Deterministic and probabilistic algorithms, heuristics, and unsolvable problems

A deterministic algorithm always produces the same output for the same input, while a probabilistic (randomized) algorithm uses random choices, so it may give different results on different runs; for example, randomly sampling points can estimate the value of pi, getting closer with more samples. Some problems are technically solvable but impractical: for the traveling salesperson problem, finding the shortest route through many cities by checking every possible route takes far too long as the number of cities grows, because the number of routes grows faster than exponentially. In such cases, programmers use a heuristic, a shortcut that finds a good-enough solution quickly without guaranteeing the best one, like always visiting the nearest unvisited city next. Some problems cannot be solved by any algorithm at all. In 1936 Alan Turing proved that no algorithm can correctly decide, for every possible program and input, whether that program will eventually stop or run forever; this halting problem is the classic example of an undecidable problem.

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