Creating inputs that make a subtraction-based GCD algorithm slow
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I have a GCD algorithm that is based on comparison and subtraction. The principle looks like this:
while a != b
while (a > b)
n = 1
while (a > pow(b, n))
a = a - pow(b, n)
n = n + 1
swap a with b
I want to specifically find some numbers that makes the algorithm slow (number of loops run). For example, $5$ and $15$ makes the algorithm very fast, while $2^{15}$ and $1$ makes it slow, but $2^{15}-1$ and $2^{13}-1$ makes it even slower. However, randomly-generated numbers like $30533$ and $19015$ can be slowest, among what numbers I have now.
Is there an algorithm that can find (or better, calculate) a pair of number that makes the above GCD algorithm slow, with both numbers under a given cap. In my case, I need the numbers to be smaller than (not equal to) $2^{15}$ (or $32768$).
Enumerating is not an option for me because profiling the algorithm in my environment is very slow, also if the cap gets bigger, enumerating is impractical.
algorithms greatest-common-divisor
asked Dec 4 at 11:10
iBug
1427
add a comment |
up vote
0
down vote
favorite
I have a GCD algorithm that is based on comparison and subtraction. The principle looks like this:
while a != b
while (a > b)
n = 1
while (a > pow(b, n))
a = a - pow(b, n)
n = n + 1
swap a with b
I want to specifically find some numbers that makes the algorithm slow (number of loops run). For example, $5$ and $15$ makes the algorithm very fast, while $2^{15}$ and $1$ makes it slow, but $2^{15}-1$ and $2^{13}-1$ makes it even slower. However, randomly-generated numbers like $30533$ and $19015$ can be slowest, among what numbers I have now.
Is there an algorithm that can find (or better, calculate) a pair of number that makes the above GCD algorithm slow, with both numbers under a given cap. In my case, I need the numbers to be smaller than (not equal to) $2^{15}$ (or $32768$).
Enumerating is not an option for me because profiling the algorithm in my environment is very slow, also if the cap gets bigger, enumerating is impractical.
algorithms greatest-common-divisor
asked Dec 4 at 11:10
iBug
1427
I get loop count $32767$ for $gcd(2^{15},1)$ and $2736$ for $gcd(2^{15}-1,2^{13}-1)$. What do you get?
– Somos
Dec 4 at 13:15
@Somos Sorry, I changed the actual algorithm when I kept working. I updated the question
– iBug
Dec 4 at 13:57
Copying what's currently in your post directly into Python has $gcd(2^{15}, 1)$ at 32767 and $gcd(2^{15} - 1, 2^{13} - 1)$ at 42.
– Mees de Vries
Dec 4 at 14:10
1
In fact $gcd(2^{15}, 1)$ is fairly easily seen to be (approximately) slowest. As long as one of the numbers is not one, the total $a + b$ drops with at least 2 every loop, for a total of at most $2^{15}$ steps, which you already achieve with $2^{15}, 1$.
– Mees de Vries
Dec 4 at 14:16
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a GCD algorithm that is based on comparison and subtraction. The principle looks like this:
while a != b
while (a > b)
n = 1
while (a > pow(b, n))
a = a - pow(b, n)
n = n + 1
swap a with b
I want to specifically find some numbers that makes the algorithm slow (number of loops run). For example, $5$ and $15$ makes the algorithm very fast, while $2^{15}$ and $1$ makes it slow, but $2^{15}-1$ and $2^{13}-1$ makes it even slower. However, randomly-generated numbers like $30533$ and $19015$ can be slowest, among what numbers I have now.
Is there an algorithm that can find (or better, calculate) a pair of number that makes the above GCD algorithm slow, with both numbers under a given cap. In my case, I need the numbers to be smaller than (not equal to) $2^{15}$ (or $32768$).
Enumerating is not an option for me because profiling the algorithm in my environment is very slow, also if the cap gets bigger, enumerating is impractical.
algorithms greatest-common-divisor
asked Dec 4 at 11:10
iBug
1427
I have a GCD algorithm that is based on comparison and subtraction. The principle looks like this:
while a != b
while (a > b)
n = 1
while (a > pow(b, n))
a = a - pow(b, n)
n = n + 1
swap a with b
I want to specifically find some numbers that makes the algorithm slow (number of loops run). For example, $5$ and $15$ makes the algorithm very fast, while $2^{15}$ and $1$ makes it slow, but $2^{15}-1$ and $2^{13}-1$ makes it even slower. However, randomly-generated numbers like $30533$ and $19015$ can be slowest, among what numbers I have now.
Is there an algorithm that can find (or better, calculate) a pair of number that makes the above GCD algorithm slow, with both numbers under a given cap. In my case, I need the numbers to be smaller than (not equal to) $2^{15}$ (or $32768$).
Enumerating is not an option for me because profiling the algorithm in my environment is very slow, also if the cap gets bigger, enumerating is impractical.
algorithms greatest-common-divisor
algorithms greatest-common-divisor
asked Dec 4 at 11:10
iBug
1427
asked Dec 4 at 11:10
iBug
1427
edited Dec 4 at 13:54
asked Dec 4 at 11:10
iBug
1427
asked Dec 4 at 11:10
iBug
1427
asked Dec 4 at 11:10
iBug
1427
1427
I get loop count $32767$ for $gcd(2^{15},1)$ and $2736$ for $gcd(2^{15}-1,2^{13}-1)$. What do you get?
– Somos
Dec 4 at 13:15
@Somos Sorry, I changed the actual algorithm when I kept working. I updated the question
– iBug
Dec 4 at 13:57
Copying what's currently in your post directly into Python has $gcd(2^{15}, 1)$ at 32767 and $gcd(2^{15} - 1, 2^{13} - 1)$ at 42.
– Mees de Vries
Dec 4 at 14:10
1
In fact $gcd(2^{15}, 1)$ is fairly easily seen to be (approximately) slowest. As long as one of the numbers is not one, the total $a + b$ drops with at least 2 every loop, for a total of at most $2^{15}$ steps, which you already achieve with $2^{15}, 1$.
– Mees de Vries
Dec 4 at 14:16
add a comment |
I get loop count $32767$ for $gcd(2^{15},1)$ and $2736$ for $gcd(2^{15}-1,2^{13}-1)$. What do you get?
– Somos
Dec 4 at 13:15
@Somos Sorry, I changed the actual algorithm when I kept working. I updated the question
– iBug
Dec 4 at 13:57
Copying what's currently in your post directly into Python has $gcd(2^{15}, 1)$ at 32767 and $gcd(2^{15} - 1, 2^{13} - 1)$ at 42.
– Mees de Vries
Dec 4 at 14:10
1
In fact $gcd(2^{15}, 1)$ is fairly easily seen to be (approximately) slowest. As long as one of the numbers is not one, the total $a + b$ drops with at least 2 every loop, for a total of at most $2^{15}$ steps, which you already achieve with $2^{15}, 1$.
– Mees de Vries
Dec 4 at 14:16
I get loop count $32767$ for $gcd(2^{15},1)$ and $2736$ for $gcd(2^{15}-1,2^{13}-1)$. What do you get?
– Somos
Dec 4 at 13:15
I get loop count $32767$ for $gcd(2^{15},1)$ and $2736$ for $gcd(2^{15}-1,2^{13}-1)$. What do you get?
– Somos
Dec 4 at 13:15
@Somos Sorry, I changed the actual algorithm when I kept working. I updated the question
– iBug
Dec 4 at 13:57
@Somos Sorry, I changed the actual algorithm when I kept working. I updated the question
– iBug
Dec 4 at 13:57
Copying what's currently in your post directly into Python has $gcd(2^{15}, 1)$ at 32767 and $gcd(2^{15} - 1, 2^{13} - 1)$ at 42.
– Mees de Vries
Dec 4 at 14:10
Copying what's currently in your post directly into Python has $gcd(2^{15}, 1)$ at 32767 and $gcd(2^{15} - 1, 2^{13} - 1)$ at 42.
– Mees de Vries
Dec 4 at 14:10
1
1
In fact $gcd(2^{15}, 1)$ is fairly easily seen to be (approximately) slowest. As long as one of the numbers is not one, the total $a + b$ drops with at least 2 every loop, for a total of at most $2^{15}$ steps, which you already achieve with $2^{15}, 1$.
– Mees de Vries
Dec 4 at 14:16
In fact $gcd(2^{15}, 1)$ is fairly easily seen to be (approximately) slowest. As long as one of the numbers is not one, the total $a + b$ drops with at least 2 every loop, for a total of at most $2^{15}$ steps, which you already achieve with $2^{15}, 1$.
– Mees de Vries
Dec 4 at 14:16
add a comment |
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I get loop count $32767$ for $gcd(2^{15},1)$ and $2736$ for $gcd(2^{15}-1,2^{13}-1)$. What do you get?
– Somos
Dec 4 at 13:15
@Somos Sorry, I changed the actual algorithm when I kept working. I updated the question
– iBug
Dec 4 at 13:57
Copying what's currently in your post directly into Python has $gcd(2^{15}, 1)$ at 32767 and $gcd(2^{15} - 1, 2^{13} - 1)$ at 42.
– Mees de Vries
Dec 4 at 14:10
1
In fact $gcd(2^{15}, 1)$ is fairly easily seen to be (approximately) slowest. As long as one of the numbers is not one, the total $a + b$ drops with at least 2 every loop, for a total of at most $2^{15}$ steps, which you already achieve with $2^{15}, 1$.
– Mees de Vries
Dec 4 at 14:16