Can I do this in C++?
if (4<5<6)
cout<<"valid"<<endl;
i.e a double comparison? Since I know that I can
bool a;
a = 1+2<3+4<5>6;//etc
Yes, you can do it, but it won't be what you expect. It's parsed as
if ( (4<5) < 6 )
which yields
if ( 1 < 6 )
because 4<5 evaluates to true which is promoted to 1, which yields, obviously, true.
You'll need
if ( (4<5) && (5<6) )
Also, yes, you can do
a = 1+2<3+4<5>6;
but that as well is parsed as
a = ((1+2)<((3+4)<5))>6;
which will evaluate to false since (1+2)<((3+4)<5) yields a boolean, which is always smaller than 6.
It compiles but won't do what you expect -
if( 4 < 5 < 2)
same as
if( (4 < 5) < 2)
same as
if( (1 < 2) ) //1 obtained from cast to boolean
which is of course true, even though I imagine you were expecting something quite different.
It may be clumsy but this will work:
int i, j, k;
i = 4; j = 5; k = 6;
if ( (i < j) && (j < k) )
{
cout << "Valid!" << endl;
}
Related
I am learning Dafny, attempting to write a specification for the hamming weight problem, aka the number of 1 bits in a number. I believe I have gotten the specification correct, but it still doesn't verify. For speed of verification I limited it to 8 bit numbers;
problem definition: https://leetcode.com/problems/number-of-1-bits/
function method twoPow(x: bv16): bv16
requires 0 <= x <= 16
{
1 << x
}
function method oneMask(n: bv16): bv16
requires 0 <= n <= 16
ensures oneMask(n) == twoPow(n)-1
{
twoPow(n)-1
}
function countOneBits(n:bv8): bv8 {
if n == 0 then 0 else (n & 1) + countOneBits(n >> 1)
}
method hammingWeight(n: bv8) returns (count: bv8 )
ensures count == countOneBits(n)
{
count := 0;
var i := 0;
var n' := n;
assert oneMask(8) as bv8 == 255; //passes
while i < 8
invariant 0 <= i <= 8
invariant n' == n >> i
invariant count == countOneBits(n & oneMask(i) as bv8);
{
count := count + n' & 1;
n' := n' >> 1;
i := i + 1;
}
}
I have written the same code in javascript to test the behavior and example the invariant values before and after the loop. I don't seen any problems.
function twoPow(x) {
return 1 << x;
}
function oneMask(n) {
return twoPow(n)-1;
}
function countOneBits(n) {
return n === 0 ? 0 : (n & 1) + countOneBits(n >> 1)
}
function hammingWeight(n) {
if(n < 0 || n > 256) throw new Error("out of range")
console.log(`n: ${n} also ${n.toString(2)}`)
let count = 0;
let i = 0;
let nprime = n;
console.log("beforeloop",`i: ${i}`, `n' = ${nprime}`, `count: ${count}`, `oneMask: ${oneMask(i)}`, `cb: ${countOneBits(n & oneMask(i))}`)
console.log("invariants", i >= 0 && i <= 8, nprime == n >> i, count == countOneBits(n & oneMask(i)));
while (i < 8) {
console.log("");
console.log('before',`i: ${i}`, `n' = ${nprime}`, `count: ${count}`, `oneMask: ${oneMask(i)}`, `cb: ${countOneBits(n & oneMask(i))}`)
console.log("invariants", i >= 0 && i <= 8, nprime == n >> i, count == countOneBits(n & oneMask(i)));
count += nprime & 1;
nprime = nprime >> 1;
i++;
console.log('Afterloop',`i: ${i}`, `n' = ${nprime}`, `count: ${count}`, `oneMask: ${oneMask(i)}`, `cb: ${countOneBits(n & oneMask(i))}`)
console.log("invariants", i >= 0 && i <= 8, nprime == n >> i, count == countOneBits(n & oneMask(i)));
}
return count;
};
hammingWeight(128);
All invariants evaluate as true. I must be missing something. it says invariant count == countOneBits(n & oneMask(i) as bv8); might not be maintained by the loop. Running the javascript shows that they are all true. Is it due to the cast of oneMask to bv8?
edit:
I replaced the mask function with one that didn't require casting and that still not resolve the problem.
function method oneMaskOr(n: bv8): bv8
requires 0 <= n <= 8
ensures oneMaskOr(n) as bv16 == oneMask(n as bv16)
{
if n == 0 then 0 else (1 << (n-1)) | oneMaskOr(n-1)
}
One interesting thing I found is that it shows me a counter example where it has reached the end of the loop and the final bit of the input variable n is set, so values 128 or greater. But when I add an assertion above the loop that value equals the count at the end of the loop it then shows me the another value of n.
assert 1 == countOneBits(128 & OneMaskOr(8)); //counterexample -> 192
assert 2 == countOneBits(192 & OneMaskOr(8)); //counterexample -> 160
So it seems like it isn't evaluating the loop invariant after the end of loop? I thought the whole point of the invariants was to evaluate after the end of loop.
Edit 2:
I figured it out, apparently adding the explicit decreases clause to the while loop fixed it. I don't get it though. I thought Dafny could figure this out.
while i < 8
invariant 0 <= i <= 8
invariant n' == n >> i
invariant count == countOneBits(n & oneMask(i) as bv8);
decreases 8 - i
{
I see one line in the docs for loop termination saying
If the decreases clause of a loop specifies *, then no termination check will be performed. Use of this feature is sound only with respect to partial correctness.
So is if the decreases clause is missing does it default to *?
After playing around, I did find a version which passes though it required reworking countOneBits() so that its recursion followed the order of iteration:
function countOneBits(n:bv8, i: int, j:int): bv8
requires i ≥ 0 ∧ i ≤ j ∧ j ≤ 8
decreases 8-i {
if i == j then 0
else (n&1) + countOneBits(n >> 1, i+1, j)
}
method hammingWeight(n: bv8) returns (count: bv8 )
ensures count == countOneBits(n,0,8)
{
count ≔ 0;
var i ≔ 0;
var n' ≔ n;
//
assert count == countOneBits(n,0,i);
//
while i < 8
invariant 0 ≤ i ≤ 8;
invariant n' == n >> i;
invariant count == countOneBits(n,0,i);
{
count ≔ (n' & 1) + count;
n' ≔ n' >> 1;
i ≔ i + 1;
}
}
The intuition here is that countOneBits(n,i,j) returns the number of 1 bits between i (inclusive) and j (exclusive). This then reflects what the loop is doing as we increase i.
This is code -
for (i = 1; i<=1000000 ; i++ ) {
for ( j = 1; j<= 1000000; j++ ) {
for ( k = 1; k<= 1000000; k++ ) {
if (i % j == k && j%k == 0)
count++;
}
}
}
or is it better to reduce any % operation that goes upto million times in any programme ??
edit- i am sorry ,
initialized by 0, let say i = 1 ok!
now, if i reduce the third loop as #darshan's answer then both the first
&& second loop can run upto N times
and also it calculating % , n*n times. ex- 2021 mod 2022 , then 2021 mod 2023..........and so on
so my question is- % modulus is twice (and maybe more) as heavy as compared to +, - so there's any other logic can be implemented here ?? which is alternate for this question. and gives the same answer as this logic will give..
Thank you so much for knowledgeable comments & help-
Question is:
3 integers (A,B,C) is considered to be special if it satisfies the
following properties for a given integer N :
A mod B=C
B mod C=0
1≤A,B,C≤N
I'm so curious if there is any other smartest solution which can greatly reduces time complexity.
A much Efficient code will be the below one , but I think it can be optimized much more.
First of all modulo (%) operator is quite expensive so try to avoid it on a large scale
for(i = 0; i<=1000000 ; i++ )
for( j = 0; j<= 1000000; j++ )
{
a = i%j;
for( k = 0; k <= j; k++ )
if (a == k && j%k == 0)
count++;
}
We placed a = i%j in second loop because there is no need for it to be calculated every time k changes as it is independent of k and for the condition j%k == 0 to be true , k should be <= j hence change looping restrictions
First of all, your code has undefined behavior due to division by zero: when k is zero then j%k is undefined, so I assume that all your loops should start with 1 and not 0.
Usually the % and the / operators are much slower to execute than any other operation. It is possible to get rid of most invocations of the % operators in your code by several simple steps.
First, look at the if line:
if (i % j == k && j%k == 0)
The i % j == k has a very strict constrain over k which plays into your hands. It means that it is pointless to iterate k at all, since there is only one value of k that passes this condition.
for (i = 1; i<=1000000 ; i++ ) {
for ( j = 1; j<= 1000000; j++ ) {
k = i % j;
// Constrain k to the range of the original loop.
if (k <= 1000000 && k > 0 && j%k == 0)
count++;
}
}
To get rid of "i % j" switch the loop. This change is possible since this code is affected only by which combinations of i,j are tested, not in the order in which they are introduced.
for ( j = 1; j<= 1000000; j++ ) {
for (i = 1; i<=1000000 ; i++ ) {
k = i % j;
// Constrain k to the range of the original loop.
if (k <= 1000000 && k > 0 && j%k == 0)
count++;
}
}
Here it is easy to observe how k behaves, and use that in order to iterate on k directly without iterating on i and so getting rid of i%j. k iterates from 1 to j-1 and then does it again and again. So all we have to do is to iterate over k directly in the loop of i. Note that i%j for j == 1 is always 0, and since k==0 does not pass the condition of the if we can safely start with j=2, skipping 1:
for ( j = 2; j<= 1000000; j++ ) {
for (i = 1, k=1; i<=1000000 ; i++, k++ ) {
if (k == j)
k = 0;
// Constrain k to the range of the original loop.
if (k <= 1000000 && k > 0 && j%k == 0)
count++;
}
}
This is still a waste to run j%k repeatedly for the same values of j,k (remember that k repeats several times in the inner loop). For example, for j=3 the values of i and k go {1,1}, {2,2}, {3,0}, {4,1}, {5,2},{6,0},..., {n*3, 0}, {n*3+1, 1}, {n*3+2, 2},... (for any value of n in the range 0 < n <= (1000000-2)/3).
The values beyond n= floor((1000000-2)/3)== 333332 are tricky - let's have a look. For this value of n, i=333332*3=999996 and k=0, so the last iteration of {i,k}: {n*3,0},{n*3+1,1},{n*3+2, 2} becomes {999996, 0}, {999997, 1}, {999998, 2}. You don't really need to iterate over all these values of n since each of them does exactly the same thing. All you have to do is to run it only once and multiply by the number of valid n values (which is 999996+1 in this case - adding 1 to include n=0).
Since that did not cover all elements, you need to continue the remainder of the values: {999999, 0}, {1000000, 1}. Notice that unlike other iterations, there is no third value, since it would set i out-of-range.
for (int j = 2; j<= 1000000; j++ ) {
if (j % 1000 == 0) std::cout << std::setprecision(2) << (double)j*100/1000000 << "% \r" << std::flush;
int innerCount = 0;
for (int k=1; k<j ; k++ ) {
if (j%k == 0)
innerCount++;
}
int innerLoopRepeats = 1000000/j;
count += innerCount * innerLoopRepeats;
// complete the remainder:
for (int k=1, i= j * innerLoopRepeats+1; i <= 1000000 ; k++, i++ ) {
if (j%k == 0)
count++;
}
}
This is still extremely slow, but at least it completes in less than a day.
It is possible to have a further speed up by using an important property of divisibility.
Consider the first inner loop (it's almost the same for the second inner loop),
and notice that it does a lot of redundant work, and does it expensively.
Namely, if j%k==0, it means that k divides j and that there is pairK such that pairK*k==j.
It is trivial to calculate the pair of k: pairK=j/k.
Obviously, for k > sqrt(j) there is pairK < sqrt(j). This implies that any k > sqrt(j) can be extracted simply
by scanning all k < sqrt(j). This feature lets you loop over only a square root of all interesting values of k.
By searching only for sqrt(j) values gives a huge performance boost, and the whole program can finish in seconds.
Here is a view of the second inner loop:
// complete the remainder:
for (int k=1, i= j * innerLoopRepeats+1; i <= 1000000 && k*k <= j; k++, i++ ) {
if (j%k == 0)
{
count++;
int pairI = j * innerLoopRepeats + j / k;
if (pairI != i && pairI <= 1000000) {
count++;
}
}
}
The first inner loop has to go over a similar transformation.
Just reorder indexation and calculate A based on constraints:
void findAllSpecial(int N, void (*f)(int A, int B, int C))
{
// 1 ≤ A,B,C ≤ N
for (int C = 1; C < N; ++C) {
// B mod C = 0
for (int B = C; B < N; B += C) {
// A mod B = C
for (int A = C; A < N; A += B) {
f(A, B, C);
}
}
}
}
No divisions not useless if just for loops and adding operations.
Below is the obvious optimization:
The 3rd loop with 'k' is really not needed as there is already a many to One mapping from (I,j) -> k
What I understand from the code is that you want to calculate the number of (i,j) pairs such that the (i%j) is a factor of j. Is this correct or am I missing something?
Consider the following
int main() {
int a = 8;
int b = 10;
while (true) {
if (a /= 2 && b < 12) {
b++;
std::cout << b << std::endl;
}
else break;
}
return 0;
}
Now c++ is not my main language, but how does c++ evaluate this if statement?
In this case, when b>=12, the compiler throws the "division by zero" exception, but why?
Now if i wrap the states in parentheses i do not get the exception.
if( (a /= 2) && (b < 12))
Does this have something to do with how c++ evaluates the statements?
If evaluation is not the problem:
I am aware of that
a = (a/2 && b<12)
would not hold either.
P Λ Q does not hold for P Λ ¬Q but the state of P should not be affected? Why is it P gets blamed instead of ¬Q?
if (a /= 2 && b < 12)
is the same as:
if (a /= (2 && b < 12))
so:
2 is evaluated, which is converted to true in the context of an operand to &&. This does not trigger short-circuit evaluation so we continue...
b < 12, which in the case you're talking about is false
So 2 && b < 12 evaluates to false overall
a /= 2 && b < 12 is therefore equivalent to a /= false here, which is equivalent to a /= 0.
I came accross this expression, and can't understand the meaning of line 3 in the following snippet:
int A=0, B=0;
std::cout << A << B << "\n"; // Prints 0, 0
A += B++ == 0; // how does this exp work exactly?
std::cout << A << B << "\n"; // Prints 1, 1
A adds B to it, and B is Post incremented by 1, what does the "==0" mean?
Edit:
Here's the actual code:
int lengthOfLongestSubstringKDistinct(string s, int k) {
int ctr[256] = {}, j = -1, distinct = 0, maxlen = 0;
for (int i=0; i<s.size(); ++i) {
distinct += ctr[s[i]]++ == 0; //
while (distinct > k)
distinct -= --ctr[s[++j]] == 0;
maxlen = max(maxlen, i - j);
}
return maxlen;
}
B++ == 0
This is a boolean expression resulting in true or false. In this case the result is true, true is then added to A. The value of true is 1 so the (rough) equivalent would be:
if(B == 0)
A += 1;
++B;
Note that this isn't particulary good or clear to read code and the person who wrote this should be thrown into the Gulags.
Lets break this expression into pieces: A += value, whereas value = B++ == 0. As later cout suggests, value == 1. Why is that? Here is why: value is result of comparison of B++ and 0, but ++ (increment) operation, when written after operand, is being processed after the comparison, i.e. if you write A += ++B == 0 the later cout should (and does) print 0, 1.
I've got this pseudocode:
COMPARE-EXCHANGE(A,i,j)
if A[i] > A[j]
exchange A[i] with A[j]
INSERTION-SORT(A)
for j = 2 to A.length
for i = j-1 downto 1
COMPARE-EXCHANGE(A,i,i+1)
I would interpret it as:
void insertSort( )
{
int tmp;
for( int j = 2 ; j < MAX ; ++j )
{
for( int i = j - 1 ; i > 0 ; --i )
{
if( unsortedArr[i] > unsortedArr[i + 1] )
{
tmp = unsortedArr[i];
unsortedArr[i] = unsortedArr[i + 1];
unsortedArr[i + 1] = tmp;
}
}
}
}
However that would skip unsortedArr[0].
Which means it won't work.
Changing the 2nd for to:
for( int i = j - 1 ; i >= 0 ; --i )
Will make it run as intended.
Is there a mistake in the pseudocode or was my first try of interpreting it wrong?
However that would skip unsortedArr[0]. Which means it won't work.
It is nearly universal for pseudocode to number array elements from 1, not from zero, as in C/C++
Changing the 2nd for to:
for( int i = j - 1 ; i >= 0 ; --i )
Will make it run as intended.
That is not enough: you also need to start j at 1 rather than 2 in the outer loop.
Also note that the C++ standard library offers a std::swap function which takes care of exchanging the elements of the array for you:
if( unsortedArr[i] > unsortedArr[i + 1] )
{
std::swap(unsortedArr[i], unsortedArr[i+1]);
}
I think your pseudo code is assuming that arrays start with index one [1] -- where in C & C++ they start at zero [0].
I'm guessing that the pseudocode is using 1-based indexing, rather than the 0-based indexing that C++ uses.