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"""

Problem Statement

=================

Find the length of the longest Bitonic Sequence in a given sequence of numbers. A Bitonic sequence is a sequence of

numbers which are increasing and then decreasing.

Video

-----

* https://youtu.be/TWHytKnOPaQ

Analysis

--------

* Runtime O(n)

Reference

---------

* http://www.geeksforgeeks.org/dynamic-programming-set-15-longest-bitonic-subsequence/

"""

def longest_bitonic(sequence):

length_of_input = len(sequence)

increasing_sequence = [1] * length_of_input

decreasing_sequence = [1] * length_of_input

for i in range(1, length_of_input):

for j in range(0, i):

if sequence[i] > sequence[j]:

increasing_sequence[i] = max(increasing_sequence[i], increasing_sequence[j] + 1)

for i in range(length_of_input - 2, -1, -1):

for j in range(length_of_input - 1, i, -1):

if sequence[i] > sequence[j]:

decreasing_sequence[i] = max(decreasing_sequence[i], decreasing_sequence[j] + 1)

max_value = 0

for i in range(len(sequence)):

bitonic_sequence_length = increasing_sequence[i] + decreasing_sequence[i] - 1

max_value = max(max_value, bitonic_sequence_length)

return max_value

if __name__ == '__main__':

max_value = longest_bitonic([1, 4, 3, 7, 2, 1, 8, 11, 13, 0])

assert 7 == max_value # 1, 4, 7, 8, 11, 13, 0