For example, 'how many of the first 100 integers are prime? Optimization problems require the identification of the best solution to a search problem from a given set of solutions. Info) ,mentioned in the page on Abstraction, is a good example of this.
Some classes have lessons spread unevenly across the cycle, some teachers teach more or fewer lessons than their expected allocations, some classes are shared between teachers, some lessons may have to be scheduled outside of normal time etc.
All of these things are, strictly speaking, not working to the original constraints for the problem.
There is some explanation of the software in the site.
The input would be the current state of the cube (with the facility to ignore the orientation and/or position of some of the cubies) and a few parameters like restrictions on which faces can be turned. In addition to the search component, the program must also solve the decision problem ('is this solution the shortest for the given inputs? If a problem is algorithmic and computable, being able to produce a solution may depend on the size of the input or the limitations of the hardware used to implement it.
When n is small, it's easy to see how this approach would miss quicker routes if it having to start with a long journey.
Heuristic algorithms are based on the value of knowledge, experience and judgement in solving intractable problems. If a problem can be solved in polynomial time, it is said to belong to the P class of problems. For some intractable problems, you can verify that the solution is correct using a P-type algorithm.What we do is pick a place to start from, try every possible route, store the total cost of each journey and compare them when all routes have been found.For a map with 4 cities, it's quite easy to see what we would have to do, All this means that there are (n-1)! Fine if we start with a small number of cities but not workable when the number of cities increases.Algorithms of the class O(n are solutions to tractable problems.Some problems can only be solved with algorithms whose execution time grows too quickly in relation to their input to be solved in polynomial time. The Travelling Salesperson Problem is the example most used to describe intractability.In such cases, sub-optimal solutions can be found more quickly - in polynomial time.Almost any school timetable will, for this reason, contain some oddities. There are problems which lend themselves to abstraction, can be broken down into rules or steps such that an algorithm can be written that would solve the problem. An example of a non-algorithmic problem would be 'What is happiness? These kind of problems are riddled with difficulty - there is a lot of material written on the subject - some of it even scientific in its methodology.If you explore this a little on the WWW, you will get a sense of just how fundamentally difficult that question is to answer without posing yet more questions of the answer.A decision problem is a problem for which the answer for every valid input is yes or no.Deciding whether a number is prime, odd or even are both examples of decision problems.