I was wondering if there is a problem solvable on a 2D grid using square root decomposition :-) or if anyone used it before

I tried to solve problem e in round 199 xenia and tree using heavy light decomposition but I couldn't figure out the whole idea anyone ?

in this problem (http://codeforces.com/contest/165/problem/D) there is a condition which makes it easy which is "there exists no more than one vertex, whose degree is more than two" so what if this condition didnot exist ?

i read the tutorial and other solutions on the internet but none of them described the lazy propagation , it seems a little bit complicated anyone ?

Some people recommend solving on usaco for ioier's so i wanted to ask if this really helps . if you have any experience share it please :) .

i have recently participated in ioi and my score wasnot very good (98/600) , and i just wanted to ask for adivce , what is the plan or what does it take to get a medal in ioi or let me say how to have better understanding for problem solving before the ioi i tried to solve problems on techniques but i think that what i did was wrong cuz i didnot practise on thinking in general also i have to study well to prepare for the college so i need to work in a smart way and ideas ?

i find this a very interesting problem but i need a little help anyone ?

i took part in this contest and i wanted to see if anyone can give me a hand in this problem .

the link ->http://cms.ioi-jp.org/contest/tasks/fortune_telling2/description