Problem A. Sherlock and Parentheses

Category: Ad-Hoc, Math, Greedy

Let's define n = min(L,R). Then the claim is that the sequence ()()...() consisting of n pairs of "()" will give us the maximum answer n*(n+1) / 2

Proof:

Let's look at some sequence T = (S) where S is also a balanced parenthesis sequence of length m. Let's denote by X(S), the number of sub-strings of S that are also balanced and by X'(S) the number of balanced parenthesis sub-strings that include the first and last characters of S (In fact this will be just 1). Then X(T) = X(S) + X'(S).

It should be easy to prove that X'(S) <= X(S) (simply because X'(S) is a constrained version of X(S))

If we rearrange our sequence T to make another sequence T' = "S()", then X(T') >= X(S) + X'(S). The greater than equal to sign comes because we might have balanced sub-strings consisting of a suffix of S and the final pair of brackets.

Thus, we have that X(T') >= X(S) + X'(S) = X(T). Thus by rearranging the sequence in this manner, we will never perform worse. If we keep applying this operation on each string, we will finally end up with a sequence of the form "()()...()" possibly followed by non-zero number of "(" or ")"

Problem B. Sherlock and Watson Gym Secrets

Let's look at all valid pairs (i,j) in which i is fixed. Then, if i^a mod k = r then j^b mod k = (k-r) mod k.

There are only k possible values of the remainder after division. Since k is small enough, if we can quickly determine how many numbers i exist such that i^a mod k = r and j^b mod k = (k-r) mod k then we can simply multiply these quantities and get our answer. Thus the reduced problem statement is —

Find the number of numbers i <= N such that i ^ a mod k = r for all r = 0 to k-1.

Let's denote by dp1[r] = number of i <= N such that i ^ a mod k = r. We will try to compute dp1 from r = 0 to k-1 in time O(K log a)

Let's make another observation. If i ^ a mod k = (i + k) ^ a mod k = (i + 2k) ^ a mod k = (i + x*k) ^ a mod k.

Proof:

(i + x*k) ^ a mod k = (i mod k + (x * k) mod k) ^ a mod k

= (i + 0) ^ a mod k = i ^ a mod k

Now i + x * k <= n. This implies that x <= (n — i) / k.