ple: Input: [10,9,2,5,3,7,101,18] Output: 4 Explanation: The longest increasing subsequence is [2,3,7,101], therefore the length is 4. Time complexity: First algorithm is O(n^2). Second algorithm is O(nlogx) where x is the max element in the list Third algorithm is O(nlogn) Space complexity: First algorithm is O(n) Second algorithm is O(x) where

Given an unsorted array of integers, find the length of
longest increasing subsequence.
Example:
Input: [10,9,2,5,3,7,101,18]
Output: 4
Explanation: The longest increasing subsequence is [2,3,7,101], therefore the
length is 4.
Time complexity:
First algorithm is O(n^2).
Second algorithm is O(nlogx) where x is the max element in the list
Third algorithm is O(nlogn)
Space complexity:
First algorithm is O(n)
Second algorithm is O(x) where x is the max element in the list
Third algorithm is O(n)
"""

def longest_increasing_subsequence(sequence):
"""
Dynamic Programming Algorithm for
counting the length of longest increasing subsequence
type sequence: list[int]
rtype: int
"""
length = len(sequence)
counts = [1 for _ in range(length)]
for i in range(1, length):
for j in range(0, i):
if sequence[i] > sequence[j]:
counts[i] = max(counts[i], counts[j] + 1)
print(counts)
return max(counts)

def longest_increasing_subsequence_optimized(sequence):
"""
Optimized dynamic programming.