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[NEW VIDEO] NP-COMPLETENESS - One Problem To Solve Them All
Added 2023-03-31 13:22:12 +0000 UTCHello! This video was especially challenging to make. It was very time consuming and expensive. It made me very grateful to have you supporting me and the channel. Truly, these videos would not be possible without you. As always, I hope you like it!
Comments
I have been traveling and needed time to hear everything more than once. I will try to clean up the line breaks without losing it all. Since time immemorial ~~ cooks, royalty and staff have fretted over seating charts. The Venn-like diagram 8 minutes into your NP video could be done quite differently. NP-complete problems could sit like a dense set of rational points amongst a greater sea of transcendental points, the exponentially difficult problems. NP-complete problems lay down as a grid that can surround any Exp point as closely as one wishes. "Give me an epsilon and I'll show you a delta ...." For each Exp problem, one can construct a sequence of NP-complete problems that asymptotically closes in on an intractable Exp problem; with an agreed upon cutoff. The feat can be accomplished using algorithms indexed by integers whose run-times scale no faster than polynomial time. In the cut-off neighborhood of any Exponentially difficult problem, one can find a NP-complete method that can match the Exp's output within any desired degree of accuracy. The discarded remainder contains high order differential relations one cannot discern, very much like the complex terms one ignores in a three hundred year old Taylor series. NP-complete problems form a grid over exponentially difficult problems. NP-complete problems have indices that can take on only integer values. They form a dense rational grid over the entire complex plane. The exponentially difficult problems remain disjoint from the NP-complete problems. Disjoint, set wise; yet infinitesimally close to each other, no matter where you look, or from where one starts. Using mathematical induction, one can always go from one vertex on a NP-complete Rational Grid to get to another vertex. Likewise, one could start from most anywhere in Euclid's Elements, and with a few definitions and translations of symbols, one can get from one of Euclid's elements to another (in polynomial time). When the Great Library of Alexandria burned, we lost thousands of fine scrolls. Euclid bequeaths a method that cannot be burned, a means for getting around and digging into fertile ground connecting different elements ~~~ indivisible vertices ~~ no matter from where one starts. If lost, bring out again your magnetic compass and the original Euclidean one twirling on her toe with tall legs and separated to sweep out a circle, one step, one radian out from the fixed point of a toe. With a twirling compass and an unruled rule, a tangent line, one can start from anywhere in Euclid's Elements and get to any other vertex, proof or theorem, Newton's entire Principia. With a few navigational tools and definitions, one can sail from most any chapter 'n verse and get to any other one. Euclid gave us a way of thinking. His whole Library can be derived and rise from the ashes, no matter from which page one starts. Euclid's flowing indivisible points, lines and surfaces know what to do as they show themselves as the cross-sections of volumes. The dense packing of rational vertices among transcendental points shows itself in Hawking's Information Paradox. A warm bath in a thermodynamic state of equilibrium plateaus at a state of maximum entropy. Thousands of states almost-at-equilibrium hover around those that have the greatest amount of entropy. The thermalized states have joined the transcendentals and become veiled at the end of infinite sequences and sums. In the immediate neighborhood of one of these transcendental limit points, one can always produce a rational pair of numbers that lies as close to a limit point as one wishes. The rational solutions remain everywhere dense around the exponentially difficult hard to solve problems. For all practical purposes, anything can be solved in polynomial time. Mixed states gets surrounded by pure states with a countable set of eigenstates. Good luck on the million dollar prize and whatever else befits a ringbearer. A lucky find can be like having talisman ~ an image of our solar system. With a heliocentric system populated with planets, moons and rings, and everything moving with a conserved 4-momentum and orientation, given such a right and potent vision, one has a polynomial way of discovering equal areas in equal time Keplerian orbits and even the shadows of Eclipses. Without heliocentricity and the mass of the sun as computational constraints, the celestial mechanics has an exponential degree of complexity. Without Apollo, Kepler could never have unraveled the tables of Tycho Brahe.
Scott Ready
2023-04-13 19:45:52 +0000 UTCAs a software developer myself having worked on some NP-complete and NP-hard problems, I especially enjoyed this one! Good work!
Vincent Zalzal
2023-04-02 21:30:03 +0000 UTCThis was a good one Jade! You always have a great way of making complex subjects clear, and this one really peaked my interest. I particularly liked your entrances and exits to certain scenes. Very much in keeping with demonstrating the process. 😄
John Shioli
2023-04-01 21:55:10 +0000 UTCNo comments? How about a probabilistic approach instead of a deterministic one?
Jim Allen
2023-04-01 02:29:43 +0000 UTC
