nodes and no edge directly links ghost nodes. {| class="wikitable" |- style="text-align:center;" class="wikitable" |+ Symmetric path weights ! !! A !! B !! C !! A′ !! B′ !! C′ |- style="text-align:center;" ! A | || || || −w || 6 || 5 |- style="text-align:center;" ! B | || || || 1 || −w || 4 |- style="text-align:center;" ! C | || || || 2 || 3 || −w |- style="text-align:center;" ! A′ | −w || 1 || 2 || || || |- style="text-align:center;" ! B′ | 6 || −w || 3 || || || |- style="text-align:center;" ! C′ | 5 || 4 || −w || || || |} The weight −w of the "ghost" edges linking the ghost nodes to the corresponding original nodes must be low enough to ensure that all ghost edges must belong to any optimal symmetric TSP solution on the new graph (w=0 is not always low enough). As a consequence, in the optimal symmetric tour, each original node appears next to its ghost node (e.g. a possible path is ) and by merging the original and ghost nodes again we get an (optimal) solution of the original asymmetric problem (in our example, ). Analyst's problem There is an analogous problem in geometric measure theory which asks the following: under what conditions may a subset E of Euclidean space be contained in a rectifiable curve (that is, when is there a curve with finite length that visits every point in E)? This problem is known as the analyst's travelling salesman problem. Path length for random sets of points in a square Suppose are independent random variables with uniform distribution in the square , and let be the shortest path length (i.e. TSP solution) for this set of points, according to the usual Euclidean distance. It is known that, almost surely, where is a positive constant that is not known explicitly. Since (see below), it follows from bounded convergence theorem that , hence lower and upper bounds on follow from bounds on . The almost sure limit as may not exist if the independent locations are replaced with observations from a stationary ergodic process with uniform marginals. Upper bound One has , and therefore , by using a naive path which visits monotonically the points inside each of slices of width in the square. Few proved , hence , later improved by Karloff (1987): . Fietcher