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## Discussion of Problem 1003. Parity

Solution
Posted by Alikhan Zimanov 12 Jul 2020 13:56
Let a, ... , a[n] be the sequence written by the friend and pref[i] be the xor of the prefix a[1...i]. Then a question (l, r, t) (which means the parity of the number of ones on the segment from l to r is t) can be represented as (pref[r] xor pref[l - 1] = t). Let's build a graph with two vertices for each prefix (thus, there will be 2 * (n + 1) vertices). Using compression, we actually need only at most 2 * m vertices (where m is the number of questions). Let the vertices corresponding to the prefix i be f0(i) and f1(i). Now, let's add all edges f0(i) ~ f1(i). For a question (pref[r] xor pref[l - 1] = t) if t = 1, we add two edges f0(l - 1) ~ f0(r) and f1(l - 1) ~ f1(r), else if t = 0, we add edges f0(l - 1) ~ f1(r) and f1(l - 1) ~ f0(r). To check whether a sequence of questions from 1 to x is achievable we just need to check whether our current graph is bipartite. This can be done by simply doing dfs after each question.
Re: Solution
Posted by springWaltz 31 Aug 2021 19:27
Nice!
Re: Solution
Posted by Abdul Matin 18 Jun 2022 14:50
Can you please elaborate in a few words? I actually didn't get the significance of having two vertices for each prefix and how it's solving the problem.