We revisit classical problems about searching in totally monotone and Monge matrices, which have many applications in computational geometry and other areas. We present a number of new results, including the following:

- A randomized algorithm that finds the row minima in an n x n Monge staircase matrix in O(n) expected time; this improves a longstanding O(n alpha(n)) bound by Klawe and Kleitman (1990) for totally monotone staircase matrices.
- A randomized algorithm that reports the K smallest elements (in an arbitrary order) in an n x n Monge (complete or staircase) matrix in O(n + K) expected time; this improves and extends a previous O(n + K log n) algorithm by Kravets and Park [SODA'90].
- A randomized algorithm that reports the K smallest elements (in an arbitrary order) in an n x n totally monotone (complete) matrix in O(n + K log^* n) expected time.
- A randomized algorithm that reports the k_i smallest elements in the i-th row, for every i, in an n x n totally monotone (complete) matrix in O((n + K) log^* n) expected time, where K = sum_i k_i.
- A randomized algorithm that finds the row minima in an n x n totally monotone "v-matrix" in O(n alpha(n) log^* n loglog n) expected time; this answers an open question by Klawe [SODA'90]. The log^* n factor can be removed in the Monge case.

We revisit classical problems about searching in totally monotone matrices, which have many applications in computational geometry and other areas. In a companion paper, we gave new (near-)linear-time algorithms for a number of such problems. In the present paper, we describe new subquadratic results for more basic problems, including the following:

- A randomized algorithm to select the K-th smallest element in an n x n totally monotone matrix in O(n^{4/3} polylog n) expected time; this improves previous O(n^{3/2} polylog n) algorithms by Alon and Azar [SODA'92], Mansour, Park, Schieber, and Sen (1993), and Agarwal and Sen (1996).
- A near-matching lower bound of Omega(n^{4/3}) for the problem (which holds even for Monge matrices).
- A similar result for selecting the k_i-th smallest in the i-th row for all i.
- In the case when all k_i's are the same, an improvement of the running time to O(n^{6/5} polylog n).
- Variants of all these bounds that are sensitive to K (or sum_i k_i).

These matrix searching problems are intimately related to problems about arrangements of pseudo-lines. In particular, our selection algorithm implies an O(n^{4/3} polylog n) algorithm for computing incidences between n points and n pseudo-lines in the plane. This improves, extends, and simplifies a previous method by Agarwal and Sharir [SODA'02].

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Timothy Chan (Last updated Aug 2023)