Here we introduce the "longest increasing subsequence" problem, which is, given an array A, try to find the longest length of any subset of A that is strictly increasing. I do an example in the video. It's easily shown then that the LIS problem can be solved in O(2^n) time by examining every subset. However, we can use dynamic programming here by noticing that "close" subsets are not going to have wildly different LIS values. So we develop a recurrence for the LIS problem, and then make an "iterative" algorithm that saves previous calculations in a table. We then show that it can be solved in O(n^2) time. Chapters: 0:00 - Intro 0:39 - What is the problem? 5:04 - Brute force algorithm 10:05 - Can we go faster? 12:39 - Defining the recurrence 22:09 - How the memoization works here 22:57 - Iterative algorithm 33:05 - What is the runtime of the DP algorithm? Donation (appears on streams): https://streamlabs.com/easytheory1/tip Paypal: https://paypal.me/easytheory Patreon: https://www.patreon.com/easytheory Discord: https://discord.gg/SD4U3hs Youtube Live Streaming (Sundays) - subscribe for when these occur. Merch: Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122 Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1 Gold Supporters: Micah Wood Silver Supporters: Timmy Gy ▶SEND ME THEORY QUESTIONS◀ ryan.e.dougherty@icloud.com ▶ABOUT ME◀ I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes. ... https://www.youtube.com/watch?v=ggUN8TijNiU

Here we present a research problem, which is determining the 5th Busy Beaver number. This problem asks how long a Turing Machine with 5 states can run before it eventually stops, and only a few machines remain. Let's solve it together! Here is the github repo of the current best-known results: https://github.com/danbriggs/Turing Thanks to the following supporters of the channel for helping support this video. If you want to contribute, links are below. Dolev Abuhazira, Josh Hibschman, Micah Wood, Morgan Jones, Patrik Keinonen, Simone Glinz, Tao Su, Timothy Gorden, unit220, Valentine Eben Easy Theory Website: https://www.easytheory.org Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join Donation (appears on streams): https://streamlabs.com/easytheory1/tip Paypal: https://paypal.me/easytheory Patreon: https://www.patreon.com/easytheory Discord: https://discord.gg/SD4U3hs Merch: Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122 Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1 ▶ABOUT ME◀ I am a professor of Computer Science, and am passionate about it. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes. The views expressed in this video are not reflective of any of my current or former employers. ... https://www.youtube.com/watch?v=31IXLjRoMw4

Here we give an overview of the Bellman-Ford algorithm for computing shortest paths in graphs where there are negative-weights. Note that if there is a negative-weight cycle in the graph, then no shortest paths can exist. What the B-F algorithm does is find the shortest weights, as well as detect if a negative-weight cycle exists. Thanks to the following supporters of the channel for helping support this video. If you want to contribute, links are below. Names are listed in alphabetical order by surname. Platinum: Micah Wood Silver: Timmy Gy, Josh Hibschman, Patrik Keinonen, Travis Schnider, and Tao Su Easy Theory Website: https://www.easytheory.org Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join Donation (appears on streams): https://streamlabs.com/easytheory1/tip Paypal: https://paypal.me/easytheory Patreon: https://www.patreon.com/easytheory Discord: https://discord.gg/SD4U3hs Merch: Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122 Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1 ▶SEND ME THEORY QUESTIONS◀ ryan.e.dougherty@icloud.com ▶ABOUT ME◀ I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes. ... https://www.youtube.com/watch?v=KaiV5v8hc8E

Full Theory of Computation Lecture playlist: https://www.youtube.com/watch?v=OPaB-rpKhZ0&list=PLylTVsqZiRXMiTARmrsxCWU2RahyKB_Ae&index=1&t=1s Lecture "a la carte" playlist: https://www.youtube.com/watch?v=gO9Ho9Dpu8k&list=PLylTVsqZiRXP51EfJWv8cxD6wIFTZQD9_&index=1 Here we prove that regular languages are closed under complement (i.e., if R is a regular language, then Sigma* - R is also a regular language). This is proved because a DFA for R has exactly one computation on any string, so flipping the final and non-final states of the DFA results in a DFA that has the complement of R as its language. Easy Theory Website: https://www.easytheory.org Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join Donation (appears on streams): https://streamlabs.com/easytheory1/tip Paypal: https://paypal.me/easytheory Patreon: https://www.patreon.com/easytheory Discord: https://discord.gg/SD4U3hs #easytheory Youtube Live Streaming (Sundays) - subscribe for when these occur. Social Media: Facebook Page: https://www.facebook.com/easytheory/ Facebook group: https://www.facebook.com/groups/easytheory/ Twitter: https://twitter.com/EasyTheory Merch: Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122 Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1 ▶SEND ME THEORY QUESTIONS◀ ryan.e.dougherty@icloud.com ▶ABOUT ME◀ I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes. ... https://www.youtube.com/watch?v=xgldUh2M_Ac