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I am trying to implement this code and this website has kindly provided their algorithm but I am trying to Find out what is "N" I understood what "I" and "M" is but not "N", is "N" the Total input(in the below example 5 because there are 5 letters)?
Algorithm:
Combinations are generated in lexicographical order. The algorithm uses indexes of the elements of the set. Here is how it works on example: Suppose we have a set of 5 elements with indexes 1 2 3 4 5 (starting from 1), and we need to generate all combinations of size m
= 3.
First, we initialize the first combination of size m - with indexes in ascending order
1 2 3
Then we check the last element (i = 3). If its value is less than n - m + i, it is incremented by 1.
1 2 4
Again we check the last element, and since it is still less than n - m
i, it is incremented by 1.
1 2 5
Now it has the maximum allowed value: n - m + i = 5 - 3 + 3 = 5, so we move on to the previous element (i = 2).
If its value less than n - m + i, it is incremented by 1, and all following elements are set to value of their previous neighbor plus 1
1 (2+1)3 (3+1)4 = 1 3 4
Then we again start from the last element i = 3
1 3 5
Back to i = 2
1 4 5
Now it finally equals n - m + i = 5 - 3 + 2 = 4, so we can move to first element (i = 1) (1+1)2 (2+1)3 (3+1)4 = 2 3 4
And then,
2 3 5
2 4 5
3 4 5
and it is the last combination since all values are set to the maximum possible value of n - m + i.
Input:
A
B
C
D
E
Output:
A B C
A B D
A B E
A C D
A C E
A D E
B C D
B C E
B D E
C D E
Take a look at the very first paragraf of the link you provided.
It states that
This combinations calculator generates all possible combinations of m elements from the set of n elements.
So yes, n is the number of elements or letters that the algorithm needs to use.
N here is the size of the set of set from which you generate the combinations. In the given example, "Suppose we have a set of 5 elements with indexes 1 2 3 4 5 (starting from 1)", N is 5.
Combinations are usually symbolized with nCm, or n choose m. So n is the total set size(in this example 5) and m is the number chosen(3).
Imagine we have a loop with index variable n counting from 1 to 6. As I got through this loop I want to print the values
1
2
3
3
2
1
Is there a elegant mathematical operation (i.e. avoiding if statements) which can implement this? I know for example if I print
1 + modulo(n-1,3)
it would print
1
2
3
1
2
3
but I want that second half reversed. The loop will always have an even number of iterations if that helps. I am writing in Fortran 90.
How about the following:
do i=1,n
print *, int(abs(i-n/2.0-0.5)+0.5)
end do
This will print the expected result for even n and introduces a ZERO for odd n.
Is the following elegant enough? I think this is much clearer than obfuscated maths
ian-admin#agon ~/work/stack $ cat merge.f90
Program test
Implicit None
Integer :: n
Integer :: i
Write( *, * ) 'n?'
Read ( *, * ) n
Do i = 1, n
Write( *, * ) Merge( i, n - ( i - 1 ), i <= n / 2 )
End Do
End Program test
ian-admin#agon ~/work/stack $ gfortran -std=f2003 -Wall -Wextra merge.f90
ian-admin#agon ~/work/stack $ ./a.out
n?
6
1
2
3
3
2
1
ian-admin#agon ~/work/stack $ ./a.out
n?
7
1
2
3
4
3
2
1
ian-admin#agon ~/work/stack $
Consider
min(n, 7-n)
This expression produces 1, 2, 3, 3, 2, 1 as n goes from 1 to 6. More generally, if N is the (even) upper bound for the variable n, the formula becomes:
min(n, N+1-n)
which produces
1, 2, ..., N/2, N/2, N/2 - 1, ..., 2, 1
as n runs between 1 and N.
In the case where N is odd, the same formula generates the sequence
1, 2, ..., (N+1)/2, (N-1)/2, ..., 2, 1.
For example:
1, 2, 3, 4, 3, 2, 1
for N=7.
n = 6
for i in range(1,n+1):
print(min(i, n + 1 - i))
or (// is integer division equivalent to bitshift right)
print(n//2 - abs(n + 1 - 2 * i)//2)
>> 1 2 3 3 2 1
For odd length - the first approach works as is, the second way requires changing to (n+1)//2 (it does not break work for even length)
Bit trick to find min of two 32-bit integers:
min = b+((a-b)&((a-b)>>31))
We are given an array A of integers. I want to find 2 contiguous subarrays of the largest length(both subarrays must be equal in length) that have the same weighted average. The weights are the positions in the subarray. For example
A=41111921111119
Subarrays:: (11119) and (11119)
Ive tried to find the weighted average of all subarrays by DP and then sorting them columnwise to find 2 with same length.But I cant proceed further and my approach seems too vague/bruteforce.I would appreciate any help. Thanks in advance.
The first step should be to sort the array. Any pairs of equal values can then be identified and factored out. The remaining numbers will all be different, like this:
2, 3, 5, 9, 14, 19 ... etc
The next step would be to compare pairs to their center:
2 + 5 == 2 * 3 ?
3 + 9 == 2 * 5 ?
5 + 14 == 2 * 9 ?
9 + 19 == 2 * 14 ?
The next step is to compare nested pairs, meaning if you have A B C D, you compare A+D to B+C. So for the above example it would be:
2+9 == 3+5 ?
3+15 == 5+9 ?
5+19 == 9+14 ?
Next you would compare triples to the two inside values:
2 + 3 + 9 == 3 * 5 ?
2 + 5 + 9 == 3 * 3 ?
3 + 5 + 14 == 3 * 9 ?
3 + 9 + 14 == 3 * 5 ?
5 + 9 + 19 == 3 * 14 ?
5 + 14 + 19 == 3 * 9 ?
Then you would compare pairs of triples:
2 + 3 + 19 == 5 + 9 + 14 ?
2 + 5 + 19 == 3 + 9 + 14 ?
2 + 9 + 19 == 3 + 5 + 14 ?
and so on. There are different ways to do the ordering. One way is to create an initial bracket, for example, given A B C D E F G H, the initial bracket is ABGH versus CDEF, ie the outside compared to the center. Then switch values according to the comparison. For example, if ABGH > CDEF, then you can try all switches where the left value is greater than the right value. In this case G and H are greater than E and F, so the possible switches are:
G <-> E
G <-> F
H <-> E
H <-> F
GH <-> EF
First, as the length of the two subarray must be equal, you can consider the length from 1 to n step by step.
For length i, you can calculate the weighted sum of every subarray in a total complexity of O(n). Then sort the sums to determine if there's an equal pair.
Because you sort n times the time would be O(n^2 log n) while the space is O(n).
Maybe I just repeated your solution mentioned in the question? But I don't think it can be optimized any more...
How to find out the units digit of a certain number (e.g. 3 power 2011). What logic should I use to find the answer to this problem?
For base 3:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
3^7 = 2187
...
That is the units digit has only 4 possibilities and then it repeats in ever the same cycle.
With the help of Euler's theorem we can show that this holds for any integer n, meaning their units digit will repeat after at most 4 consecutive exponents. Looking only at the units digit of an arbitrary product is equivalent to taking the remainder of the multiplication modulo 10, for example:
2^7 % 10 = 128 % 10 = 8
It can also be shown (and is quite intuitive) that for an arbitrary base, the units digit of any power will only depend on the units digit of the base itself - that is 2013^2013 has the same units digit as 3^2013.
We can exploit both facts to come up with an extremely fast algorithm (thanks for the help - with kind permission I may present a much faster version).
The idea is this: As we know that for any number 0-9 there will be at most 4 different outcomes, we can as well store them in a lookup table:
{ 0,0,0,0, 1,1,1,1, 6,2,4,8, 1,3,9,7, 6,4,6,4,
5,5,5,5, 6,6,6,6, 1,7,9,3, 6,8,4,2, 1,9,1,9 }
That's the possible outcomes for 0-9 in that order, grouped in fours. The idea is now for an exponentiation n^a to
first take the base mod 10 => := i
go to index 4*i in our table (it's the starting offset of that particular digit)
take the exponent mod 4 => := off (as stated by Euler's theorem we only have four possible outcomes!)
add off to 4*i to get the result
Now to make this as efficient as possible, some tweaks are applied to the basic arithmetic operations:
Multiplying by 4 is equivalent to shifting two to the left ('<< 2')
Taking a number a % 4 is equivalent to saying a&3 (masking the 1 and 2 bit, which form the remainder % 4)
The algorithm in C:
static int table[] = {
0, 0, 0, 0, 1, 1, 1, 1, 6, 2, 4, 8, 1, 3, 9, 7, 6, 4, 6, 4,
5, 5, 5, 5, 6, 6, 6, 6, 1, 7, 9, 3, 6, 8, 4, 2, 1, 9, 1, 9
};
int /* assume n>=0, a>0 */
unit_digit(int n, int a)
{
return table[((n%10)<<2)+(a&3)];
}
Proof for the initial claims
From observing we noticed that the units digit for 3^x repeats every fourth power. The claim was that this holds for any integer. But how is this actually proven? As it turns out that it's quite easy using modular arithmetic. If we are only interested in the units digit, we can perform our calculations modulo 10. It's equivalent to say the units digit cycles after 4 exponents or to say
a^4 congruent 1 mod 10
If this holds, then for example
a^5 mod 10 = a^4 * a^1 mod 10 = a^4 mod 10 * a^1 mod 10 = a^1 mod 10
that is, a^5 yields the same units digit as a^1 and so on.
From Euler's theorem we know that
a^phi(10) mod 10 = 1 mod 10
where phi(10) is the numbers between 1 and 10 that are co-prime to 10 (i.e. their gcd is equal to 1). The numbers < 10 co-prime to 10 are 1,3,7 and 9. So phi(10) = 4 and this proves that really a^4 mod 10 = 1 mod 10.
The last claim to prove is that for exponentiations where the base is >= 10 it suffices to just look at the base's units digit. Lets say our base is x >= 10, so we can say that x = x_0 + 10*x_1 + 100*x_2 + ... (base 10 representation)
Using modular representation it's easy to see that indeed
x ^ y mod 10
= (x_0 + 10*x_1 + 100*x_2 + ...) ^ y mod 10
= x_0^y + a_1 * (10*x_1)^y-1 + a_2 * (100*x_2)^y-2 + ... + a_n * (10^n) mod 10
= x_0^y mod 10
where a_i are coefficients that include powers of x_0 but finally not relevant since the whole product a_i * (10 * x_i)^y-i will be divisible by 10.
You should look at Modular exponentiation. What you want is the same of calculating n^e (mod m) with m = 10. That is the same thing as calculating the remainder of the division by ten of n^e.
You are probably interested in the Right-to-left binary method to calculate it, since it's the most time-efficient one and the easiest not too hard to implement. Here is the pseudocode, from Wikipedia:
function modular_pow(base, exponent, modulus)
result := 1
while exponent > 0
if (exponent & 1) equals 1:
result = (result * base) mod modulus
exponent := exponent >> 1
base = (base * base) mod modulus
return result
After that, just call it with modulus = 10 for you desired base and exponent and there's your answer.
EDIT: for an even simpler method, less efficient CPU-wise but more memory-wise, check out the Memory-efficient section of the article on Wikipedia. The logic is straightforward enough:
function modular_pow(base, exponent, modulus)
c := 1
for e_prime = 1 to exponent
c := (c * base) mod modulus
return c
I'm sure there's a proper mathematical way to solve this, but I would suggest that since you only care about the last digit and since in theory every number multiplied by itself repeatedly should generate a repeating pattern eventually (when looking only at the last digit), you could simply perform the multiplications until you detect the first repetition and then map your exponent into the appropriate position in the pattern that you built.
Note that because you only care about the last digit, you can further simplify things by truncating your input number down to its ones-digit before you start building your pattern mapping. This will let you to determine the last digit even for arbitrarily large inputs that would otherwise cause an overflow on the first or second multiplication.
Here's a basic example in JavaScript: http://jsfiddle.net/dtyuA/2/
function lastDigit(base, exponent) {
if (exponent < 0) {
alert("stupid user, negative values are not supported");
return 0;
}
if (exponent == 0) {
return 1;
}
var baseString = base + '';
var lastBaseDigit = baseString.substring(baseString.length - 1);
var lastDigit = lastBaseDigit;
var pattern = [];
do {
pattern.push(lastDigit);
var nextProduct = (lastDigit * lastBaseDigit) + '';
lastDigit = nextProduct.substring(nextProduct.length - 1);
} while (lastDigit != lastBaseDigit);
return pattern[(exponent - 1) % pattern.length];
};
function doMath() {
var base = parseInt(document.getElementById("base").value, 10);
var exp = parseInt(document.getElementById("exp").value, 10);
console.log(lastDigit(base, exp));
};
console.log(lastDigit(3003, 5));
Base: <input id="base" type="text" value="3" /> <br>
Exponent: <input id="exp" type="text" value="2011"><br>
<input type="button" value="Submit" onclick="doMath();" />
And the last digit in 3^2011 is 7, by the way.
We can start by inspecting the last digit of each result obtained by raising the base 10 digits to successive powers:
d d^2 d^3 d^4 d^5 d^6 d^7 d^8 d^9 (mod 10)
--- --- --- --- --- --- --- --- ---
0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1
2 4 8 6 2 4 8 6 2
3 9 7 1 3 9 7 1 3
4 6 4 6 4 6 4 6 4
5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6
7 9 3 1 7 9 3 1 7
8 4 2 6 8 4 2 6 8
9 1 9 1 9 1 9 1 9
We can see that in all cases the last digit cycles through no more than four distinct values. Using this fact, and assuming that n is a non-negative integer and p is a positive integer, we can compute the result fairly directly (e.g. in Javascript):
function lastDigit(n, p) {
var d = n % 10;
return [d, (d*d)%10, (d*d*d)%10, (d*d*d*d)%10][(p-1) % 4];
}
... or even more simply:
function lastDigit(n, p) {
return Math.pow(n % 10, (p-1) % 4 + 1) % 10;
}
lastDigit(3, 2011)
/* 7 */
The second function is equivalent to the first. Note that even though it uses exponentiation, it never works with a number larger than nine to the fourth power (6561).
The key to solving this type of question lies in Euler's theorem.
This theorem allows us to say that a^phi(m) mod m = 1 mod m, if and only if a and m are coprime. That is, a and m do not divide evenly. If this is the case, (and for your example it is), we can solve the problem on paper, without any programming what so ever.
Let's solve for the unit digit of 3^2011, as in your example. This is equivalent to 3^2011 mod 10.
The first step is to check is 3 and 10 are co-prime. They do not divide evenly, so we can use Euler's theorem.
We also need to compute what the totient, or phi value, is for 10. For 10, it is 4. For 100 phi is 40, 1000 is 4000, etc.
Using Euler's theorem, we can see that 3^4 mod 10 = 1. We can then re-write the original example as:
3^2011 mod 10 = 3^(4*502 + 3) mod 10 = 3^(4*502) mod 10 + 3^3 mod 10 = 1^502 * 3^3 mod 10 = 27 mod 10 = 7
Thus, the last digit of 3^2011 is 7.
As you saw, this required no programming whatsoever and I solved this example on a piece of scratch paper.
You ppl are making simple thing complicated.
Suppose u want to find out the unit digit of abc ^ xyz .
divide the power xyz by 4,if remainder is 1 ans is c^1=c.
if xyz%4=2 ans is unit digit of c^2.
else if xyz%4=3 ans is unit digit of c^3.
if xyz%4=0
then we need to check whether c is 5,then ans is 5
if c is even ans is 6
if c is odd (other than 5 ) ans is 1.
Bellow is a table with the power and the unit digit of 3 to that power.
0 1
1 3
2 9
3 7
4 1
5 3
6 9
7 7
Using this table you can see that the unit digit can be 1, 3, 9, 7 and the sequence repeats in this order for higher powers of 3. Using this logic you can find that the unit digit of (3 power 2011) is 7. You can use the same algorithm for the general case.
Here's a trick that works for numbers that aren't a multiple of a factor of the base (for base 10, it can't be a multiple of 2 or 5.) Let's use base 3. What you're trying to find is 3^2011 mod 10. Find powers of 3, starting with 3^1, until you find one with the last digit 1. For 3, you get 3^4=81. Write the original power as (3^4)^502*3^3. Using modular arithmetic, (3^4)^502*3^3 is congruent to (has the same last digit as) 1^502*3^3. So 3^2011 and 3^3 have the same last digit, which is 7.
Here's some pseudocode to explain it in general. This finds the last digit of b^n in base B.
// Find the smallest power of b ending in 1.
i=1
while ((b^i % B) != 1) {
i++
}
// b^i has the last digit 1
a=n % i
// For some value of j, b^n == (b^i)^j * b^a, which is congruent to b^a
return b^a % B
You'd need to be careful to prevent an infinite loop, if no power of b ends in 1 (in base 10, multiples of 2 or 5 don't work.)
Find out the repeating set in this case, it is 3,9,7,1 and it repeats in the same order for ever....so divide 2011 by 4 which will give you a reminder 3. That is the 3rd element in the repeating set. This is the easiest way to find for any given no. say if asked for 3^31, then the reminder of 31/4 is 3 and so 7 is the unit digit. for 3^9, 9/4 is 1 and so the unit will be 3. 3^100, the unit will be 1.
If you have the number and exponent separate it's easy.
Let n1 is the number and n2 is the power. And ** represents power.
assume n1>0.
% means modulo division.
pseudo code will look like this
def last_digit(n1, n2)
if n2==0 then return 1 end
last = n1%10
mod = (n2%4).zero? ? 4 : (n2%4)
last_digit = (last**mod)%10
end
Explanation:
We need to consider only the last digit of the number because that determines the last digit of the power.
it's the maths property that count of possibility of each digits(0-9) power's last digit is at most 4.
1) Now if the exponent is zero we know the last digit would be 1.
2) Get the last digit by %10 on the number(n1)
3) %4 on the exponent(n2)- if the output is zero we have to consider that as 4 because n2 can't be zero. if %4 is non zero we have to consider %4 value.
4) now we have at most 9**4. This is easy for the computer to calculate.
take the %10 on that number. You have the last digit.
Let's say I have a number of base 3, 1211. How could I check this number is divisible by 2 without converting it back to base 10?
Update
The original problem is from TopCoder
The digits 3 and 9 share an interesting property. If you take any multiple of 3 and sum its digits, you get another multiple of 3. For example, 118*3 = 354 and 3+5+4 = 12, which is a multiple of 3. Similarly, if you take any multiple of 9 and sum its digits, you get another multiple of 9. For example, 75*9 = 675 and 6+7+5 = 18, which is a multiple of 9. Call any digit for which this property holds interesting, except for 0 and 1, for which the property holds trivially.
A digit that is interesting in one base is not necessarily interesting in another base. For example, 3 is interesting in base 10 but uninteresting in base 5. Given an int base, your task is to return all the interesting digits for that base in increasing order. To determine whether a particular digit is interesting or not, you need not consider all multiples of the digit. You can be certain that, if the property holds for all multiples of the digit with fewer than four digits, then it also holds for multiples with more digits. For example, in base 10, you would not need to consider any multiples greater than 999.
Notes
- When base is greater than 10, digits may have a numeric value greater than 9. Because integers are displayed in base 10 by default, do not be alarmed when such digits appear on your screen as more than one decimal digit. For example, one of the interesting digits in base 16 is 15.
Constraints
- base is between 3 and 30, inclusive.
This is my solution:
class InterestingDigits {
public:
vector<int> digits( int base ) {
vector<int> temp;
for( int i = 2; i <= base; ++i )
if( base % i == 1 )
temp.push_back( i );
return temp;
}
};
The trick was well explained here : https://math.stackexchange.com/questions/17242/how-does-base-of-a-number-relate-to-modulos-of-its-each-individual-digit
Thanks,
Chan
If your number k is in base three, then you can write it as
k = a0 3^n + a1 3^{n-1} + a2 3^{n-2} + ... + an 3^0
where a0, a1, ..., an are the digits in the base-three representation.
To see if the number is divisible by two, you're interested in whether the number, modulo 2, is equal to zero. Well, k mod 2 is given by
k mod 2 = (a0 3^n + a1 3^{n-1} + a2 3^{n-2} + ... + an 3^0) mod 2
= (a0 3^n) mod 2 + (a1 3^{n-1}) mod 2 + ... + an (3^0) mod 2
= (a0 mod 2) (3^n mod 2) + ... + (an mod 2) (3^0 mod 2)
The trick here is that 3^i = 1 (mod 2), so this expression is
k mod 2 = (a0 mod 2) + (a1 mod 2) + ... + (an mod 2)
In other words, if you sum up the digits of the ternary representation and get that this value is divisible by two, then the number itself must be divisible by two. To make this even cooler, since the only ternary digits are 0, 1, and 2, this is equivalent to asking whether the number of 1s in the ternary representation is even!
More generally, though, if you have a number in base m, then that number is divisible by m - 1 iff the sum of the digits is divisible by m. This is why you can check if a number in base 10 is divisible by 9 by summing the digits and seeing if that value is divisible by nine.
You can always build a finite automaton for any base and any divisor:
Normally to compute the value n of a string of digits in base b
you iterate over the digits and do
n = (n * b) + d
for each digit d.
Now if you are interested in divisibility you do this modulo m instead:
n = ((n * b) + d) % m
Here n can take at most m different values. Take these as states of a finite automaton, and compute the transitions depending on the digit d according to that formula. The accepting state is the one where the remainder is 0.
For your specific case we have
n == 0, d == 0: n = ((0 * 3) + 0) % 2 = 0
n == 0, d == 1: n = ((0 * 3) + 1) % 2 = 1
n == 0, d == 2: n = ((0 * 3) + 2) % 2 = 0
n == 1, d == 0: n = ((1 * 3) + 0) % 2 = 1
n == 1, d == 1: n = ((1 * 3) + 1) % 2 = 0
n == 1, d == 2: n = ((1 * 3) + 2) % 2 = 1
which shows that you can just sum the digits 1 modulo 2 and ignore any digits 0 or 2.
Add all the digits together (or even just count the ones) - if the answer is odd, the number is odd; if it's even, the nmber is even.
How does that work? Each digit from the number contributes 0, 1 or 2 times (1, 3, 9, 27, ...). A 0 or a 2 adds an even number, so no effect on the oddness/evenness (parity) of the number as a whole. A 1 adds one of the powers of 3, which is always odd, and so flips the parity). And we start from 0 (even). So by counting whether the number of flips is odd or even we can tell whether the number itself is.
I'm not sure on what CPU you have a number in base-3, but the normal way to do this is to perform a modulus/remainder operation.
if (n % 2 == 0) {
// divisible by 2, so even
} else {
// odd
}
How to implement the modulus operator is going to depend on how you're storing your base-3 number. The simplest to code will probably be to implement normal pencil-and-paper long division, and get the remainder from that.
0 2 2 0
_______
2 ⟌ 1 2 1 1
0
---
1 2
1 1
-----
1 1
1 1
-----
0 1 <--- remainder = 1 (so odd)
(This works regardless of base, there are "tricks" for base-3 as others have mentioned)
Same as in base 10, for your example:
1. Find the multiple of 2 that's <= 1211, that's 1210 (see below how to achieve it)
2. Substract 1210 from 1211, you get 1
3. 1 is < 10, thus 1211 isn't divisible by 2
how to achieve 1210:
1. starts with 2
2. 2 + 2 = 11
3. 11 + 2 = 20
4. 20 + 2 = 22
5. 22 + 2 = 101
6. 101 + 2 = 110
7. 110 + 2 = 112
8. 112 + 2 = 121
9. 121 + 2 = 200
10. 200 + 2 = 202
... // repeat until you get the biggest number <= 1211
it's basically the same as base 10 it's just the round up happens on 3 instead of 10.