How to calculate this long recursion - c++

Here is a code doing recursion which crashes for larger values:
int rec(int m,int n)
{
if(m==0)
return n+1;
if(m>0 && n==0)
return rec(m-1,1);
if(m>0 && n>0)
return rec(m-1,rec(m,n-1));
}
If I call the function rec(m,n):
with m=1 and n=2, the result I get is 4
with m=2 and n=2, it is 7,
with m=3 and n=2, it is 29
But it crashes for m=4 and m=2. Is there an alternate way to calculate it?

Ackermann's function can be stated non-recursively as
Ack( m, n ) = (2 opm (n + 3)) - 3
where opm is the m'th arithmetic operation in the order addition, multiplication, exponentiation, tower function, ...
In other words it explodes in value pretty darn instantaneously, with no way to represent those numbers as C++ integers.
For an explanation see my homepage from the 1990's :),
(http://web.archive.org/web/20120315031240/http://members.fortunecity.com/alf_steinbach/content/programming/narrow_topics/ackermann/ackermann.html)

rec(4,2)
-> rec(3, rec(4, 1))
->rec(3, rec(4, 0)
->rec(3, 1)
->rec(2, rec(3, 0))
->rec(2, 1)
->rec(1, rec(2, 0)) -->rec(1, 2) //return 4
->rec(1, 0) //return 2 -->rec(0, rec(1, 1)) //return 4
->rec(0, 1) //return 2 -->rec(0, rec(1, 0)) //return 3
-->return 2
this boils down to:
rec(4,2)
-> rec(3, rec(4, 1))
->rec(3, rec(4, 0)
->rec(3, 1)
->rec(2, rec(3, 0))
->rec(2, 1)
->rec(1, 4)
This can be solved further but the space is insufficient and will take a lot of time.
I am predicting that your application either crashes due to stackoverflow or due to reaching the limit of allowable recursions.
But I cannot say for sure...
#Cheers and hth. - Alf explains it mathematically in much more subtle manner ;)

Related

Music Chairs problem implementation in C++

I am currently practicing algorithms and DS. I have stumbled upon a question that I can't figure out how to solve. So the question's link is there:
In summary, it says that there is a number of chairs in a circle, and the position of the person (relative to a certain chair), and how many M movements he should make.
So the input is as following:
3 integer numbers N, M, X , The number of chairs, the number of times the boy should move and the first chair he will start from respectively ( 1  ≤  X  ≤  N < 2^63 , 0  ≤  M < 2^63 )
So, what have I done so far? I thought about the following:
So I thought that the relative position after M movements is (x+m) % n, and since this can cause Integer overflow, I have done it like that, ((x%n) + (m%n)) % n. I have figured out that if the person has reached the last index of chair, it will be 0 so I handled that. However, it passes only 2 tests. I don't need any code to be written, I want to directed in the right way of thinking. Here is my code so far:
#include <iostream>
using namespace std;
int main() {
long long n, m, x;
cin >> n >> m >> x;
// After each move, he reaches (X+1).
// X, N chairs.
// ((X % N) + (M % N)) % N;
// Odd conideration.
if ( m % 2 == 1) {
m += 1;
}
long long position = (x % n + m % n) % n;
if (position == 0) {
position = n;
}
cout << position;
return 0;
}
If the question required specific error handling, it should have stated so (so don't feel bad).
In every real-world project, there should be a standard to dictate what to do with weird input. Do you throw? Do you output a warning? If so, does it have to be translated to the system language?
In the absence of such instructions I would err toward excluding these values after reading them. Print an error to std::cerr (or throw an exception). Do this as close to where you read them as possible.
For overflow detection, you can use the methods described here. Some may disagree, and for a lab-exercise, it's probably not important. However, there is a saying in computing "Garbage in == Garbage out". It's a good habit to check for garbage before processing, rather than attempting to "recycle" garbage as you process.
Here's the problem:
Say the value of N is 2^63-1, and X and M are both 2^63 - 2.
When your program runs untill the ((X % N) + (M % N)) % N part,
X % N evaluates into 2^63 - 2 (not changed), and so does M % N.
Then, the addition between the two results occurs, 2^63 - 2 + 2^63 - 2 there is the overflow happening.
After the comment of #WBuck, the answer is actually rather easy which is to change the long long to unsigned because there are no negative numbers and therefore, increase the MAX VALUE of long long (when using unsigned).
Thank you so much.

wrote function to calculate set bits from 0 to n,

in this function i check if the no. is one less than the power of 2 and then make recursive calls for 2^b - 1 and n - 2^b(this call keeps happening till the no. here is one less than a power of 2)
now I know the code is wrong but why does it give segmentation fault.
int countSetBits(int n)
{
if (n == 0)
return 0;
int b = floor(log2(n));
if ( (n + 1) & n == 0 ) {
return (1<<b)* floor(log2(n + 1));
}
return (n - 1<<b + 1) + countSetBits(n - 1<<b) + countSetBits(1<<b - 1);
}
The infinite number of (recursive) function calls causes stack overflow. The program's stack has become too large and tries to "overflow" into the next memory segment. This is not allowed, and hence the segfault.
The reasons for the errors are two-fold:
You are using floating point numbers, in floor and log2. Those are imprecise and won't give you the exactness you need for this task.
You are left shifting your numbers (making them bigger). I didn't follow the logic you wanted, but usually, the approach is to take the least bits out and then right-shift the numbers (>>)
Lastly, either use a debugger, or introduce debug couts to see what your program is doing, that will make it overall much easier.

when this code executes, it shows no output. Why is that? [closed]

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Edit the question to include desired behavior, a specific problem or error, and the shortest code necessary to reproduce the problem. This will help others answer the question.
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Following code has no output when it executes. Can anyone explain the following code?
int main() {
int i, j;
for (i = 0; i < 10; i++) {
if (i % 2)
printf("%d\t", i);
else
break;
}
}
0 % 2 gives false so the loop terminates at the first iteration without calling the printf.
Your code has two issues:
1.
if (i % 2)
The condition of (i % 2) evaluates to false because the calculation - 0 divided by 2 - result in 0 -> 0 / 2 = 0. The remainder is also 0.
2.
else break;
That provides the result if (i % 2) is not true (which is the case with 0 at the first iteration) you immediately will break out of the for loop. Omit the break; statement in general, if you want to proof if all values from 0 to 9 have an remainder when divided by 2 or not.
Note that to pack break; in an separate else statement is nonetheless redundant.
Side note:
j has no use in your code.
I guess what you want is something like that:
#include <stdio.h>
int main (void) {
int i, j;
for ( i = 0; i < 10; i++ ) {
if ( ( j = i % 2 ) )
printf("(%d / 2) has a remainder of %d.\n", i, j);
else
printf("(%d / 2) has no remainder.\n", i);
}
}
Output:
(0 / 2) has no remainder.
(1 / 2) has a remainder of 1.
(2 / 2) has no remainder.
(3 / 2) has a remainder of 1.
(4 / 2) has no remainder.
(5 / 2) has a remainder of 1.
(6 / 2) has no remainder.
(7 / 2) has a remainder of 1.
(8 / 2) has no remainder.
(9 / 2) has a remainder of 1.
Try this code online.
Note that by the division 1 / 2, the result 0.5 is raised to the nearest upper integral value 1 because of the implicit double to int conversion.
i % 2 gives the remainder of the Euclidian division of i by 2. It does NOT mean "i is divisible by 2". If the latter is true, then i % 2 gives 0 which implicitely converts to false. If it is false, then i % 2 gives 1 which implicitely converts to true. Therefore, in the first iteration, the conditional ex^pression evaluates to false so you break out of the loop immediately.
If you want to check the divisibility of i by 2, you must use i % 2 == 0.

Determining if a number is either a multiple of ten or within a particular set of ranges

I have a few loops that I need in my program. I can write out the pseudo code, but I'm not entirely sure how to write them logically.
I need -
if (num is a multiple of 10) { do this }
if (num is within 11-20, 31-40, 51-60, 71-80, 91-100) { do this }
else { do this } //this part is for 1-10, 21-30, 41-50, 61-70, 81-90
This is for a snakes and ladders board game, if it makes any more sense for my question.
I imagine the first if statement I'll need to use modulus. Would if (num == 100%10) be correct?
The second one I have no idea. I can write it out like if (num > 10 && num is < 21 || etc.), but there has to be something smarter than that.
For the first one, to check if a number is a multiple of use:
if (num % 10 == 0) // It's divisible by 10
For the second one:
if(((num - 1) / 10) % 2 == 1 && num <= 100)
But that's rather dense, and you might be better off just listing the options explicitly.
Now that you've given a better idea of what you are doing, I'd write the second one as:
int getRow(int num) {
return (num - 1) / 10;
}
if (getRow(num) % 2 == 0) {
}
It's the same logic, but by using the function we get a clearer idea of what it means.
if (num is a multiple of 10) { do this }
if (num % 10 == 0) {
// Do something
}
if (num is within 11-20, 31-40, 51-60, 71-80, 91-100) { do this }
The trick here is to look for some sort of commonality among the ranges. Of course, you can always use the "brute force" method:
if ((num > 10 && num <= 20) ||
(num > 30 && num <= 40) ||
(num > 50 && num <= 60) ||
(num > 70 && num <= 80) ||
(num > 90 && num <= 100)) {
// Do something
}
But you might notice that, if you subtract 1 from num, you'll have the ranges:
10-19, 30-39, 50-59, 70-79, 90-99
In other words, all two-digit numbers whose first digit is odd. Next, you need to come up with a formula that expresses this. You can get the first digit by dividing by 10, and you can test that it's odd by checking for a remainder of 1 when you divide by 2. Putting that all together:
if ((num > 0) && (num <= 100) && (((num - 1) / 10) % 2 == 1)) {
// Do something
}
Given the trade-off between longer but maintainable code and shorter "clever" code, I'd pick longer and clearer every time. At the very least, if you try to be clever, please, please include a comment that explains exactly what you're trying to accomplish.
It helps to assume the next developer to work on the code is armed and knows where you live. :-)
If you are using GCC or any compiler that supports case ranges you can do this, but your code will not be portable.
switch(num)
{
case 11 ... 20:
case 31 ... 40:
case 51 ... 60:
case 71 ... 80:
case 91 ... 100:
// Do something
break;
default:
// Do something else
break;
}
This is for future visitors more so than a beginner. For a more general, algorithm-like solution, you can take a list of starting and ending values and check if a passed value is within one of them:
template<typename It, typename Elem>
bool in_any_interval(It first, It last, const Elem &val) {
return std::any_of(first, last, [&val](const auto &p) {
return p.first <= val && val <= p.second;
});
}
For simplicity, I used a polymorphic lambda (C++14) instead of an explicit pair argument. This should also probably stick to using < and == to be consistent with the standard algorithms, but it works like this as long as Elem has <= defined for it. Anyway, it can be used like this:
std::pair<int, int> intervals[]{
{11, 20}, {31, 40}, {51, 60}, {71, 80}, {91, 100}
};
const int num = 15;
std::cout << in_any_interval(std::begin(intervals), std::end(intervals), num);
There's a live example here.
The first one is easy. You just need to apply the modulo operator to your num value:
if ( ( num % 10 ) == 0)
Since C++ is evaluating every number that is not 0 as true, you could also write:
if ( ! ( num % 10 ) ) // Does not have a residue when divided by 10
For the second one, I think this is cleaner to understand:
The pattern repeats every 20, so you can calculate modulo 20.
All elements you want will be in a row except the ones that are dividable by 20.
To get those too, just use num-1 or better num+19 to avoid dealing with negative numbers.
if ( ( ( num + 19 ) % 20 ) > 9 )
This is assuming the pattern repeats forever, so for 111-120 it would apply again, and so on. Otherwise you need to limit the numbers to 100:
if ( ( ( ( num + 19 ) % 20 ) > 9 ) && ( num <= 100 ) )
With a couple of good comments in the code, it can be written quite concisely and readably.
// Check if it's a multiple of 10
if (num % 10 == 0) { ... }
// Check for whether tens digit is zero or even (1-10, 21-30, ...)
if ((num / 10) % 2 == 0) { ... }
else { ... }
You basically explained the answer yourself, but here's the code just in case.
if((x % 10) == 0) {
// Do this
}
if((x > 10 && x < 21) || (x > 30 && x < 41) || (x > 50 && x < 61) || (x > 70 && x < 81) || (x > 90 && x < 101)) {
// Do this
}
You might be overthinking this.
if (x % 10)
{
.. code for 1..9 ..
} else
{
.. code for 0, 10, 20 etc.
}
The first line if (x % 10) works because (a) a value that is a multiple of 10 calculates as '0', other numbers result in their remainer, (b) a value of 0 in an if is considered false, any other value is true.
Edit:
To toggle back-and-forth in twenties, use the same trick. This time, the pivotal number is 10:
if (((x-1)/10) & 1)
{
.. code for 10, 30, ..
} else
{
.. code for 20, 40, etc.
}
x/10 returns any number from 0 to 9 as 0, 10 to 19 as 1 and so on. Testing on even or odd -- the & 1 -- tells you if it's even or odd. Since your ranges are actually "11 to 20", subtract 1 before testing.
A plea for readability
While you already have some good answers, I would like to recommend a programming technique that will make your code more readable for some future reader - that can be you in six months, a colleague asked to perform a code review, your successor, ...
This is to wrap any "clever" statements into a function that shows exactly (with its name) what it is doing. While there is a miniscule impact on performance (from "function calling overhead") this is truly negligible in a game situation like this.
Along the way you can sanitize your inputs - for example, test for "illegal" values. Thus you might end up with code like this - see how much more readable it is? The "helper functions" can be hidden away somewhere (the don't need to be in the main module: it is clear from their name what they do):
#include <stdio.h>
enum {NO, YES, WINNER};
enum {OUT_OF_RANGE=-1, ODD, EVEN};
int notInRange(int square) {
return(square < 1 || square > 100)?YES:NO;
}
int isEndOfRow(int square) {
if (notInRange(square)) return OUT_OF_RANGE;
if (square == 100) return WINNER; // I am making this up...
return (square % 10 == 0)? YES:NO;
}
int rowType(unsigned int square) {
// return 1 if square is in odd row (going to the right)
// and 0 if square is in even row (going to the left)
if (notInRange(square)) return OUT_OF_RANGE; // trap this error
int rowNum = (square - 1) / 10;
return (rowNum % 2 == 0) ? ODD:EVEN; // return 0 (ODD) for 1-10, 21-30 etc.
// and 1 (EVEN) for 11-20, 31-40, ...
}
int main(void) {
int a = 12;
int rt;
rt = rowType(a); // this replaces your obscure if statement
// and here is how you handle the possible return values:
switch(rt) {
case ODD:
printf("It is an odd row\n");
break;
case EVEN:
printf("It is an even row\n");
break;
case OUT_OF_RANGE:
printf("It is out of range\n");
break;
default:
printf("Unexpected return value from rowType!\n");
}
if(isEndOfRow(10)==YES) printf("10 is at the end of a row\n");
if(isEndOfRow(100)==WINNER) printf("We have a winner!\n");
}
For the first one:
if (x % 10 == 0)
will apply to:
10, 20, 30, .. 100 .. 1000 ...
For the second one:
if (((x-1) / 10) % 2 == 1)
will apply for:
11-20, 31-40, 51-60, ..
We basically first do x-1 to get:
10-19, 30-39, 50-59, ..
Then we divide them by 10 to get:
1, 3, 5, ..
So we check if this result is odd.
As others have pointed out, making the conditions more concise won't speed up the compilation or the execution, and it doesn't necessarily help with readability either.
It can help in making your program more flexible, in case you decide later that you want a toddler's version of the game on a 6 x 6 board, or an advanced version (that you can play all night long) on a 40 x 50 board.
So I would code it as follows:
// What is the size of the game board?
#define ROWS 10
#define COLUMNS 10
// The numbers of the squares go from 1 (bottom-left) to (ROWS * COLUMNS)
// (top-left if ROWS is even, or top-right if ROWS is odd)
#define firstSquare 1
#define lastSquare (ROWS * COLUMNS)
// We haven't started until we roll the die and move onto the first square,
// so there is an imaginary 'square zero'
#define notStarted(num) (num == 0)
// and we only win when we land exactly on the last square
#define finished(num) (num == lastSquare)
#define overShot(num) (num > lastSquare)
// We will number our rows from 1 to ROWS, and our columns from 1 to COLUMNS
// (apologies to C fanatics who believe the world should be zero-based, which would
// have simplified these expressions)
#define getRow(num) (((num - 1) / COLUMNS) + 1)
#define getCol(num) (((num - 1) % COLUMNS) + 1)
// What direction are we moving in?
// On rows 1, 3, 5, etc. we go from left to right
#define isLeftToRightRow(num) ((getRow(num) % 2) == 1)
// On rows 2, 4, 6, etc. we go from right to left
#define isRightToLeftRow(num) ((getRow(num) % 2) == 0)
// Are we on the last square in the row?
#define isLastInRow(num) (getCol(num) == COLUMNS)
// And finally we can get onto the code
if (notStarted(mySquare))
{
// Some code for when we haven't got our piece on the board yet
}
else
{
if (isLastInRow(mySquare))
{
// Some code for when we're on the last square in a row
}
if (isRightToLeftRow(mySquare))
{
// Some code for when we're travelling from right to left
}
else
{
// Some code for when we're travelling from left to right
}
}
Yes, it's verbose, but it makes it clear exactly what's happening on the game board.
If I was developing this game to display on a phone or tablet, I'd make ROWS and COLUMNS variables instead of constants, so they can be set dynamically (at the start of a game) to match the screen size and orientation.
I'd also allow the screen orientation to be changed at any time, mid-game - all you need to do is switch the values of ROWS and COLUMNS, while leaving everything else (the current square number that each player is on, and the start/end squares of all the snakes and ladders) unchanged.
Then you 'just' have to draw the board nicely, and write code for your animations (I assume that was the purpose of your if statements) ...
You can try the following:
// Multiple of 10
if ((num % 10) == 0)
{
// Do something
}
else if (((num / 10) % 2) != 0)
{
// 11-20, 31-40, 51-60, 71-80, 91-100
}
else
{
// Other case
}
I know that this question has so many answers, but I will thrown mine here anyway...
Taken from Steve McConnell's Code Complete, 2nd Edition:
"Stair-Step Access Tables:
Yet another kind of table access is the stair-step method. This access method isn’t as direct as an index structure, but it doesn’t waste as much data space. The general idea of stair-step structures, illustrated in Figure 18-5, is that entries in a table are valid for ranges of data rather than for distinct data points.
Figure 18-5 The stair-step approach categorizes each entry by determining the level at which it hits a “staircase.” The “step” it hits determines its category.
For example, if you’re writing a grading program, the “B” entry range might be from 75 percent to 90 percent. Here’s a range of grades you might have to program someday:
To use the stair-step method, you put the upper end of each range into a table and then write a loop to check a score against the upper end of each range. When you find the point at which the score first exceeds the top of a range, you know what the grade is. With the stair-step technique, you have to be careful to handle the endpoints of the ranges properly. Here’s the code in Visual Basic that assigns grades to a group of students based on this example:
Although this is a simple example, you can easily generalize it to handle multiple students, multiple grading schemes (for example, different grades for different point levels on different assignments), and changes in the grading scheme."
Code Complete, 2nd Edition, pages 426 - 428 (Chapter 18).

Modulus operator in c

I need to check the divisibility of a number in c. How can I use the modulus operatpr in C to check if a number is divisible by another number? I tried doing this:
if (file_int % 3) {
printf("Hoppity \n");
}
It didn't work, although file_int is equal to 9.
What did I do wrong?
Thanks
It didn't work because the operation will return 0 which will be treated as false.
You actually need:
if(!(file_int % 3)) {
printf("Hoppity \n");
}
if (file_int % 3) is the same as if (file_int % 3 != 0), which is the opposite of what you want.
if (file_int % 3 == 0) {
printf("Hoppity \n");
}
// or
if (!(file_int % 3)) {
printf("Hoppity \n");
}
If the result of the modulus is 0, it is evenly divisible. It would appear you are looking for it to be not divisible by 3 to continue the loop, though your code snippet is not sufficient to confidently assume your intent.
because if it is divisible by 3 file_int % 3 will be equal to 0, and the if block won't execute.
Try
if(file_int % 3 == 0) {
// do stuff
}
The mod operator returns the remainder resulting from the division... since 9 is divisible by three with no remainder, the return would be zero.
However, conditional statements evaluates to true if non-zero, false if zero. You need to change it to (file_int % 3 == 0).