Geometric separator theorems and applications

We find a large number of “geometric separator theorems” such as: I: Given N disjoint isooriented squares in the plane, there exists a rectangle with ⩽2N/3 squares inside, ⩽2N/3 squares outside, and ⩽(4+0(1))√N partly in & out. II: There exists a rectangle that is crossed by the minimal spanning tree of N sites in the plane at ⩽(4·3^1/4+0(1))√N points, having ⩽2N/3 sites inside and outside. These theorems yield a large number of applications, such as subexponential algorithms for traveling salesman tour and rectilinear Steiner minimal tree in R^d , new point location algorithms, and new upper and lower bound proofs for “planar separator theorems”