Computing the Greedy Spanner in Linear Space

The greedy spanner is a high-quality spanner: its total weight, edge count and maximal degree are asymptotically optimal and in practice significantly better than for any other spanner with reasonable construction time. Unfortunately, all known algorithms that compute the greedy spanner of n points use Omega(n^2) space, which is impractical on large instances. To the best of our knowledge, the largest instance for which the greedy spanner was computed so far has about 13,000 vertices. We present a O(n)-space algorithm that computes the same spanner for points in R^d running in O(n^2 log^2 n) time for any fixed stretch factor and dimension. We discuss and evaluate a number of optimizations to its running time, which allowed us to compute the greedy spanner on a graph with a million vertices. To our knowledge, this is also the first algorithm for the greedy spanner with a near-quadratic running time guarantee that has actually been implemented

The bone morphogenetic protein pathway is inactivated in the majority of sporadic colorectal cancers

Background & Aims: The finding of bone morphogenetic protein (BMP) receptor la mutations in juvenile polyposis suggests that BMPs are important in colorectal cancer (CRC). We investigated the BMP pathway in sporadic CRC. Methods: We investigated BMP receptor (BMPR) expression using immuno-blotting and sequenced BMPR2 in CRC cell lines. We assessed the expression of BMPRs, SMAD4, and pSMAD1/5/8 in 72 sporadic CRCs using a tissue microarray and immunohistochemistry. We assessed the effect of reintroduction of wild-type BMPR2 on BMP pathway activity and the effect of wild-type or mutated BMPR2 3' untranslated region (UTR) sequences on protein expression by attachment to pCMV-Luc. Results: BMPR2 and SMAD4 protein expression is abrogated in microsatellite unstable (MSI) and microsatellite stable (MSS) cell lines, respectively. BMPR2 3'UTR is mutated in all MSI and in none of the MSS cell lines. Mutant BMPR2 3'UTR sequences reduced luciferase expression 10-fold compared with wild-type BMPR2 3'UTR. BMPR2 expression is impaired more frequently in MSI CRCs than MSS (85% vs 29%; P <.0001) and shows a mutually exclusive pattern of impaired expression compared with SMAD4. Nine of 11 MSI cancers with impaired expression of BMPR2 have microsatellite mutations. The BMP pathway is inactivated, as judged by nuclear pSMAD1/5/8 expression, in 70% of CRCs, and this correlates with BMPR and SMAD4 loss. Conclusions: Our data suggest that the BMP pathway is inactivated in the majority of sporadic CRCs. In MSI CRC this is associated predominantly,with impaired BMPR2 expression and in MSS CRC with impaired SMAD4 expression