Near-Optimal Critical Sink Routing Tree Constructions

40 is a near branch, an analogous argument again shows that Pin(b) is q's lower-right quadrant. Thus, q is the closest connection between p, Pin(a) and Pin(b). Except for redundancies and pruning of sub-optimal trees, BB-SORT-C searches over all possible ways to construct a Steiner tree sequentially, such that each sink is added by a closest connection to some edge in the current tree. Thus, we have: Theorem B1: For any positive linear combination of sink delays, f = k X i=1 i t(n i), i > 0 8i, algorithm BB-SORT-C returns a Steiner tree T which minimizes f. 39 path in T will contain only edges in M, edges in branches oo of M, or edges in a sequence of far branches oo of branches of M. (For example, consider the paths from q to sinks n 1 and n 2 in Figure 21(c)-(e).) Thus, Pin(a) and p cannot be on the same side of a line that passes through q and is perpendicular to M. Consequently, q will be the closest connection between edge (p; q) and Pin(a). The second case is when q is a degree-3 Steiner node in T i. Let a and b be the children of q in T such that Pin(a) and Pin(b) are q's children in T i. Without loss of generality, we assume that Pin(q) = Pin(a) and n i = Pin(b). We must show that q is located at the closest connection between nodes p, Pin(a), and Pin(b). There are four possible conngurations for connections at q, as shown in parts (a)-(d) of Figure 21. In Figure 21(a), edge (p; q) is L-shaped and both Pin(a) and Pin(b) (denoted by n 1 and n 2 in the gure) must be on maximalsegments with entry point q; it is easy to see that q is the closest connection between p, Pin(a), and Pin(b). In Figure 21(b)-(d), edge (p; q) is a a straight edge. Let M be the MS containing (p; q), and let M 0 be the MS perpendicular to M with entry point q. In Figure 21(b), edge (q; a) is L-shaped and edge (q; b) is on the MS M 0. By Lemma B4, M 0 must contain a sink, which will be contained in subtree T b. Thus, Pin(b) (n 2 in the Figure) is located on M 0. Node a is the entry point for two …