DAGs and SCC

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Small-to-Large

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The Vanilla Method Let's take the given problem: We have some sets as U = {S1,S2,S3....Sn} where U is the "container" set. Initially each set contains a single element. After each operation each set will contain some numbers. We define the cost of a set as the number of unique elements in the set. For each query, we merge two unmerged set and then report the sum of cost of all the sets.  The most brute force approach is for each query, add all numbers in one of the sets and put them in the other set. This will take approximately O(nlogn) time per merge operation and since there can be about O(N) mergings (because after atmost N mergings there will be only 1 set left) this will take time proportional to O(n^2logn) time. For the report, part, we just return the sum of the sizes of the set. This part is O(1) as we only have to check the size of the new merged set as the sizes of the other sets haven't changed. This is obviously a poor algorithm for a seem...
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Segment Tree Beats - An introduction

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Getting prepared for RMO

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Regional Mathematics Olympiad The Regional Mathematics Olympiad (RMO) is the second stage in the selection for the selection of the Indian team to the IMO, the first being the pre-RMO. It is one of the tougher olympiads conducted across India, wherein only about 300 students are selected to sit for the Indian National Mathematics Olympiad(INMO). The main thing that makes RMO tough, at least for the common students, is twofolds -- firstly, the syllebus is quite a bit different than the normal class X or XI curriculum that is followed in schools, especially in areas like combinatorics and number theory, the latter being a topic that is scarcely touched upon. Secondly, in topics that are known to us, like geometry, the level of questions, even if they don't require advanced theorems and results, they need a level of creativity that cannot be mastered overnight. From hereon I will say what I did to clear it last year.. when I was studying in class IX. Firstly, I focused heavil...
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