PROBLEM LINK:

Practice
Contest

Author:Sunny Aggarwal
Tester:Shiplu Hawlader
Editorialist:Lalit Kundu

DIFFICULTY:

MEDIUM

PRE-REQUISITES:

Maths

PROBLEM:

K numbers denoted by array B from set S = [1,2,...N] are removed. Find the minimum number X such that X cannot be formed by picking a set of numbers from S.
1 ≤ N≤ 10⁹
1 ≤ K ≤ 5*10⁵

EXPLANATION:

If the minimum X is odd, second player wins, else first player wins.
So, we just need to find X.

If k == 0, then all numbers from 1 to (N * (N + 1)) / 2 are possible to form.
Consider a special case, X = 1 if 1 has been removed from set [1..N].

Based on this observation, we can first sort the array B in ascending order.
Fact: Let's say all numbers from 1 to i are available, then we can form every number till (i * (i + 1)) / 2.

Let's consider last reachable number till now is M. So now we want to form numbers M + 1, M + 2 and so on.
We have generated all numbers till M now and now we want to generate M + 1 and the new number that available number we get is say B_i + 1, we can't generate M + 1, if B_i + 1 > M + 1. In such a case X will be M + 1.
Or else, we know that all numbers between B_i + 1 and B_i+1 - 1(both inclusive) are available. So, we know that all numbers from M + 1 to M + S are also available, where S = sum of all numbers between B_i + 1 and $B_{i+1}$ - 1(both inclusive).

We do this for all unavailable numbers in sorted traversal to get the maximum unachievable sum M.

Complexity: O(K log K)

You might want to read this, for better clarity.

SOLUTIONS:

Setter's solution
Tester's solution