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Represent a $d$-dimensional box by its side-length vector and allow axis permutations. A box fits another when its permuted dimensions are all strictly smaller. Sorting each vector reduces containment to coordinatewise strict comparison, by the sorted-coordinate criterion for box containment. For an input-order-preserving subsequence, the longest chain ending at position $j$ has one plus the greatest chain length at an earlier fitting position. This last-box decomposition proves the dynamic programming recurrence. Sorting all dimensions and testing all ordered index pairs takes $O(nd\log d+n^{2}d)$ running time. Arbitrary reordering is a different convention, handled by the containment Directed acyclic graph.

Sort the two dimension vectors separately into ascending order, $a_1\le \cdots \le a_d$ and $b_1\le \cdots \le b_d$. Then return true exactly when $a_i<b_i$ for every $i$. Sorting costs $O(d\log d)$ and the coordinate comparisons cost $O(d)$.

Sufficiency follows by matching corresponding ranks and translating that rank matching back to the original axes. For necessity, suppose $a_k\ge b_k$ for some $k$. The $d-k+1$ dimensions $a_k,\dots ,a_d$ must all be assigned to strictly larger target dimensions. None of the first $k$ sorted target dimensions is large enough, leaving at most $d-k$ eligible slots. The pigeonhole principle gives a contradiction. This proves the sorted-coordinate criterion for box containment, including ties: equality fails the required strict fit.