[...]... a one-to-one correspondence between the edges of G and edge nodes in the corresponding DCEL An edge node consists of four information elds V 1 V 2 F 1 and F 2, and two pointer elds P 1 and P 2 The elds V 1 and V 2 contain the starting vertex and ending vertex of the edge, respectively (So we give each edge of the PSLG G an orientation This orientation can be de ned arbitrarily.) The elds F 1 and F... sides Therefore, the simple polygon P is the convex hull CH(S) 3.2 Proximity problems The examples of proximity problems include CLOSEST-PAIR, ALLNEAREST-NEIGHBORS, EUCLIDEAN-MINIMUM-SPANNING-TREE, TRIANGULATION, and MAXIMUM-EMPTY-CIRCLE Proximity problems arise in many applications where physical or mathematical objects are represented as points in space Examples include the following: clustering: a... as fundamental to computational geometry as sorting to general algorithms It is also a vehicle for the solution of a number of apparently unrelated questions arising in computational geometry The construction of the convex hull of a nite set of points has also found applications in many areas, such as in pattern recognition, in image processing, in Robotics, and in stock cutting and allocation Theorem... ELSE IF p is an intersection point of segments Si and Sj such that Si is on the left of Sj in STATUS GEOMETRIC SWEEPING 34 REPORT(p) swap the positions of Si and Sj in STATUS Let Sk be the segment left to Sj and let Sh be the segment right to Si in STATUS IF Sk and Sj intersect at p1 and x(p1) > x(p) INSERT p1 into EVENT IF Sh and Si intersect at p2 and x(p2) > x(p) INSERT p2 into EVENT END {WHILE}... is subdivided into an interior region and an exterior region such that every curve connecting a point in the interior region and a point in the exterior region must intersect the curve C The k-dimensional Euclidean (x1 xk ) of real numbers xi , 1 p1 = (x1 xk) and p2 = (y1 by space E k is the space of all k-tuples i k The distance between two points yk ) in the k-dimensional space is de ned d(p1 p2)... by splicing the tree T1(i;2) and the subtrees in si;1 , and the subtrees in si;1 have height h(si;1 ) Thus, the height of the 2-3 tree T1(i;1) is at least h(si;1 ) Now we prove the rest inequalities Since the 2-3 tree T1(1) is obtained by splicing the subtrees in the segment s1, and segment s1 contains at most two subtrees, both of height h(s1) Thus, the height of the 2-3 tree T1(1) is at most h(s1... constructing the 2-3 tree T1(i) from the 2-3 tree T1(i;1) and the trees in the segment si According to Lemma 2.3.2, we have h(T1(i;1)) h(si) Thus, if si is a single subtree ti , then according the analysis of the time complexity of the algorithm SPLICE, the time of splicing T1(i;1) and ti is bounded by a constant times h(si ) ; h(T1(i;1)) On the other hand, if si consists of two subtrees t0i and t00 , then... closest (classi ed) neighbor and air-tra c control: the two airplanes that are closest are the two most in danger We will restrict ourselves to 2-dimensions The input to these problems is a set S of n points in the plane The distance between points in S will be the Euclidean distance between the points CLOSEST-PAIR Find a pair of points in the set S which are closest ALL-NEAREST-NEIGHBORS For every point... EUCLIDEAN-MINIMUM-SPANNING-TREE Find an interconnecting tree of minimum total length whose vertices are the points in the set S TRIANGULATION Join the points in the set S by non-intersecting straight line segments so that every region interior to the convex hull of S is a triangle PROXIMITY PROBLEMS 27 p i p H(pi , pj ) j Figure 3.1: The points that are closer to pi than to pj MAXIMUM-EMPTY-CIRCLE... are less than any elements in T2, and that the height of T1 is at most that of T2 Other cases can be dealt with similarly.} BEGIN IF height(T1) = height(T2) THEN make T a parent of T1 and T2 ELSE WHILE height(T2 )-1 > height(T1) DO ALGORITHMIC FOUNDATIONS 12 T2 := child1(T2) Call ADDSON(T2, T1) END 2.3.6 Split By splitting a given 2-3 tree T into two 2-3 trees, T1 and T2, at a given element x, we mean