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Recently I was trying some problems related to XOR operation, And I found this problem somewhere(Can't find out now), there is a function defined like this,

$f(i,j)$ = $1$ if i=$0$ or j = $0$

$f(i,j)$ = $f(i-1,j)$ $XOR$ $f(i,j-1)$ else

And we have to calculate $f(i,j)$ for really big numbers which can not be solved by just iteration. Can consider $i,j$ <= $10^{15}$

My Effort:

I was trying to observe something in terms of patterns, that I formed after printing matrix by just iterating over $i$ and $j$, for $20*20$ matrix. I found that it's forming inverted triangles in kind of nested form and of size $2^n-1$, but I still don't know how could I formulate or find a better way to calculate this function in lesser complexity than $O(i*j)$. I am expecting some logarithmic complexity which I am assuming is possible because of above patterns.

Kautsya Kanu
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1 Answers1

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Note that $a\mathbin{\sf xor}b = a+b \bmod 2$ when $a,b\in\{0,1\}$. So your matrix is Pascal's triangle modulo 2: $$ f(i,j) = \binom{i+j}{i} \bmod 2 $$

So in order to calculate it, you just need to be able to find out whether $\binom{m}{n} = \frac{m!}{(m-n)!n!} $ is even or odd. You can do that by computing how many factors of $2$ there are in each of the factorials and checking if there are as many in the denominator as in the numerator.

For counting factors of $2$ in a factorial, see Highest power of a prime $p$ dividing $n!$.

hmakholm left over Monica
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  • And finding out the number of factors of 2 (or any number) in a factorial (and consequently the binomial coefficient) can be done in logarithmical time as OP speculates and is a fun technique worth looking into – cirpis Sep 01 '17 at 22:44
  • Thanks guys.. Btw.. How did you see it's correlation with Pascal Triangle?? and how did you find $f(i,j)$ from that..?? – Kautsya Kanu Sep 01 '17 at 23:06
  • @KautsyaKanu: As for how I noticed, I don't think I have an explanation other than it comes with experience from thinking about related problems. – hmakholm left over Monica Sep 01 '17 at 23:32
  • @HenningMakholm Ok, Thank you for help.. :) – Kautsya Kanu Sep 02 '17 at 00:08
  • @HenningMakholm I was thinking to solve this problem further... Can you guide me how can I write $i^{th}$ row of this matrix(formed from $f(i,j)$) with help of previous rows in some logarithmic approach? Like how we represent a number in any base.. 5 = 1$2^2$ + 0$2^1$ + 1*$2^0$ , I read some articles on Pascal's binary matrix but none help for this one.. – Kautsya Kanu Sep 05 '17 at 01:13