a, b + 1):
s = "{}\t{}".format(s, i)
f.write("{}\n".format(s))
for i in range(a, b + 1):
s = "{}".format(i)
for j in range (a, b + 1):
k = gcd(i, j)
s = "{}\t{}".format(s, k)
h[i, j] = k
f.write("{}\n".format(s))
f.close()
return h
And here is another with a list of lists
def printgcd3(name, a, b):
f = open(name, "wt")
s = ""
u = [ ]
for i in range(a, b + 1):
s = "{}\t{}".format(s, i)
f.write("{}\n".format(s))
for i in range(a, b + 1):
v = [ ]
s = "{}".format(i)
for j in range (a, b + 1):
k = gcd(i, j)
s = "{}\t{}".format(s, k)
v.append(k)
f.write("{}\n".format(s))
u.append(v)
f.close()
return u