Div2 894E — Ralph and Mushrooms
as we know we can find cycle and contract to a point to make the graph become DAG
but tutorial dont explain about finding mushrooms in the cycle by math knowledge
I have read some one's code and the math is as the follow
IF I HAVE MISTAKE WELLCOME TO TELL ME! THX
walk through the cycle me can get the max mushrooms as we could
because we can still pass it though we get 0 mushroom so we can get all the mushrooms
mushrooms decrease by 1,2,3...n
for the initial mushrooms w
i-th time you get w-i(i+1)/2
for every w suppose you can get mushrooms n time
w,w-1,w-1-2,w-1-2-3...w-(1+2..n)
which is w*(n+1)-[i(i+1)/2 i for 0 to n]
-> w(n+1)-i(i+1)(i+2)/6
see here https://en.wikipedia.org/wiki/Tetrahedral_number
and the way to find n
we have w already
to make it simple
consider w is one of the i(i+1)/2
8*w
=4*i(i+1)
=4i^2+4i
8*w+1=(2i+1)^2
sqrt(8*w+1)-1=2i
[sqrt(8*w+1)-1]/2=2i/2=i
which is what we want,i is the max times to get mushroom
for any w get floor for the ans
c++
n is the max times to get mushroom
int n= floor((sqrt(1+8*w)-1)/2);
sum_mushroom=(n+1)*w — n*(n+1)*(n+2)/6;
i.e.
w=9
9,8,6,3,-1
9,9,9,9,9 (+ plus
0,1,3,6,10 (- minus
n is 4
9*(4+1)-(0+1+(1+2)+(1+2+3)+(1+2+3+4))
=9+8+6+3=26