What is linear time algorithm and polynomial time algorithm? Explain with examples?
1. Well linear time means that the algorithm time complexity is O(n). This means roughly that if you double the size of the input, the algorithm will run twice as long. Linear searching is a very simple example of a linear time algorithm. 2. A polynomial time algorithm has a time complexity of O(n^k). Again if you double the size of the input, the algorithm will take 2^k as much time. Bubblesort e.g. is an O(n²) time complexity algorithm while the Floyd-Warshall path algorithm is O(n³).
What do you think about the paper "A Polynomial Time Algorithm for the Hamilton Circuit Problem"?
That particular author is publishing different versions of the algorithm since 2009. You can find him as #53 in the list on the P-versus-NP page, among lots and lots of similar papers claiming either P=NP or P!=NP. It is almost certainly wrong, just like all the previous versions were.
Is there any polynomial-time algorithm which when given a valid/attainable state of chess board as input, outputs any one valid sequence of moves required to reach the input state?
It is somewhat problematic to make statements regarding asymptotic complexity for games where there are large constants which can be used to generate a (useless) constant time solution to the problem.Standard rules of chess allow for an infinite game (with a repeating sequence). The board size gives an upper bound of [math]13^{64}[/math], but most of these positions will not be valid. We can deal with the infinite game as long as the problem definition says shortest sequence of moves. AFAIK there is no computationally easy decision function to determine if a position is valid or invalid, though certainly some positions are trivially invalid. That is I suspect that finding the sequence of moves is the same as determining validity.Overall I think that if we discount the constant bounds as “cheating” then I think there is no polynomial time algorithm (*). That is we are searching a tree with a very high branching factor for a sequence which maybe very long. As the game removes pieces from the board their maybe little information available in the input to prune non-viable paths.(*) This isn’t a decision problem, so the possibility that [math]P=NP[/math] is not relevant.
What are Polynomial Time Algorithms?
An algorithm is said to be of polynomial time if its running time has an polynomial upper bound, which is in the size of the input for the algorithm, i.e., [math]T(n)=O(n^k)[/math] for some positive integer [math]k[/math]. Problems for which a polynomial time algorithm exists belong to the complexity class P, which is central in the field of computational complexity theory. Polynomial time algorithms are said to be "fast". Most familiar mathematical operations such as addition, subtraction, multiplication, and division, as well as computing square roots, powers, and logarithms, can be performed in polynomial time.
Abstract Algebra - Division Algorithm for Polynomials?
Using synthetic division: (2i).|.1...0...(3+i)...1 ...........2i......-4..(-2-2i) .....--------------------------- .......1..2i...(-1+i)..(-1-2i). So, q(x) = x^2 + (2i)x + (-1+i), and r(x) = -1 - 2i. I hope this helps!
Is it possible to show that no polynomial time algorithm exists for solving a given problem?
Yes. There are many problems that can be proven to not be in P; anything in exptime-hard or exptime-complete, for example. See http://cs.stackexchange.com/a/25... for one example.Neither; NP has a definition. NP (complexity) NP is the class of problems we can verify a solution when handed to us in polynomial time.See EXPTIME as well: Problems that are EXPTIME-complete might be thought of as the hardest problems in EXPTIME. Notice that although we don't know if NP is equal to P or not, we do know that EXPTIME-complete problems are not in P; it has been proven that these problems cannot be solved in polynomial time, by the time hierarchy theorem.
Are there any algorithms of the order O(sqrt(n))?
Yes, and this is not just a technicality.Technicalities first: For example, [math]O(\sqrt{n})[/math] is the time complexity of the naive algorithm that tests whether [math]n[/math] is a prime by checking all divisors from 2 to [math]\lfloor\sqrt{n}\rfloor[/math].Why do I call the above case a technicality? Because in that case the variable [math]n[/math] is not the actual input size. The input size is proportional to [math]\log n[/math], and thus the above algorithm isn’t even polynomial in the input size.But even if the [math]n[/math] in your question is the input size, the answer remains “yes”. There is quite a lot of theory behind algorithms that use a sublinear amount of time and/or space. Such algorithms can actually do many useful thing.For example, suppose you have a collection of elements. The number of elements is [math]n[/math], and [math]n[/math] is really really large. One question you may ask is the question whether all elements in your collection are distinct. Obviously, the worst-case time complexity of any exact algorithm must be linear, as you have to examine all elements to be sure that the answer is “yes, they are all distinct”. Still, we can do something useful in sublinear time: with high probability we can tell apart two situations that are sufficiently distinct. In particular, we can distinguish the following two cases:all [math]n[/math] elements are distinctthe number of distinct elements is at most [math](1-\varepsilon)n[/math] (that is, duplicates form at least a small constant fraction of the population)How can we do that? It turns out that cleverly sampling [math]O(\sqrt{n})[/math] elements from the population is sufficient. The intuition behind the proof is closely related to the birthday paradox. For more details, see lecture 1 in 6.893 Materials.
What are some interesting facts about POLYNOMIALS?
Polynomials have grisly stories behind it. Before I tell you the story of the many deaths and feuds of those who studied polynomials, I will tell you some elementary but important theorems of polynomials. Factor theorem: For some polynomial p, if p(k) is 0, then (x-k) divides p(x) for all x. Eisenstein's criterion: If a polynomial p(x)=(a_k)x^k + (a_k-1)x^k-1 + ... + (a_0) with integers a_i, and there exists a prime p such that p divides a_j for all j=0,1,..,k-1 and p does not divide a_k and p^2 does not divide a_0, then p(x) is irreducible over the integers. Abel-Ruffini theorem: Quintic equations have no general solutions (unlike cubic and quadratic equations). Cohn's irreducibility criterion: If there exists a polynomial p(x)=(a_k)x^k + (a_k-1)x^k-1 + ... + (a_0) and there exists a prime q which can be expressed in base 10 as q=(a_k)10^k + (a_k-1)10^k-1 + ... + (a_0), then the polynomial is irreducible over the integers. Now for the story. Once upon a time, there was a mathematician who was particularly in polynomials. He had found the general solution to cubic polynomial equation, and nobody else knew about it. With this formula, he went to a particular mathematics competition for which two people would race to solve a given cubic polynomial. He made a lot of money from it, but there came a time when he had to tell someone close to him this formula. He did, upon the promise of keeping this formula a secret. Soon this friend broke this promise, and the mathematician's career in the polynomial competitions spiraled down. He swore vendetta against his ex-friend, and got him killed. A few months later, the mathematician died too. Many years later, two mathematicians Abel and Ruffini teamed up to prove that quintic equations have no general solutions. This would be a fantastic discovery and it would bring them fame and glory. Unfortunately, Abel died before they could complete the problem, and Ruffini died too. He left behind an incomplete proof, which was very close to its result. Polynomials bring bad luck. Caution must be held when studying them.
Math/Algorithms geniuses need your HELP!!!?
The problem of searching for cycles in graphs arises naturally in financial trading applications. Consider a firm that trades shares in n different companies. For each pair i != j, they maintain a trade ratio Rij, meaning that one share of i trades fo Rij shares of j. Here we allow the rate R to be fractional: that is, Rij = 2/3 means that you can trade three shares of i to get two shares of j. A trading cycle for a sequence of shares i1, i2, ..., ik consists of successively trading shares in company i1 for shares in company i2, then shares in company i2 for shares i3, and so on, finally trading shares in ik back to shares in company i1. After such a sequence of trades, one ends up with shares in the same company i1 that one starts with. Trading around a cycle is usually a bad idea, as you tend to end up with fewer shares than you started with. But occasionally, for short periods of time, there are opportunities to increase shares. We will call such a cycle an opportunity cycle, if trading along the cycle increases the number of shares. This happens exactly if the product of the ratios along the cycle is above 1. In analyzing the state of the market, a firm engaged in trading would like to know if there are any opportunity cycles. Give a polynomial-time algorithm that finds such an opportunity cycle, if one exists. HINT: Bellman-Ford Algorithm I am completely lost on this problem. If someone would be kind enough to not only solve this but tell me how you solved it. Thanks