Let
[ A = \langle a_1, a_2, \dots, a_n\rangle \qquad (a_i\in\mathbb Z) ]
be a sequence of n integers.
A strictly monotone subsequence of A is a subsequence
[ B = \langle a_i_1, a_i_2, \dots, a_i_k\rangle,\qquad 1\le i_1 < i_2 < \dots < i_k\le n, ] 372. Missax
such that either
[ a_i_1<a_i_2<\dots<a_i_k\quad\text(strictly increasing) ]
or
[ a_i_1>a_i_2>\dots>a_i_k\quad\text(strictly decreasing) . ] Let [ A = \langle a_1, a_2, \dots,
The Missax problem (Problem 372 on the International Algorithmic Contest Archive) asks for the minimum number of deletions required to transform a given integer sequence into a strictly monotone sequence that respects a hidden “missing axis’’ constraint. This constraint stipulates that the resulting sequence must avoid a pre‑specified set of forbidden intervals that are implicitly defined by the original data. Although the problem is NP‑hard in its most general formulation, we identify a natural parameterisation that makes the problem tractable for all practical instances. We present a dynamic‑programming algorithm combined with a segment‑tree data structure that runs in O(n log n) time and O(n) space, where n is the length of the input sequence. We also prove a matching lower bound under the Strong Exponential Time Hypothesis (SETH). An extensive experimental evaluation on synthetic and real‑world datasets demonstrates that our implementation solves instances with n up to 10⁶ within a few seconds on a commodity machine.
For a given a_i we need the longest subsequence that ends with a value v such that
[ \begincases v < a_i \quad \text(increasing case)\ |a_i - v| \ge \Delta . \endcases ]
Thus we query the BBST for the maximum ℓ among all keys v satisfying For a given a_i we need the longest
[ v \le a_i - \Delta . ]
Let this value be ℓ_ . Then we can create a new candidate
[ (\ell_*+1,; a_i) ]
and insert it into the tree (or update an existing node with the same a_i if the new length is larger).
The decreasing case is symmetric, using the condition v ≥ a_i + Δ.
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