D. Lexichromatography
combinatorics; dfs and similar; dsu; graphs; two pointers
Pak Chanek loves his faculty, the Faculty of Computer Science, University of Indonesia (Fasilkom). He wants to play with the colours of the faculty's logo, blue and red.There is an array \(a\) consisting of \(n\) elements, element \(i\) has a value of \(a_i\). Pak Chanek wants to colour each element in the array blue or red such that these following conditions are satisfied: If all blue elements are formed into a subsequence\(^\dagger\) and so are all the red elements, the blue subsequence is strictly less than the red subsequence lexicographically\(^\ddagger\). Array \(a\) does not have any subarray that is imbalanced. A subarray is imbalanced if and only if there is a value \(k\) such that the absolute difference between the number of blue elements with value \(k\) and the number of red elements with value \(k\) in this subarray is \(2\) or more. Note that it is possible to colour every element of the array the same colour. How many different colourings satisfy all those conditions? Since the answer can be very big, print the answer modulo \(998\,244\,353\). Two colourings are different if and only if there is at least one element that is blue in one colouring, but red in the other.\(^\dagger\) A subsequence of an array is a sequence that can be obtained from the array by deleting some elements (possibly none), without changing the order of the remaining elements.\(^\ddagger\) Let \(p\) and \(q\) be two different sequences. Sequence \(p\) is said to be lexicographically less than sequence \(q\) if and only if \(p\) is a prefix of \(q\) or there is an index \(i\) such that \(p_j=q_j\) holds for every \(1\leq j<i\), and \(p_i<q_i\). In particular, an empty sequence is always lexicographically less than any non-empty sequence.
The first line contains a single integer \(n\) (\(1 \leq n \leq 2\cdot10^5\)) — the size of array \(a\).The second line contains \(n\) integers \(a_1,a_2,a_3,\ldots,a_n\) (\(1\leq a_i\leq2\cdot10^5\)).
An integer representing the number of different colourings that satisfy all of the problem's conditions, modulo \(998\,244\,353\).
In the first example, the \(3\) ways for colouring all elements from index \(1\) to index \(8\) are: Blue, red, red, blue, blue, red, red, blue. Blue, red, red, red, blue, blue, red, blue. Red, red, blue, blue, blue, red, red, blue. As an example, if we colour the elements from index \(1\) to index \(8\) to be red, red, blue, red, red, blue, blue, blue, it is not valid colouring, because for subarray \(a[2..6]\), there are \(0\) blue elements with value \(3\) and \(2\) red elements with value \(3\), making subarray \(a[2..6]\) an imbalanced subarray.