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**Longest** **increasing** **subsequence** or LIS problem is a classical dynamic programming problem which refers to finding the length of the **longest** **subsequence** from an array such that all the elements of the sequence are in strictly **increasing** order. **Longest Increasing Subsequence** [This section was originally written by Anand Sarwate] 33.1 Introduction In this paper we will investigate the connection between **random** matrices and ﬁnding the **longest increasing subsequence** of a **permutation**. We will introduce a model for the problem using a simple card game. a **longest** **increasing** **subsequence**, since they ... **subsequence** of the original **permutation**! An interacting particle process Start with zero particles. At each step, pick a **random** point U in [0,1]; simultaneously, let the nearest particle (if any) to the right of U disappear. Lemma 1 implies: the number of particles af-. Given [10, 9, 2, 5, 3, 7, 101, 18], The **longest increasing subsequence** is [2, 3, 7, 101], therefore the length is 4 After you are done modifying the input array in-place, return the new length of the array Similarly, decreasing order sequence is considered Bitonic with the **increasing** part as empty This function searches for the **longest** ascending **subsequence** of a **permutation** using a dynamic.