A Framework for Computing the Greedy Spanner

The highest quality geometric spanner (e.g. in terms of edge count, both in theory and in practice) known to be computable in polynomial time is the greedy spanner. The state-of-the-art in computing this spanner are a O(n^2 log n) time, O(n^2) space algorithm and a O(n^2 log^2 n) time, O(n) space algorithm, as well as the `improved greedy' algorithm, taking O(n^3 log n) time in the worst case and O(n^2) space but being faster in practice thanks to a caching strategy. We identify why this caching strategy gives speedups in practice. We formalize this into a framework and give a general efficiency lemma. From this we obtain many new time bounds, both on old algorithms and on new algorithms we introduce in this paper. Interestingly, our bounds are in terms of the well-separated pair decomposition, a data structure not actually computed by the caching algorithms. Specifically, we show that the `improved greedy' algorithm has a O(n^2 log n log Phi) running time (where Phi is the spread of the point set) and a variation has a O(n^2 log^2 n) running time. We give a variation of the linear space state-of-the-art algorithm and an entirely new algorithm with a O(n^2 log n log Phi) running time, both of which improve its space usage by a factor O(1/(t-1)). We present experimental results comparing all the above algorithms. The experiments show that - when using low t - our new algorithm is up to 200 times more space efficient than the existing linear space algorithm, while being comparable in running time and much easier to implement

Distribution-sensitive construction of the greedy spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take O(n 2) time, limiting its use on large data sets. We observe that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly, and few or no ‘long’ edges that can usually be determined quickly using local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets: we give a geometric property that holds with high probability on such point sets. This property implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives a O(n log2 n log2 logn) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give a O((n¿+¿|E|) log2 n loglogn) expected time algorithm on uniformly distributed points that determines if E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E

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 on n points use O(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 linear-space algorithm that computes the same spanner for points in R^d running in O(n^2 log^2n) 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. Keywords: Geometric spanner; Dilation; Stretch factor; Greedy algorithm; Computational geometr

Distribution-sensitive construction of the greedy spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take $\varOmega (n^2)$ time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly. To find the usually few remaining ‘long’ edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives an $O(n \log ^2 n \log ^2 \log n)$ expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an $O((n + |E|) \log ^2 n \log \log n)$ expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E

Distribution-sensitive construction of the greedy spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take O(n 2) time, limiting its use on large data sets. We observe that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly, and few or no ‘long’ edges that can usually be determined quickly using local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets: we give a geometric property that holds with high probability on such point sets. This property implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives a O(n log2 n log2 logn) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give a O((n¿+¿|E|) log2 n loglogn) expected time algorithm on uniformly distributed points that determines if E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E

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 O(n2) 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 Rd running in O(n2 log2n) 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

Safety of hospital discharge before return of bowel function after elective colorectal surgery

Background: Ileus is common after colorectal surgery and is associated with an increased risk of postoperative complications. Identifying features of normal bowel recovery and the appropriateness for hospital discharge is challenging. This study explored the safety of hospital discharge before the return of bowel function. Methods: A prospective, multicentre cohort study was undertaken across an international collaborative network. Adult patients undergoing elective colorectal resection between January and April 2018 were included. The main outcome of interest was readmission to hospital within 30 days of surgery. The impact of discharge timing according to the return of bowel function was explored using multivariable regression analysis. Other outcomes were postoperative complications within 30 days of surgery, measured using the Clavien\u2013Dindo classification system. Results: A total of 3288 patients were included in the analysis, of whom 301 (9\ub72 per cent) were discharged before the return of bowel function. The median duration of hospital stay for patients discharged before and after return of bowel function was 5 (i.q.r. 4\u20137) and 7 (6\u20138) days respectively (P < 0\ub7001). There were no significant differences in rates of readmission between these groups (6\ub76 versus 8\ub70 per cent; P = 0\ub7499), and this remained the case after multivariable adjustment for baseline differences (odds ratio 0\ub790, 95 per cent c.i. 0\ub755 to 1\ub746; P = 0\ub7659). Rates of postoperative complications were also similar in those discharged before versus after return of bowel function (minor: 34\ub77 versus 39\ub75 per cent; major 3\ub73 versus 3\ub74 per cent; P = 0\ub7110). Conclusion: Discharge before return of bowel function after elective colorectal surgery appears to be safe in appropriately selected patients

Safety of hospital discharge before return of bowel function after elective colorectal surgery

© 2020 BJS Society Ltd Published by John Wiley & Sons LtdBackground: Ileus is common after colorectal surgery and is associated with an increased risk of postoperative complications. Identifying features of normal bowel recovery and the appropriateness for hospital discharge is challenging. This study explored the safety of hospital discharge before the return of bowel function. Methods: A prospective, multicentre cohort study was undertaken across an international collaborative network. Adult patients undergoing elective colorectal resection between January and April 2018 were included. The main outcome of interest was readmission to hospital within 30 days of surgery. The impact of discharge timing according to the return of bowel function was explored using multivariable regression analysis. Other outcomes were postoperative complications within 30 days of surgery, measured using the Clavien–Dindo classification system. Results: A total of 3288 patients were included in the analysis, of whom 301 (9·2 per cent) were discharged before the return of bowel function. The median duration of hospital stay for patients discharged before and after return of bowel function was 5 (i.q.r. 4–7) and 7 (6–8) days respectively (P < 0·001). There were no significant differences in rates of readmission between these groups (6·6 versus 8·0 per cent; P = 0·499), and this remained the case after multivariable adjustment for baseline differences (odds ratio 0·90, 95 per cent c.i. 0·55 to 1·46; P = 0·659). Rates of postoperative complications were also similar in those discharged before versus after return of bowel function (minor: 34·7 versus 39·5 per cent; major 3·3 versus 3·4 per cent; P = 0·110). Conclusion: Discharge before return of bowel function after elective colorectal surgery appears to be safe in appropriately selected patients