The key observation is that in an optimal (m,r)-light salesman path, the portions of the path inside each of the four subsquares must also be optimal. We maintain a table that contains optimal solutions to all subproblems specified by (a) a quadtree square, (b) a multiset of portals on the edges of the quadtree square, and (c) a valid order in which the portals in (b) are visited. The number of ways to choose a multiset of portals τ is at most (m+3)4r and the number of ways to traverse a multiset of portals without self-intersection is no greater than (4r)!. Thus the size of the table is at most O(nlogn (m+3)4r (4r)!).
The table is filled up in a bottom-up fashion, starting at the leaves of the quadtree. A square at a leaf contains at most one single node. Given a multiset of portals and a valid order, we can solve the subproblem optimally in O(2r) time because the single node can be traversed along one of the O(2r) paths in the square. A square at a non-leaf node has four internal edges (one from each of its four subsquares). The number of ways to choose a multiset of portals on the four internal edges is at most (m+3)4r and the number of ways to traverse the selected portals is at most (2r)4r(4r)! (where the term (2r)4r comes from the fact that each portal can be traversed along one of the O(2r) paths in the square). Altogether there are no more than (m+3)4r(2r)4r(4r)! possibilities. We can compute the cost of each possibility and obtain the optimal solution from the optimal solutions of the four subsquares. The time complexity of constructing the table is O(nlog n (m+4)8r(2r)4r((4r)!)2), which is O(n(log n)O(c)) for m=O(clog n) and r=O(c).
Theorem 1 (Structure Theorem) Let the minimum nonzero internode distance in a TSP instance be 8 and L be the size of the bounding box. Let shifts 0 ≤ a,b ≤ L be picked randomly. Then with probability at least 1/2, the dissection with shift (a,b) has an associated (m,r)-light salesman path of cost at most (1+1/c)OPT, where m=O(clogL) and r=O(c).
Our strategy is to modify the optimal salesman tour to a (m,r)-light salesman path and analyze the cost of the modification. Denote the optimal salesman tour by π and let tl be the number of times that π crosses a dissection line l. Note that ∑l:verticaltl + ∑l:horizontaltl ≤ 2OPT when the nonzero distance between two nodes is at least 4, where OPT is the cost of the optimal salesman tour.
(i) Moving every point the optimal tour crosses a dissection line to its nearest portal.
Consider a line l at level i. l has a spacing of L/(2i+2(m+1)) between two adjacent portals and moving a crossing on l increases the cost by at most L/(2i+2(m+1)). Thus the expected cost of moving every crossing on l to its nearest portal is at most
∑0 ≤ i < logL 2i/(L-1) × tl × L/(2i+2m) ≤ tl log L / m ,
which is at most 1/2r when m ≥ 2r log L.
(ii) Restricting the tour to cross an edge of a dissection square at most times r We use a bottom-up technique, starting from the finest squares, if the tour crosses an edge more than r times, we apply the Patching Lemma (given below) to reduce the number of crossings to 4 (4 instead of 2 because an edge of a square may be broken into two disjoint segments as a result of shifting the dissection).
Lemma 1 (Patching Lemma) There is a constant g > 0 such that the following is true. Let S be any line segment of length l and π be a closed path that crosses S at least thrice. Then there exist line segments on S whose total length is g× l and whose addition to π changes it into a closed path that crosses S at most twice.
Proof Sketch We can break the tour at the points it crosses S, and add a cycle of length ≤ 2l and a matching of length ≤ l on each side of S. The result is a Eulerian tour that crosses S at most twice. The increase in length is at most 6l.
Let cj be the number of edges of level-j squares on a line l that we apply Patching Lemma. Note that ∑j ≥ 0Cj ≤ tl / (r-3) because each application of Patching Lemma replaces at least r+1 crossings by at most 4.
The expected cost of patching on a dissection line is
≤ ∑i ≥ 0 2i/L ∑j ≥ i cj g L / 2j
≤ g ∑j ≥ 0 cj / 2j ∑i ≤ j2i
≤ g ∑j ≥ 0 2 cj
≤ 2gtl / (r-3)
where g is the constant appearing in the Patching Lemma.
Thus the expected cost of making the final salesman path (m,r)-light at line l is
2gtl/(r-3) + tl/2r ≤ 3gtl/r
for r > 15.
By linearity of expectations, it follows that the expected increase in the cost of the tour is
≤ ∑l:vertical 3gtl/r + ∑l:horizontal 3gtl/r
≤ 6gOPT / r
≤ OPT / 2c
for r ≥ 12gc .
Markov’s inequality implies that with probability at least 1/2 this increase is no more than OPT / c. Therefore, we conclude that with probability at least 1/2 the cost of the best (m,r)-light salesman path for the shifted dissection is at most (1 + 1/c) OPT.