The.Algorithm.Design.Manual.Springer-Verlag.1998

Correctness
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Correctness
It is seldom obvious whether a given algorithm correctly solves a given problem. This is why correct algorithms usually come
with a proof of correctness, which is an explanation of why we know that the algorithm correctly takes all instances of the
problem to the desired result. In this book, we will not stress formal proofs of correctness, primarily because they take
substantial mathematical maturity to properly appreciate and construct. However, before we go further it is important to
demonstrate why ``it's obvious'' never suffices as a proof of correctness and usually is flat-out wrong.
To illustrate, let us consider a problem that often arises in manufacturing, transportation, and testing applications. Suppose we
are given a robot arm equipped with a tool, say a soldering iron. In manufacturing circuit boards, all the chips and other
components must be fastened onto the substrate. More specifically, each chip has a set of contact points (or wires) that must
be soldered to the board. To program the robot arm to perform this soldering job, we must first construct an ordering of the
contact points, so that the robot visits (and solders) the first contact point, then visits the second point, third, and so forth until
the job is done. The robot arm must then proceed back to the first contact point to prepare for the next board, thus turning the
tool-path into a closed tour, or cycle.
Since robots are expensive devices, we want to find the tour that minimizes the time it takes to assemble the circuit board. A
reasonable assumption is that the robot arm moves with fixed speed, so that the time it takes to travel between two points is
the same as its distance. In short, we must solve the following algorithm problem:
Input: A set S of n points in the plane.
Output: What is the shortest cycle tour that visits each point in the set S? You are given the job of programming the robot arm.
Stop right now and think about an algorithm to solve this problem. I'll be happy to wait until you find one.
Several possible algorithms might come to mind to solve this problem. Perhaps the most popular idea is the nearest-neighbor
heuristic. Starting from some point , we walk first to its nearest neighbor . From , we walk to its nearest unvisited
neighbor, thus excluding only as a candidate. We now repeat this process until we run out of unvisited points, at which time
we return to to close off the tour. In pseudocode, the nearest-neighbor heuristic looks like this:
Figure: A good example for the nearest neighbor heuristic
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