Good evening Codeforces, let me briefly describe solutions for all problems of today's morning yandex algorithm round. Thanks everyone who participated, I apologize for making problems a bit too tough and viscous for a 100 minutes contest. Anyway, I hope everyone found something interesting.

I would like to thank lperovskaya for organising this competition and managing Yandex.contest, snarknews and SergeiFedorov for their help with the problemset, Endagorion, PavelKunyavskiy, AleX, glebushka98, gustokashin, map and boris for testing. Special thanks to Gassa and my girlfriend Marina Kruglikova for fixing mistakes and disambiguations in English and Russian statements.

Let's get it started.

Problem A. Odysseus Sails Home.

There is no tricky idea behind this problem: one just needs to check if the vector (x_f - x_s, y_f - y_s) can be represented as a convex combination of vectors (x_i, y_i). One of the easiest approaches for the general case is to try all pairs of wind vectors and check if the target vector lies inside the cone they form. However, the devil is in the details. One shouldn't forget to:

• Check if it's possible to get to Ithaca using only one wind direction;

• Special if for case (x_f, y_f) = (x_s, y_s);

• Ignore wind vectors (x_i, y_i) = (0, 0);

• Avoid usage of doubles — everything fits in long long.

Time complexity: O(n²). O(nlogn) solution already exists.

Problem B. Chariot Racing.

For the fixed value t = const we can easily calculate the intersection of all segments (chariots) as max(0, min(b_i + v_i * t) - max(a_i + v_i * t)). The problem was to find maximum for t ≥ 0.

Both min(b_i + v_i * t) and max(a_i + v_i * t) are convex functions. min(b_i + v_i * t) is concave upward, because it's derivative only decreases, as faster chariots overtake slower one. Similar max(a_i + v_i * t) is convex down. This means function min(b_i + v_i * t) - max(a_i + v_i * t) is concave upward, and this, in turn is sufficient condition for applying ternary search.

Ternary search is enough to solve the problem, but the solution which does binary search on derivative is faster and more stable. We need to find the maximum t such that the chariot i with minimum b_i + v_i * t goes faster then the chariot j with maximum a_j + v_j * t.

The only special case (for some solutions only) was n = 1.

Time complexity: O(nlogMaxC).

Problem C. Equality and Roads.

First check if the graph is connected. If no, print <<\t{NO}>> for all queries.

For connected graph count the number of connected components if only 0-edges are allowed (denote it as a) and the number of connected components if only 1-edges are allowed (denote it as b). Then, for the given x it's possible to construct the desired span if and only if condition b - 1 ≤ x ≤ n - 1 - a holds.

Lets proof the above statement. It's pretty obvious that this condition is necessary, but the sufficiensy isn't that clear. In my opinion, the easiest way is to present the algorithm. It consists of five steps:

1. Create DSU, add all 1-edges.

2. Add all 0-edges, remember which of them caused joins. There will be b - 1 such edges.

3. Clear the DSU, add all 0-edges remembered on step 2.

4. Add 0-edges until there are exactly x of them in the current span.

5. Add 1-edges until the graphs becomes connected. That will always happen because of the step 1 and 2.

Time complexity: O(n).

Problem D. Sequences of Triangles.

Thanks to snarknews — the author and developer of this problem.

Let f(x) be the longest sequence that ends on the triangle with the hypotenuse of length x. If we generate

Problem E. Strong Squad.

Thanks to SergeiFedorov — the author of this problem.

Problem F. Lexicographically Smallest String.