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Motivation: The Dictionary of Interfaces in Proteins (DIP) is a database collecting the 3D structure of interacting parts of proteins that are called patches. It serves as a repository, in which patches similar to given query patches can be found. The computation of the similarity of two patches is time consuming and traversing the entire DIP requires some hours. In this work we address the question of how the patches similar to a given query can be identified by scanning only a small part of DIP. The answer to this question requires the investigation of the distribution of the similarity of patches.
Results: The score values describing the similarity of two patches can roughly be divided into three ranges that correspond to different levels of spatial similarity. Interestingly, the two iso-score lines separating the three classes can be determined by two different approaches. Applying a concept of the theory of random graphs reveals significant structural properties of the data in DIP. These can be used to accelerate scanning the DIP for patches similar to a given query. Searches for very similar patches could be accelerated by a factor of more than 25. Patches with a medium similarity could be found 10 times faster than by brute-force search.
The weighted matching problem is to find a matching in a weighted graph
that has maximum weight. The fastest known algorithm for this problem has running time
O(nm +n2 log n). Many real world problems require graphs of such large size that this running
time is too costly. We present a linear time approximation algorithm for the weighted
matching problem with a performance ratio of 2
3 ???? ". This improves the previously best
performance ratio of 1
2 .
Recently two different linear time approximation algorithms for the weighted matching problem in graphs have been suggested [5][17]. Both these algorithms have a performance ratio of 1/2. In this paper we present a set of local improvement operations and prove that it guarantees a performance ratio of 2/3. We show that a maximal set of these local improvements can be found in linear time.
To see how these local improvements behave in practice we conduct an experimental comparison of four different approximation algorithms for calculating maximum weight matchings in weighted graphs. One of these algorithms is the commonly used Greedy algorithm which achieves a performance ratio of 1/2 but has O(m log n) runtime. The other three algorithms all have linear runtime. Two of them are the above mentioned 1/2 approximation algorithms. The third algorithm may have an arbitrarily bad performance ratio but in practice produces reasonably good results. We compare the quality of the algorithms on a test set of weighted graphs and study the improvement achieved by our local improvement operations. We also do a comparison of the runtimes of all algorithms.
We present a linear time approximation algorithm with a performance ratio of 1/2 for finding a maximum weight matching in an arbitrary graph. Such a result is already known and is due to Preis [STACS'99, Lecture Notes in Comput. Sci., Vol. 1563, 1999, pp. 259–269]. Our algorithm uses a new approach which is much simpler than the one given by Preis and needs no amortized analysis for its running time.
We investigate the problem of colouring random graphs G ? G(n; p)
in polynomial expected time. For the case p ? 1.01/n, we present an algorithm
that finds an optimal colouring in linear expected time. For
p ?? ln6(n)/n, we give algorithms which approximate the chromatic
number within a factor of O(? np). We also obtain an O(?
np/ ln(np))-
approximation algorithm for the independence number. As an application,
we propose an algorithm for deciding satisfiability of random 2k-
SAT formulas (with sufficiently many clauses) in
polynomial expected time.
We investigate the problem of colouring random graphs G ? G(n, p) in polynomial expected time. For the case p < 1.01/n, we present an algorithm that finds an optimal colouring in linear expected time. For suficiently large values of p, we give algorithms which approximate the chromatic number within a factor of O(?np). As a byproduct, we obtain an O(?np/ ln(np))-approximation algorithm for the independence number which runs in polynomial expected time provided p ? ln6 n/n.