Gold mine problem. Following sequence for loop is giving correct result.
//see link for other code
static int getMaxGold(int gold[][], int m, int n) {
//see link for other code
for (int col = n-1; col >= 0; col--) {
for (int row = 0; row < m; row++) {
int right = (col == n-1) ? 0 : goldTable[row][col+1];
int right_up = (row == 0 || col == n-1) ? 0 : goldTable[row-1][col+1];
int right_down = (row == m-1 || col == n-1) ? 0 : goldTable[row+1][col+1];
goldTable[row][col] = gold[row][col] + Math.max(right, Math.max(right_up, right_down));
}
}
}
//see link for other code
While other way round does not give the expected result. For example
for (int col = 0; col < n; col ++) {
for (int row = 0; row < m; row++) {
//same code to calculate right, rightUp and rightDown
}
}
Any explanation for this behaviour?
You don't need to store a whole matrix.
When you build the table, you just need to keep the last layer you processed.
In your recursion, there is diagonally right, or right, so the layer is a column because to compute the value of some cell, you need to know three values (on its right)
You conclude (as spotted by Damien already) that you start from the rightmost column, then to compute the value of every cells of the n-1 column, you only need to know the nth column (which you luckily computed already)
In below example. m_ij refers to i-th line, j-th column. (e.g m_01 == 2, m_10 = 5)
1 2 3 4
5 6 7 8
9 1 2 3
4 5 6 3
The last column is {4,8,3,3}
To compute the max value for m_02 you need to choose between 4 and 8
3 - 4
\
8
m_02 = 3 + 8 = 11
To compute the max value of m_12 you need to choose between 4, 8 and 3
4
/
7 - 8
\
3
m_12 = 7 + 8 = 15
Skipping stuff
m_22 = 2 + 8 = 10
m_32 = 6 + 3 = 9
Now you know the max value for each square of the third column
1 2 11 .
5 6 15 .
9 1 10 .
4 5 9 .
You do the same for m_10, m_11, ...
idem
m_01 = 2 + max(11, 15) = 17
m_11 = 6 + 15
m_21 = 1 + 15
m_31 = 5 + 10
Left to process is thus
1 17
5 21
9 16
4 15
Then
1+21
5+21
9+21
4+16
And finally score = max(22, 26, 30, 20)
As you have noticed you only need to keep track of the last processed column. Not a whole table of computation. And the last processed column must start from the right and always be the rightmost one...
I don't think an implem is relevant to help you understand but in case
const s = `
1 2 3 4
5 6 7 8
9 1 2 3
4 5 6 3`
const m = s.trim().split('\n').map(x => x.trim().split(' ').map(y => parseInt(y)))
let layer = [0, 0, 0, 0]
for (let j = 3; j >= 0; --j) {
const nextLayer = []
for (let i = 0; i < 4; ++i) {
nextLayer[i] = m[i][j] + Math.max(
layer[i-1] || 0, // we default undefined value as 0 supposing s only holds positive coefficient
layer[i],
layer[i+1] || 0
)
}
layer = nextLayer
}
console.log(Math.max(...layer))
I need help in providing an algorithm for a numerical sequence which should display a series of 1 2 4 and its consecutive summations.
e.g. If my input value is 20, it should display
1 2 4 8 9 11 15 16 18
Wherein
1 = 1
2 = 1 + 1
4 = 2 + 2
8 = 4 + 4
And the summation of 1 and 2 and 4 will repeat again starting with the present number which is 8 and so on..
9 = 8 + 1
11 = 9 + 2
15 = 11 + 4
16 = 15 + 1
18 = 16 + 2
As you can see, it should not proceed to 22 (18 + 4) since our sample input value is 20. I hope you guys get my point. I'm having a problem in designing the algorithms in the for loop. What I have now which is not working is
$input = 20;
for ($i = $i; $i < $input; $i = $i+$i) {
if($i==0){
$i = 4;
$i = $i - 3;
}elseif($i % 4 == 0){
$i = $i + 1;
}
print_r("this is \$i = $i<br><br>");
}
NOTE: Only one variable and one for loop is required, it will not be accepted if we use functions or arrays. Please help me, this is one of the most difficult problems I've encountered in PHP..
you can use the code
$input = 20;
$current = 1;
$val = 1;
while($val < $input){
print_r("this is \$val = $val\n");
$val = $val + $current;
$current = ($current == 4 ? 1 : $current*2);
}
see the online compiler
Since you have mentioned Only one variable and one for loop is required
Try this,
$input = 20;
for ($i = 1; $i < $input; $i) {
if($i>$input) break;
print_r("this is \$i = $i<br><br>");
$i=$i+1;
if($i>$input) break;
print_r("this is \$i = $i<br><br>");
$i=$i+2;
if($i>$input) break;
print_r("this is \$i = $i<br><br>");
$i=$i+4;
}
Online Compiler
def getSeq(n):
if n == 1:
return [1]
temp = [1]
seq = [ 1, 2, 4]
count, current, prev = 0, 0, 1
while True:
current = prev + seq[count]
if current > n:
break
prev = current
temp += [current]
count = (count + 1) % 3
return temp
print getSeq(20)
I'm pretty sure that this one is going to work
the case that we have to take care of is n == 1 and return a static result [1].
in other cases the second value is repeating circularly and adding up to previous value.
This Python solution should be implementable in any reasonable language:
limit = 20
n = 1 << 2
while n >> 2 < limit:
print(n >> 2)
n = (((n >> 2) + (2 ** (n & 3))) << 2) + ((n & 3) + 1) % 3
Perl Equivalent (using the style of for loop you expect):
$limit = 20;
for ($n = 1 << 2; $n >> 2 < $limit; $n = ((($n >> 2) + (2 ** ($n & 3))) << 2) + (($n & 3) + 1) % 3) {
print($n >> 2, "\n");
}
OUTPUT
1
2
4
8
9
11
15
16
18
EXPLANATION
The basic solution is this:
limit = 20
n = 1
i = 0
while n < limit:
print(n)
n = n + (2 ** i)
i = (i + 1) % 3
But we need to eliminate the extra variable i. Since i only cycles through 0, 1 and 2 we can store it in two bits. So we shift n up two bits and store the value for i in the lower two bits of n, adjusting the code accordingly.
Not only one variable and one for loop, no if statements either!
I have a matrix in Rcpp (C++ for R) which is stored in column order in memory. Ie, it looks like:
[,1] [,2] [,3] [,4] [,5]
[1,] 1 6 11 16 21
[2,] 2 7 12 17 22
[3,] 3 8 13 18 23
[4,] 4 9 14 19 24
[5,] 5 10 15 20 25
Now, I have a single for loop that runs from i = 1 to 25 (bear in mind, it is all zero based, but here I am just saying one for convenience).
For every element of the matrix, I want its Moore neighbourhood. This is easy for the elements that are not on the edge. So if our selected index is idx and the size of the square matrix is nrow then we have
leftmid = idx - nrow
lefttop = (idx - nrow) - 1
leftbot = (idx - nrow) + 1
rightmid = idx + nrow
righttop = (idx + nrow) - 1
rightbot = (idx + nrow) + 1
midtop = idx - 1
midbot = idx + 1
But i cant figure out how to deal with the edge cases. For example, if idx = 3, then i want the neighbours:
leftmid = 23
lefttop = 22
leftbot = 24
rightmid = 8
righttop = 7
rightbot = 9
midtop = 2
midbot = 4
It's a little bit more complicated at the corner cases as well. My goal here is to reduce time. I am currently running my program with a double for loop which works, but is slower than reasonable. I want to change it into a single for loop to improve performance.
Edit: I realized the left and right boundaries can be obtained by modulus. So 3 - 5 %% 25 = 23. But I still have the top and bottom edge cases.
It appears you're interested in "cyclic" boundary conditions, where the matrix has a toroidal topology, i.e. the top wraps around to the bottom and the right wraps around to the left.
It might be easier to iterate with four loops, one each over the row and column, and then one each over the row and column of the neighborhood. Something like this should work:
int mooreNeighbors[3][3];
int nRows = 5;
int nCols = 5;
// Loop over the rows and columns of the matrix
for (int i = 0; i < nRows; ++i) {
for (int j = 0; j < nCols; ++j) {
// Loop over the cyclic Moore neighborhood
for (int mnI = 0; mnI < 3; ++mnI) {
for (int mnJ = 0; mnJ < 3; ++mnJ) {
// Sub-matrix indices
int subI = (i - mnI - 1) % nRows;
int subJ = (j - mnJ - 1) % nCols;
// Index into column-dominant matrix
int idx = subI + subJ*nRows;
mooreNeighbors[mnI][mnJ] = matrix[idx];
}
}
}
}
I haven't tried compiling this, but it should be close to correct and clear enough to correct any mistakes. Think of it as pseudo-code.
Also, I'm preferring clarity over optimality. For example, you don't have to do everything in the inner-most loop.
Add all the values between 100 and 4000000 inclusively that are divisable by 3 or 5 but not both 3 and 5
Can't figure out how to implement second part of that stipulation. Here's what I have so far:
var sum = 0;
for (var i = 100; i < 4000001; i++) {
if (i % 3 || i % 5 === 0) {
sum = sum + i;
}
}
You can compute the sum without any loop, using the formula for the sum of an arithmetic progression: We have
3 + 5 + 6 + 9 + 10 + 12 + 18 + 20 + ...
= 3 + 6 + 9 + 12 + 15 + 18 + ...
+ 5 + 10 + 15 + 20 + ...
- 2*(15 + 30 + 45 + ...)
Note that we add all the multiples of 3 and 5 but then subtract the multiples of 15 twice, because they were counted twice as multiples of both 3 and 5.
Let g(n) be the sum of integers from 1 to n. We have g(n) = n*(n+1)/2.
Let f(n) be the sum of integers between 1 and n that are divisible by 3 or 5, but not both. Then we have
f(n) = 3*g(floor(n / 3)) + 5*g(floor(n/5)) - 30*g(floor(n/15))
And the sum of integers between m and n that are divisible by 3 or 5, but not both is then just f(n) - f(m - 1). This can be computed in O(1).
You simply need to escape only those part which involves division by 15, and other higher numbers(multiple of 15) will be avoided further automatically.
Note that checking divisibility by 15 should be at the top, which on being true will continue further iteration without executing the below codes of divisibility by 3 and 5. If false, then a number can only be divisible by 3 or 5 or none, but not both.
for (var i = 100; i < 4000001; i++) {
if(i % 15 == 0 )
continue;
if (i % 3 == 0) {
sum = sum + i;
}
if (i % 5 == 0) {
sum = sum + i;
}
}
Also, note that you have used === operator which I don't think is a valid operator, probably you want ==. BTW, I am not sure whether any language supports ===, I think Javascript supports that. So, be careful at that step.
You can use != instead of || since this is exactly what you want. Only divisible by 3 or 5 but not by both.
var sum = 0;
for (var i = 100; i < 4000001; i++) {
if ((i % 3 == 0) != (i % 5 == 0)) {
sum = sum + i;
}
}
var sum = 0;
for (var i = 100; i < 4000001; i++) {
if (i % 3 === 0 ^ i % 5 === 0) {
sum = sum + i;
}
}
use the exclusive OR , XOR ^ returns true only when one of the conditions not both is true.
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];