Continued from yesterday.
Mathematicians sometimes employ the imagery of a three-dimensional landscape to represent the challenge of solving complex problems. Imagine such a landscape, replete with peaks and valleys, stretching to infinity. Now imagine we want to find the highest point on that landscape, but we can only see the points that we’ve visited.
We’re standing on the highest point we’ve found, but are we as high as we could be? Are we really on a molehill, thinking it’s a mountain only because we’ve not yet discovered what real heights look like? How can we explore this terrain without spending an eternity randomly trying new locations?
Mathematicians have conjured a variety of mechanisms for grappling with this problem. They deploy algorithms, for example, that search for higher ground. They grapple with not only “solving” the problem represented by the theoretical landscape, but of inventing methods that can solve a variety of such problems more efficiently than a random walk. [Read more...]