Related
Context:
I have a hydraulic erosion algorithm that needs to receive an array of droplet starting positions. I also already have a pattern replicating algorithm, so I only need a good pattern to replicate.
The Requirements:
I need an algorism that produces a set of n^2 entries in a set of format (x,y) or [index] that describe cells in an nxn grid (where n = 2^i where i is any positive integer).
(as a set it means that every cell is mentioned in exactly one entry)
The pattern [created by the algorism ] should contain zero to none clustering of "visited" cells at any stage.
The cell (0,0) is as close to (n-1,n-1) as to (1,1), this relates to the definition of clustering
Note
I was/am trying to find solutions through fractal-like patterns built through recursion, but at the time of writing this, my solution is a lookup table of a checkerboard pattern(list of black cells + list of white cells) (which is bad, but yields fewer artifacts than an ordered list)
C, C++, C#, Java implementations (if any) are preferred
You can use a linear congruential generator to create an even distribution across your n×n space. For example, if you have a 64×64 grid, using a stride of 47 will create the pattern on the left below. (Run on jsbin) The cells are visited from light to dark.
That pattern does not cluster, but it is rather uniform. It uses a simple row-wide transformation where
k = (k + 47) mod (n * n)
x = k mod n
y = k div n
You can add a bit of randomness by making k the index of a space-filling curve such as the Hilbert curve. This will yield the pattern on the right. (Run on jsbin)
You can see the code in the jsbin links.
I have solved the problem myself and just sharing my solution:
here are my outputs for the i between 0 and 3:
power: 0
ordering:
0
matrix visit order:
0
power: 1
ordering:
0 3 2 1
matrix visit order:
0 3
2 1
power: 2
ordering:
0 10 8 2 5 15 13 7 4 14 12 6 1 11 9 3
matrix visit order:
0 12 3 15
8 4 11 7
2 14 1 13
10 6 9 5
power: 3
ordering:
0 36 32 4 18 54 50 22 16 52 48 20 2 38 34 6
9 45 41 13 27 63 59 31 25 61 57 29 11 47 43 15
8 44 40 12 26 62 58 30 24 60 56 28 10 46 42 14
1 37 33 5 19 55 51 23 17 53 49 21 3 39 35 7
matrix visit order:
0 48 12 60 3 51 15 63
32 16 44 28 35 19 47 31
8 56 4 52 11 59 7 55
40 24 36 20 43 27 39 23
2 50 14 62 1 49 13 61
34 18 46 30 33 17 45 29
10 58 6 54 9 57 5 53
42 26 38 22 41 25 37 21
the code:
public static int[] GetPattern(int power, int maxReturnSize = int.MaxValue)
{
int sideLength = 1 << power;
int cellsNumber = sideLength * sideLength;
int[] ret = new int[cellsNumber];
for ( int i = 0 ; i < cellsNumber && i < maxReturnSize ; i++ ) {
// this loop's body can be used for per-request computation
int x = 0;
int y = 0;
for ( int p = power - 1 ; p >= 0 ; p-- ) {
int temp = (i >> (p * 2)) % 4; //2 bits of the index starting from the begining
int a = temp % 2; // the first bit
int b = temp >> 1; // the second bit
x += a << power - 1 - p;
y += (a ^ b) << power - 1 - p;// ^ is XOR
// 00=>(0,0), 01 =>(1,1) 10 =>(0,1) 11 =>(1,0) scaled to 2^p where 0<=p
}
//to index
int index = y * sideLength + x;
ret[i] = index;
}
return ret;
}
I do admit that somewhere along the way the values got transposed, but it does not matter because of how it works.
After doing some optimization I came up with this loop body:
int x = 0;
int y = 0;
for ( int p = 0 ; p < power ; p++ ) {
int temp = ( i >> ( p * 2 ) ) & 3;
int a = temp & 1;
int b = temp >> 1;
x = ( x << 1 ) | a;
y = ( y << 1 ) | ( a ^ b );
}
int index = y * sideLength + x;
(the code assumes that c# optimizer, IL2CPP, and CPP compiler will optimize variables temp, a, b out)
I am looking for an efficient algorithm for the following problem:
There is an array with values, i.e. (note that index 0 is omitted on purpose)
Index 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Value 17, 12, 5, 22, 3, 12, 6, 13, 7, 0, 2, 15
What I need to find is a subset of indices under these constraints:
The number of indices is constant (i.e. 3)
The sum of indices is constant (i.e. 20)
Each index may only appear once (so [2, 9, 9] is not a valid solution)
The sum of values is maximum.
For example if the subset length is 3 and the sum is 20, all possible solutions would be
Indices: [1, 7, 12] Sum of values: 17 + 6 + 15 = 38
Indices: [1, 8, 11] Sum of values: 17 + 13 + 2 = 32
Indices: [1, 9, 10] Sum of values: 17 + 7 + 0 = 24
Indices: [2, 6, 12] Sum of values: 12 + 12 + 15 = 39
Indices: [2, 7, 11] Sum of values: 12 + 6 + 2 = 20
Indices: [2, 8, 10] Sum of values: 12 + 13 + 0 = 25
Indices: [3, 5, 12] Sum of values: 5 + 3 + 15 = 23
Indices: [3, 6, 11] Sum of values: 5 + 12 + 2 = 19
Indices: [3, 7, 10] Sum of values: 5 + 6 + 0 = 11
Indices: [3, 8, 9] Sum of values: 5 + 13 + 7 = 25
Indices: [4, 5, 11] Sum of values: 22 + 3 + 2 = 27
Indices: [4, 6, 10] Sum of values: 22 + 12 + 0 = 34
Indices: [4, 7, 9] Sum of values: 22 + 6 + 7 = 35
Indices: [5, 6, 9] Sum of values: 3 + 12 + 7 = 22
Indices: [5, 7, 8] Sum of values: 3 + 6 + 13 = 22
of which [2, 6, 12] is the optimal solution because it has the maximum sum of values.
At the moment I run through all possible combinations using a slightly modified partition algorithm which grows exponentially as the sum of indices grows, so I wonder if there is any better way?
Solution O(I.S.K)
Let's do some naming first:
I is the greatest index (12 in your example)
S is the sum of values whose indices are selected (20 in your example)
K is the number of selected indices
V[] the array of values linked to the indices
maxsum(s, i, k) the maximal sum reachable by using k indices, all differents, whose value is less than or equal to i and whose sum is s.
Then you want to find maxsum(S, I, K)
Your problem exhibits some good properties:
optimal sub-structure
redundant sub-problems
For instance, when trying to compute maxsum(s, i, k) I can either not use index i, in which case the value is maxsum(s, i-1, k). Or I could use index i. In this case, I want to solve the sub-problem: what is the maximum sum reachable by indices less than or equal to i-1 and whose sum is s-i using k-1 such indices. This is the value: V[i] + maxsum(s-i, i-1, k-1).
As we want to reach the maximal sum we end up having: (Edit: corrected maxsum(s-i, i-1, k) to maxsum(s-i, i-1, k-1))
maxsum(s, i, k) = max{ maxsum(s, i-1, k) ; V[i] + maxsum(s-i, i-1, k-1) }
This is typical of a problem solvable by dynamic programming.
Here is an example C++ program solving the problem in O(I.S.K) (space and time).
We can improve the space complexity to O(I.S) at the price of a bigger time complexity: O(I.S.K²).
How to use the program
g++ -std=c++14 -g -Wall -O0 dp.cpp -o dp
./dp input.txt
Where input.txt is a file with the following format:
first line contains three integers: I S K
second line contains I integers, the values of the indices
Example run
---- K=1 ----
17 12 5 22 3 12 6 13 7 0 2 15
[ 1][ 2][ 3][ 4][ 5][ 6][ 7][ 8][ 9][10][11][12]
[ 1] 17 17 17 17 17 17 17 17 17 17 17 17
[ 2] 12 12 12 12 12 12 12 12 12 12 12
[ 3] 5 5 5 5 5 5 5 5 5 5
[ 4] 22 22 22 22 22 22 22 22 22
[ 5] 3 3 3 3 3 3 3 3
[ 6] 12 12 12 12 12 12 12
[ 7] 6 6 6 6 6 6
[ 8] 13 13 13 13 13
[ 9] 7 7 7 7
[10] 0 0 0
[11] 2 2
[12] 15
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
---- K=2 ----
17 12 5 22 3 12 6 13 7 0 2 15
[ 1][ 2][ 3][ 4][ 5][ 6][ 7][ 8][ 9][10][11][12]
[ 1]
[ 2] 12 12 12 12 12 12 12 12 12 12 12
[ 3] 29 29 29 29 29 29 29 29 29 29 29
[ 4] 22 22 22 22 22 22 22 22 22 22
[ 5] 17 39 39 39 39 39 39 39 39 39
[ 6] 34 34 34 34 34 34 34 34 34
[ 7] 27 27 29 29 29 29 29 29 29
[ 8] 8 24 24 24 24 24 24 24
[ 9] 25 25 25 30 30 30 30 30
[10] 34 34 34 34 34 34 34
[11] 15 28 28 28 28 28 28
[12] 9 35 35 35 35 35
[13] 18 18 29 29 29 32
[14] 25 25 25 25 27
[15] 19 19 19 24 24
[16] 13 13 13 37
[17] 20 20 20 20
[18] 13 13 27
[19] 7 15 21
[20] 9 28
---- K=3 ----
17 12 5 22 3 12 6 13 7 0 2 15
[ 1][ 2][ 3][ 4][ 5][ 6][ 7][ 8][ 9][10][11][12]
[ 1]
[ 2]
[ 3]
[ 4]
[ 5] 17 17 17 17 17 17 17 17 17 17
[ 6] 34 34 34 34 34 34 34 34 34 34
[ 7] 51 51 51 51 51 51 51 51 51
[ 8] 44 44 44 44 44 44 44 44 44
[ 9] 39 39 41 41 41 41 41 41 41
[10] 42 42 42 42 42 42 42 42
[11] 37 51 51 51 51 51 51 51
[12] 30 46 46 46 46 46 46 46
[13] 39 40 52 52 52 52 52
[14] 20 35 47 47 47 47 47
[15] 37 37 42 42 42 42 44
[16] 31 37 37 37 41 41
[17] 40 40 40 40 40 54
[18] 21 47 47 47 47 49
[19] 41 41 41 41 44
[20] 22 35 35 35 39
index: 12 sum: 20
index: 6 sum: 8
index: 2 sum: 2
max sum: 39
The source code
#include <cstdio>
#include <iomanip>
#include <iostream>
#include <limits>
#include <valarray>
#include <vector>
using namespace std;
auto const INF = numeric_limits<double>::infinity();
struct matrix {
matrix(size_t rows, size_t cols, double value)
: cells(value, rows*cols)
, rows(rows)
, cols(cols)
, value(value)
{}
double& operator() (int r, int c)
{
if(r < 0 || c < 0)
return value;
return cells[r*cols+c];
}
valarray<double> cells;
size_t rows;
size_t cols;
double value;
};
int main(int argc, char* argv[]) {
if(argc > 1)
freopen(argv[1], "r", stdin);
// I: max index
// S: sum of indices
// K: number of indices in the sum S
int I, S, K;
cin >> I >> S >> K;
// load values
vector<double> V(I+1, 0);
for(int i=1; i<=I; ++i)
cin >> V[i];
// dynamic programming:
// --------------------
// maxsum(i, s, k) is the maximal sum reachable using 'k' indices, less
// than or equal to 'i', all differents, and having a sum of 's'
//
// maxsum(i, s, k) =
// -oo if i > s
//
// -oo if i < s && k == 1
//
// V[s] if i >= s && s <= I && k == 1
// -oo if (i < s || s > I) && k == 1
//
// max { V[i] + maxsum(i-1, S-i, k-1), maxsum(i-1, S, k) }
vector<matrix> maxsum(K+1, matrix(S+1, I+1, -INF));
// initialize K=1
for(int s=0; s<=I && s<=S; ++s) {
for(int i=s; i<=I; ++i) {
maxsum[1](s, i) = V[s];
}
}
// K > 1
for(int k=2; k<=K; ++k) {
for(int s=2; s<=S; ++s) {
for(int i=1; i<=I; ++i) {
auto l = V[i] + maxsum[k-1](s-i, i-1);
auto r = maxsum[k](s, i-1);
maxsum[k](s, i) = max(l, r);
}
}
}
// display the whole dynamic programming tables (optional)
for(int k=1; k<=K; ++k) {
cout << "---- K=" << k << " ----\n";
cout << " ";
for(int i=1; i<=I; ++i) {
cout << setw(3) << V[i] << ' ';
}
cout << '\n';
cout << " ";
for(int i=1; i<=I; ++i) {
cout << '[' << setw(2) << i << ']';
}
cout << '\n';
for(int s=1; s<=S; ++s) {
cout << '[' << setw(2) << s << "] ";
for(int i=1; i<=I; ++i) {
if(maxsum[k](s, i) == -INF) {
cout << " ";
} else {
cout << setw(3) << maxsum[k](s, i) << ' ';
}
}
cout << '\n';
}
}
// output the indices belonging to the solution by working backward in the
// dynamic programming tables
int t_S = S;
int t_I = I;
for(int k=K; k>=1; --k) {
if(t_I <= 0 || t_S <= 0) {
cout << "error...\n";
break;
}
auto m = maxsum[k](t_S, t_I);
int i;
for(i=t_I; i>=1; --i) {
if(maxsum[k](t_S, i) != m)
break;
}
cout << "index: " << setw(3) << (i+1) << ' ';
cout << "sum: " << setw(3) << t_S << '\n';
t_I = i;
t_S = t_S - i - 1;
}
cout << "max sum: " << maxsum[K](S, I) << '\n';
}
Take the arrays, and sort them by value instead of by index (keeping the index-value pairs preserved). Now, starting at the end of the array, take the last k numbers in the indices array, where k is the number of indices you have to have, and sum them up. If it equals the desired sum, great- you are done. If not, take note of the difference (desired sum - actual sum), and add that to the (n - k)th index. Find that index in the index array (ordered by value, mind you), now find your new sum of values (you can optimize this by subtracting out the old index's value and add the new one, instead of recomputing the sum of all k values).
You now have one valid solution, and a lower bound. You know the indices of the rest of the valid solution that can even possibly beat this score must come after the smallest index's value in the value-sorted array. That is:
Both sorted by value-
indices: | bunch of indices | index we found | more | k-1 'random' indices |
values: | bunch of values | value for ^ | more | k-1 largest values |
So we only have to search 'more' and the k-1 largest values for valid indices that satisfy the criteria and also have values that form a larger sum. To do this, we rinse and repeat, moving the smallest of the (n-k-1) elements backwards one, so we effectively try all combination of these elements, but in the order of decreasing subset-sums of our set of k elements. This allows us to continually narrow the space we search as we find larger sums, because know for certain that any sum that contains a smaller value than that of best solution will have a smaller sum (because the rest of the set is already as large as possible).
Pseudo Code:
pair_array = input() // each pair consists of index and value
sort_by_value(pair_array)
best_sum = 0
markers = [n - (k-1) .. n] // mark the k-1 indices being summed
while True:
sum_of_indices = sum_indices(pair_array[markers])
value_sum = sum_value(pair_array[markers])
if pair_array.contains(desired_sum - sum_of_indices): // this lets us effectively reduce our search by a factor of N, given contains uses a hashtable
value_sum += pair_array(pair_array.index(desired_sum - sum_of_indices)).value
if value_sum > best_sum:
best_sum = value_sum
pair_array.remove(0 .. n - (k-1)) // this greatly reduces the combinations checked
if has_next_combination(markers, pair_array):
next_greatest_combination(markers, pair_array) // pick new markers, in a reverse-binary counting fashion (most significant bit first way)
else:
print(best_sum)
break
One small trick I can think if you try to find lets say 3 indexes, is instead of iterating for the 3 indexes you can calculate the 3rd index when you know the first two indexes. For example when you know that
p1 = 1, p2 = 7 => p3 = 20 - (p1 + p2) = 12
This can be generalized when having N indexes the last one can always be inferred from the N-1 previous indexes.
I tried this in Python:
Index = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ]
Value = [17, 12, 5, 22, 3, 12, 6, 13, 7, 0, 2, 15 ]
maxSum = 0 SumInd = 20
for p1 in range(1,len(Index)+1):
for p2 in range(p1,len(Index)+1):
if (p1 + p2 < SumInd) and (p1 != p2):
p3 = SumInd - (p1 + p2) #If you know p1 and p2 calculate p3 index
if (p3 < len(Index)+1) and (p2 != p3) and (p1 != p3):
fooSum = Value[p1-1]+Value[p2-1] + Value[p3-1]
print(p1,p2,p3 , "Sum is ",fooSum)
if maxSum < fooSum:
maxSum = fooSum
print("Max Sum is ", maxSum)
Of course you need to save the indexes when you find maxSum. Also this implementation calculate many similar pairs (e.g. [1,9,10] and [9,10,1]). Perhaps a better solution could eliminate this pairs.
EDIT: Big Improvement I found a way to eliminate most unnecessary checks. Let's say you need 3 indexes. The first one checks the whole range of possible values. Let's say it is index1.
The other two indexes must sum at 20 - ind1 let's call this rest. The list of indexes is always ordered so you can have a back index showing the smallest value(first item in the list bigger than index1) and a front index that show the biggest values (last item in the list). So index2 = backIndex , index3 = frontIndex.
If the rest is smaller than the sum of index2,3 you could increase the back index (get the next bigger value) or if it is larger you decrease the front index until the two indexes meet each other where you break and increase index1. This eliminates checking (1,7,12) and (1,12,7) two times.
The code is here in Python:
maxSum = 0
SumInd = 20
for index_1 in range(1,len(Index)):
rest = SumInd - index_1
backIndex = index_1+1
frontIndex = len(Index)
while backIndex < frontIndex:
if rest > (backIndex + frontIndex):
backIndex = backIndex + 1
elif rest < (backIndex + frontIndex):
frontIndex = frontIndex - 1
else:
fooSum = Value[index_1-1]+Value[backIndex-1] + Value[frontIndex-1]
print("Checking for ",index_1,backIndex,frontIndex,' Sum of values:',fooSum)
if maxSum < fooSum:
indList = [index_1-1,backIndex,frontIndex]
maxSum = fooSum
backIndex = backIndex + 1 #To avoid Inf loop
print("Max Sum is ", maxSum,"at",indList)
and gives these results:
Checking for 1 7 12 Sum of values: 38
Checking for 1 8 11 Sum of values: 32
Checking for 1 9 10 Sum of values: 24
Checking for 2 6 12 Sum of values: 39
Checking for 2 7 11 Sum of values: 20
Checking for 2 8 10 Sum of values: 25
Checking for 3 5 12 Sum of values: 23
Checking for 3 6 11 Sum of values: 19
Checking for 3 7 10 Sum of values: 11
Checking for 3 8 9 Sum of values: 25
Checking for 4 5 11 Sum of values: 27
Checking for 4 6 10 Sum of values: 34
Checking for 4 7 9 Sum of values: 35
Checking for 5 6 9 Sum of values: 22
Checking for 5 7 8 Sum of values: 22
Max Sum is 39 at [1, 6, 12]
This can always be generalized for N indexes. The first N-2 indexes can search the whole range of the list (like index 1 in the case above, also it should be noted that all these indexes start checking from previous index value plus one until the end of the list to eliminate many duplicate checks).
The last two indexes can be calculated like I showed in my code and avoid many duplicate checks.
I have the following Julia code and I would like to parallelize it.
using DistributedArrays
function f(x)
return x^2;
end
y = DArray[]
#parallel for i in 1:100
y[i] = f(i)
end
println(y)
The output is DistributedArrays.DArray[]. I would like to have the value of y as follows: y=[1,4,9,16,...,10000]
You can use n-dimensional distributed array comprehensions:
First you need to add some more processes, either local or remote:
julia> addprocs(CPU_CORES - 1);
Then you must use DistributedArrays at every one of the spawned processes:
julia> #everywhere using DistributedArrays
Finally you can use the #DArray macro, like this:
julia> x = #DArray [#show x^2 for x = 1:10];
From worker 2: x ^ 2 = 1
From worker 2: x ^ 2 = 4
From worker 4: x ^ 2 = 64
From worker 2: x ^ 2 = 9
From worker 4: x ^ 2 = 81
From worker 4: x ^ 2 = 100
From worker 3: x ^ 2 = 16
From worker 3: x ^ 2 = 25
From worker 3: x ^ 2 = 36
From worker 3: x ^ 2 = 49
You can see it does what you expect:
julia> x
10-element DistributedArrays.DArray{Int64,1,Array{Int64,1}}:
1
4
9
16
25
36
49
64
81
100
Remember it works with an arbitrary number of dimensions:
julia> y = #DArray [#show i + j for i = 1:3, j = 4:6];
From worker 4: i + j = 7
From worker 4: i + j = 8
From worker 4: i + j = 9
From worker 2: i + j = 5
From worker 2: i + j = 6
From worker 2: i + j = 7
From worker 3: i + j = 6
From worker 3: i + j = 7
From worker 3: i + j = 8
julia> y
3x3 DistributedArrays.DArray{Int64,2,Array{Int64,2}}:
5 6 7
6 7 8
7 8 9
julia>
This is the most julian way to do what you intended IMHO.
We can look at macroexpand output in order to see what's going on:
Note: this output has been slightly edited for readability, T stands for:
DistributedArrays.Tuple{DistributedArrays.Vararg{DistributedArrays.UnitRange{DistributedArrays.Int}}}
julia> macroexpand(:(#DArray [i^2 for i = 1:10]))
:(
DistributedArrays.DArray(
(
#231#I::T -> begin
[i ^ 2 for i = (1:10)[#231#I[1]]]
end
),
DistributedArrays.tuple(DistributedArrays.length(1:10))
)
)
Which basically is the same as manually typing:
julia> n = 10; dims = (n,);
julia> DArray(x -> [i^2 for i = (1:n)[x[1]]], dims)
10-element DistributedArrays.DArray{Any,1,Array{Any,1}}:
1
4
9
16
25
36
49
64
81
100
julia>
Hi Kira,
I am new on Julia, but facing the same problem. Try this approach and see if it fits your needs.
function f(x)
return x^2;
end
y=#parallel vcat for i= 1:100
f(i);
end;
println(y)
Regards, RN
I'm interested in finding the nth row of pascal triangle (not a specific element but the whole row itself). What would be the most efficient way to do it?
I thought about the conventional way to construct the triangle by summing up the corresponding elements in the row above which would take:
1 + 2 + .. + n = O(n^2)
Another way could be using the combination formula of a specific element:
c(n, k) = n! / (k!(n-k)!)
for each element in the row which I guess would take more time the the former method depending on the way to calculate the combination. Any ideas?
>>> def pascal(n):
... line = [1]
... for k in range(n):
... line.append(line[k] * (n-k) / (k+1))
... return line
...
>>> pascal(9)
[1, 9, 36, 84, 126, 126, 84, 36, 9, 1]
This uses the following identity:
C(n,k+1) = C(n,k) * (n-k) / (k+1)
So you can start with C(n,0) = 1 and then calculate the rest of the line using this identity, each time multiplying the previous element by (n-k) / (k+1).
A single row can be calculated as follows:
First compute 1. -> N choose 0
Then N/1 -> N choose 1
Then N*(N-1)/1*2 -> N choose 2
Then N*(N-1)*(N-2)/1*2*3 -> N choose 3
.....
Notice that you can compute the next value from the previous value, by just multipyling by a single number and then dividing by another number.
This can be done in a single loop. Sample python.
def comb_row(n):
r = 0
num = n
cur = 1
yield cur
while r <= n:
r += 1
cur = (cur* num)/r
yield cur
num -= 1
The most efficient approach would be:
std::vector<int> pascal_row(int n){
std::vector<int> row(n+1);
row[0] = 1; //First element is always 1
for(int i=1; i<n/2+1; i++){ //Progress up, until reaching the middle value
row[i] = row[i-1] * (n-i+1)/i;
}
for(int i=n/2+1; i<=n; i++){ //Copy the inverse of the first part
row[i] = row[n-i];
}
return row;
}
here is a fast example implemented in go-lang that calculates from the outer edges of a row and works it's way to the middle assigning two values with a single calculation...
package main
import "fmt"
func calcRow(n int) []int {
// row always has n + 1 elements
row := make( []int, n + 1, n + 1 )
// set the edges
row[0], row[n] = 1, 1
// calculate values for the next n-1 columns
for i := 0; i < int(n / 2) ; i++ {
x := row[ i ] * (n - i) / (i + 1)
row[ i + 1 ], row[ n - 1 - i ] = x, x
}
return row
}
func main() {
for n := 0; n < 20; n++ {
fmt.Printf("n = %d, row = %v\n", n, calcRow( n ))
}
}
the output for 20 iterations takes about 1/4 millisecond to run...
n = 0, row = [1]
n = 1, row = [1 1]
n = 2, row = [1 2 1]
n = 3, row = [1 3 3 1]
n = 4, row = [1 4 6 4 1]
n = 5, row = [1 5 10 10 5 1]
n = 6, row = [1 6 15 20 15 6 1]
n = 7, row = [1 7 21 35 35 21 7 1]
n = 8, row = [1 8 28 56 70 56 28 8 1]
n = 9, row = [1 9 36 84 126 126 84 36 9 1]
n = 10, row = [1 10 45 120 210 252 210 120 45 10 1]
n = 11, row = [1 11 55 165 330 462 462 330 165 55 11 1]
n = 12, row = [1 12 66 220 495 792 924 792 495 220 66 12 1]
n = 13, row = [1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1]
n = 14, row = [1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1]
n = 15, row = [1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1]
n = 16, row = [1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1]
n = 17, row = [1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1]
n = 18, row = [1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1]
n = 19, row = [1 19 171 969 3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876 969 171 19 1]
An easy way to calculate it is by noticing that the element of the next row can be calculated as a sum of two consecutive elements in the previous row.
[1, 5, 10, 10, 5, 1]
[1, 6, 15, 20, 15, 6, 1]
For example 6 = 5 + 1, 15 = 5 + 10, 1 = 1 + 0 and 20 = 10 + 10. This gives a simple algorithm to calculate the next row from the previous one.
def pascal(n):
row = [1]
for x in xrange(n):
row = [l + r for l, r in zip(row + [0], [0] + row)]
# print row
return row
print pascal(10)
In Scala Programming: i would have done it as simple as this:
def pascal(c: Int, r: Int): Int = c match {
case 0 => 1
case `c` if c >= r => 1
case _ => pascal(c-1, r-1)+pascal(c, r-1)
}
I would call it inside this:
for (row <- 0 to 10) {
for (col <- 0 to row)
print(pascal(col, row) + " ")
println()
}
resulting to:
.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
To explain step by step:
Step 1: We make sure that if our column is the first one we always return figure 1.
Step 2: Since each X-th row there are X number of columns. So we say that; the last column X is greater than or equal to X-th row, then the return figure 1.
Step 3: Otherwise, we get the sum of the repeated pascal of the column just before the current one and the row just before the current one ; and the pascal of that column and the row just before the current one.
Good Luck.
Let me build upon Shane's excellent work for an R solution. (Thank you, Shane!. His code for generating the triangle:
pascalTriangle <- function(h) {
lapply(0:h, function(i) choose(i, 0:i))
}
This will allow one to store the triangle as a list. We can then index whatever row desired. But please add 1 when indexing! For example, I'll grab the bottom row:
pt_with_24_rows <- pascalTriangle(24)
row_24 <- pt_with_24_rows[25] # add one
row_24[[1]] # prints the row
So, finally, make-believe I have a Galton Board problem. I have the arbitrary challenge of finding out percentage of beans have clustered in the center: say, bins 10 to 15 (out of 25).
sum(row_24[[1]][10:15])/sum(row_24[[1]])
Which turns out to be 0.7704771. All good!
In Ruby, the following code will print out the specific row of Pascals Triangle that you want:
def row(n)
pascal = [1]
if n < 1
p pascal
return pascal
else
n.times do |num|
nextNum = ((n - num)/(num.to_f + 1)) * pascal[num]
pascal << nextNum.to_i
end
end
p pascal
end
Where calling row(0) returns [1] and row(5) returns [1, 5, 10, 10, 5, 1]
Here is the another best and simple way to design a Pascal Triangle dynamically using VBA.
`1
11
121
1331
14641`
`Sub pascal()
Dim book As Excel.Workbook
Dim sht As Worksheet
Set book = ThisWorkbook
Set sht = book.Worksheets("sheet1")
a = InputBox("Enter the Number", "Fill")
For i = 1 To a
For k = 1 To i
If i >= 2 And k >= 2 Then
sht.Cells(i, k).Value = sht.Cells(i - 1, k - 1) + sht.Cell(i- 1, k)
Else
sht.Cells(i, k).Value = 1
End If
Next k
Next i
End Sub`
I used Ti-84 Plus CE
The use of –> in line 6 is the store value button
Forloop syntax is
:For(variable, beginning, end [, increment])
:Commands
:End
nCr syntax is
:valueA nCr valueB
List indexes start at 1 so that's why i set it to R+1
N= row
R= column
PROGRAM: PASCAL
:ClrHome
:ClrList L1
:Disp "ROW
:Input N
:For(R,0,N,1)
:N nCr R–>L1(R+1)
:End
:Disp L1
This is the fastest way I can think of to do this in programming (with a ti 84) but if you mean to be able to calculate the row using pen and paper then just draw out the triangle cause doing factorals are a pain!
Here's an O(n) space-complexity solution in Python:
def generate_pascal_nth_row(n):
result=[1]*n
for i in range(n):
previous_res = result.copy()
for j in range(1,i):
result[j] = previous_res[j-1] + previous_res[j]
return result
print(generate_pascal_nth_row(6))
class Solution{
public:
int comb(int n,int r){
long long c=1;
for(int i=1;i<=r;i++) { //calculates n!/(n-r)!
c=((c*n))/i; n--;
}
return c;
}
vector<int> getRow(int n) {
vector<int> v;
for (int i = 0; i < n; ++i)
v.push_back(comb(n,i));
return v;
}
};
faster than 100% submissions on leet code https://leetcode.com/submissions/detail/406399031/
The most efficient way to calculate a row in pascal's triangle is through convolution. First we chose the second row (1,1) to be a kernel and then in order to get the next row we only need to convolve curent row with the kernel.
So convolution of the kernel with second row gives third row [1 1]*[1 1] = [1 2 1], convolution with the third row gives fourth [1 2 1]*[1 1] = [1 3 3 1] and so on
This is a function in julia-lang (very simular to matlab):
function binomRow(n::Int64)
baseVector = [1] #the first row is equal to 1.
kernel = [1,1] #This is the second row and a kernel.
row = zeros(n)
for i = 1 : n
row = baseVector
baseVector = conv(baseVector, kernel) #convoltion with kernel
end
return row::Array{Int64,1}
end
To find nth row -
int res[] = new int[n+1];
res[0] = 1;
for(int i = 1; i <= n; i++)
for(int j = i; j > 0; j++)
res[j] += res[j-1];
Locked. This question and its answers are locked because the question is off-topic but has historical significance. It is not currently accepting new answers or interactions.
Generate a list of lists (or print, I don't mind) a Pascal's Triangle of size N with the least lines of code possible!
Here goes my attempt (118 characters in python 2.6 using a trick):
c,z,k=locals,[0],'_[1]'
p=lambda n:[len(c()[k])and map(sum,zip(z+c()[k][-1],c()[k][-1]+z))or[1]for _ in range(n)]
Explanation:
the first element of the list comprehension (when the length is 0) is [1]
the next elements are obtained the following way:
take the previous list and make two lists, one padded with a 0 at the beginning and the other at the end.
e.g. for the 2nd step, we take [1] and make [0,1] and [1,0]
sum the two new lists element by element
e.g. we make a new list [(0,1),(1,0)] and map with sum.
repeat n times and that's all.
usage (with pretty printing, actually out of the code-golf xD):
result = p(10)
lines = [" ".join(map(str, x)) for x in result]
for i in lines:
print i.center(max(map(len, lines)))
output:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
K (Wikipedia), 15 characters:
p:{x{+':x,0}\1}
Example output:
p 10
(1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1)
It's also easily explained:
p:{x {+':x,0} \ 1}
^ ^------^ ^ ^
A B C D
p is a function taking an implicit parameter x.
p unfolds (C) an anonymous function (B) x times (A) starting at 1 (D).
The anonymous function simply takes a list x, appends 0 and returns a result by adding (+) each adjacent pair (':) of values: so e.g. starting with (1 2 1), it'll produce (1 2 1 0), add pairs (1 1+2 2+1 1+0), giving (1 3 3 1).
Update: Adapted to K4, which shaves off another two characters. For reference, here's the original K3 version:
p:{x{+':0,x,0}\1}
J, another language in the APL family, 9 characters:
p=:!/~#i.
This uses J's builtin "combinations" verb.
Output:
p 10
1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9
0 0 1 3 6 10 15 21 28 36
0 0 0 1 4 10 20 35 56 84
0 0 0 0 1 5 15 35 70 126
0 0 0 0 0 1 6 21 56 126
0 0 0 0 0 0 1 7 28 84
0 0 0 0 0 0 0 1 8 36
0 0 0 0 0 0 0 0 1 9
0 0 0 0 0 0 0 0 0 1
Haskell, 58 characters:
r 0=[1]
r(n+1)=zipWith(+)(0:r n)$r n++[0]
p n=map r[0..n]
Output:
*Main> p 5
[[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1],[1,5,10,10,5,1]]
More readable:
-- # row 0 is just [1]
row 0 = [1]
-- # row (n+1) is calculated from the previous row
row (n+1) = zipWith (+) ([0] ++ row n) (row n ++ [0])
-- # use that for a list of the first n+1 rows
pascal n = map row [0..n]
69C in C:
f(int*t){int*l=t+*t,*p=t,r=*t,j=0;for(*t=1;l<t+r*r;j=*p++)*l++=j+*p;}
Use it like so:
int main()
{
#define N 10
int i, j;
int t[N*N] = {N};
f(t);
for (i = 0; i < N; i++)
{
for (j = 0; j <= i; j++)
printf("%d ", t[i*N + j]);
putchar('\n');
}
return 0;
}
F#: 81 chars
let f=bigint.Factorial
let p x=[for n in 0I..x->[for k in 0I..n->f n/f k/f(n-k)]]
Explanation: I'm too lazy to be as clever as the Haskell and K programmers, so I took the straight forward route: each element in Pascal's triangle can be uniquely identified using a row n and col k, where the value of each element is n!/(k! (n-k)!.
Python: 75 characters
def G(n):R=[[1]];exec"R+=[map(sum,zip(R[-1]+[0],[0]+R[-1]))];"*~-n;return R
Shorter prolog version (112 instead of 164):
n([X],[X]).
n([H,I|T],[A|B]):-n([I|T],B),A is H+I.
p(0,[[1]]):-!.
p(N,[R,S|T]):-O is N-1,p(O,[S|T]),n([0|S],R).
another stab (python):
def pascals_triangle(n):
x=[[1]]
for i in range(n-1):
x.append(list(map(sum,zip([0]+x[-1],x[-1]+[0]))))
return x
Haskell, 164C with formatting:
i l=zipWith(+)(0:l)$l++[0]
fp=map (concatMap$(' ':).show)f$iterate i[1]
c n l=if(length l<n)then c n$' ':l++" "else l
cl l=map(c(length$last l))l
pt n=cl$take n fp
Without formatting, 52C:
i l=zipWith(+)(0:l)$l++[0]
pt n=take n$iterate i[1]
A more readable form of it:
iterateStep row = zipWith (+) (0:row) (row++[0])
pascalsTriangle n = take n $ iterate iterateStep [1]
-- For the formatted version, we reduce the number of rows at the final step:
formatRow r = concatMap (\l -> ' ':(show l)) r
formattedLines = map formatRow $ iterate iterateStep [1]
centerTo width line =
if length line < width
then centerTo width (" " ++ line ++ " ")
else line
centerLines lines = map (centerTo (length $ last lines)) lines
pascalsTriangle n = centerLines $ take n formattedLines
And perl, 111C, no centering:
$n=<>;$p=' 1 ';for(1..$n){print"$p\n";$x=" ";while($p=~s/^(?= ?\d)(\d* ?)(\d* ?)/$2/){$x.=($1+$2)." ";}$p=$x;}
Scheme — compressed version of 100 characters
(define(P h)(define(l i r)(if(> i h)'()(cons r(l(1+ i)(map +(cons 0 r)(append r '(0))))))(l 1 '(1)))
This is it in a more readable form (269 characters):
(define (pascal height)
(define (next-row row)
(map +
(cons 0 row)
(append row '(0))))
(define (iter i row)
(if (> i height)
'()
(cons row
(iter (1+ i)
(next-row row)))))
(iter 1 '(1)))
VBA/VB6 (392 chars w/ formatting)
Public Function PascalsTriangle(ByVal pRows As Integer)
Dim iRow As Integer
Dim iCol As Integer
Dim lValue As Long
Dim sLine As String
For iRow = 1 To pRows
sLine = ""
For iCol = 1 To iRow
If iCol = 1 Then
lValue = 1
Else
lValue = lValue * (iRow - iCol + 1) / (iCol - 1)
End If
sLine = sLine & " " & lValue
Next
Debug.Print sLine
Next
End Function
PHP 100 characters
$v[]=1;while($a<34){echo join(" ",$v)."\n";$a++;for($k=0;$k<=$a;$k++)$t[$k]=$v[$k-1]+$v[$k];$v=$t;}
Ruby, 83c:
def p(n);n>0?(m=p(n-1);k=m.last;m+[([0]+k).zip(k+[0]).map{|x|x[0]+x[1]}]):[[1]];end
test:
irb(main):001:0> def p(n);n>0?(m=p(n-1);k=m.last;m+[([0]+k).zip(k+[0]).map{|x|x[0]+x[1]}]):[[1]];end
=> nil
irb(main):002:0> p(5)
=> [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1], [1, 5, 10, 10, 5, 1]]
irb(main):003:0>
Another python solution, that could be much shorter if the builtin functions had shorter names... 106 characters.
from itertools import*
r=range
p=lambda n:[[len(list(combinations(r(i),j)))for j in r(i+1)]for i in r(n)]
Another try, in prolog (I'm practising xD), not too short, just 164c:
s([],[],[]).
s([H|T],[J|U],[K|V]):-s(T,U,V),K is H+J.
l([1],0).
l(P,N):-M is N-1,l(A,M),append(A,[0],B),s(B,[0|A],P).
p([],-1).
p([H|T],N):-M is N-1,l(H,N),p(T,M).
explanation:
s = sum lists element by element
l = the Nth row of the triangle
p = the whole triangle of size N
VBA, 122 chars:
Sub p(n)
For r = 1 To n
l = "1"
v = 1
For c = 1 To r - 1
v = v / c * (r - c)
l = l & " " & v
Next
Debug.Print l
Next
End Sub
I wrote this C++ version a few years ago:
#include <iostream>
int main(int,char**a){for(int b=0,c=0,d=0,e=0,f=0,g=0,h=0,i=0;b<atoi(a[1]);(d|f|h)>1?e*=d>1?--d:1,g*=f>1?--f:1,i*=h>1?--h:1:((std::cout<<(i*g?e/(i*g):1)<<" "?d=b+=c++==b?c=0,std::cout<<std::endl?1:0:0,h=d-(f=c):0),e=d,g=f,i=h));}
The following is just a Scala function returning a List[List[Int]]. No pretty printing or anything. Any suggested improvements? (I know it's inefficient, but that's not the main challenge now, is it?). 145 C.
def p(n: Int)={def h(n:Int):List[Int]=n match{case 1=>1::Nil;case _=>(0::h(n-1) zipAll(h(n-1),0,0)).map{n=>n._1+n._2}};(1 to n).toList.map(h(_))}
Or perhaps:
def pascal(n: Int) = {
def helper(n: Int): List[Int] = n match {
case 1 => 1 :: List()
case _ => (0 :: helper(n-1) zipAll (helper(n-1),0,0)).map{ n => n._1 + n._2 }
}
(1 to n).toList.map(helper(_))
}
(I'm a Scala noob, so please be nice to me :D )
a Perl version (139 chars w/o shebang)
#p = (1,1);
while ($#p < 20) {
#q =();
$z = 0;
push #p, 0;
foreach (#p) {
push #q, $_+$z;
$z = $_
}
#p = #q;
print "#p\n";
}
output starts from 1 2 1
PHP, 115 chars
$t[][]=1;
for($i=1;$i<$n;++$i){
$t[$i][0]=1;
for($j=1;$j<$i;++$j)$t[$i][$j]=$t[$i-1][$j-1]+$t[$i-1][$j];
$t[$i][$i]=1;}
If you don't care whether print_r() displays the output array in the correct order, you can shave it to 113 chars like
$t[][]=1;
for($i=1;$i<$n;++$i){
$t[$i][0]=$t[$i][$i]=1;
for($j=1;$j<$i;++$j)$t[$i][$j]=$t[$i-1][$j-1]+$t[$i-1][$j];}
Perl, 63 characters:
for(0..9){push#z,1;say"#z";#z=(1,map{$z[$_-1]+$z[$_]}(1..$#z))}
My attempt in C++ (378c). Not anywhere near as good as the rest of the posts.. but I'm proud of myself for coming up with a solution on my own =)
int* pt(int n)
{
int s=n*(n+1)/2;
int* t=new int[s];
for(int i=0;i<n;++i)
for(int j=0;j<=i;++j)
t[i*n+j] = (!j || j==i) ? 1 : t[(i-1)*n+(j-1)] + t[(i-1)*n+j];
return t;
}
int main()
{
int n,*t;
std::cin>>n;
t=pt(n);
for(int i=0;i<n;++i)
{
for(int j=0;j<=i;j++)
std::cout<<t[i*n+j]<<' ';
std::cout<<"\n";
}
}
Old thread, but I wrote this in response to a challenge on another forum today:
def pascals_triangle(n):
x=[[1]]
for i in range(n-1):
x.append([sum(i) for i in zip([0]+x[-1],x[-1]+[0])])
return x
for x in pascals_triangle(5):
print('{0:^16}'.format(x))
[1]
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 4, 6, 4, 1]