2,997 research outputs found
Faster Parametric Shortest Path and Minimum Balance Algorithms
The parametric shortest path problem is to find the shortest paths in graph
where the edge costs are of the form w_ij+lambda where each w_ij is constant
and lambda is a parameter that varies. The problem is to find shortest path
trees for every possible value of lambda.
The minimum-balance problem is to find a ``weighting'' of the vertices so
that adjusting the edge costs by the vertex weights yields a graph in which,
for every cut, the minimum weight of any edge crossing the cut in one direction
equals the minimum weight of any edge crossing the cut in the other direction.
The paper presents fast algorithms for both problems. The algorithms run in
O(nm+n^2 log n) time. The paper also describes empirical studies of the
algorithms on random graphs, suggesting that the expected time for finding a
minimum-mean cycle (an important special case of both problems) is O(n log(n) +
m)
2-Vertex Connectivity in Directed Graphs
We complement our study of 2-connectivity in directed graphs, by considering
the computation of the following 2-vertex-connectivity relations: We say that
two vertices v and w are 2-vertex-connected if there are two internally
vertex-disjoint paths from v to w and two internally vertex-disjoint paths from
w to v. We also say that v and w are vertex-resilient if the removal of any
vertex different from v and w leaves v and w in the same strongly connected
component. We show how to compute the above relations in linear time so that we
can report in constant time if two vertices are 2-vertex-connected or if they
are vertex-resilient. We also show how to compute in linear time a sparse
certificate for these relations, i.e., a subgraph of the input graph that has
O(n) edges and maintains the same 2-vertex-connectivity and vertex-resilience
relations as the input graph, where n is the number of vertices.Comment: arXiv admin note: substantial text overlap with arXiv:1407.304
Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph
In this paper, we address the problem of enumerating all induced subtrees in
an input k-degenerate graph, where an induced subtree is an acyclic and
connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for
any its induced subgraph has a vertex whose degree is less than or equal to k,
and many real-world graphs have small degeneracies, or very close to small
degeneracies. Although, the studies are on subgraphs enumeration, such as
trees, paths, and matchings, but the problem addresses the subgraph
enumeration, such as enumeration of subgraphs that are trees. Their induced
subgraph versions have not been studied well. One of few example is for
chordless paths and cycles. Our motivation is to reduce the time complexity
close to O(1) for each solution. This type of optimal algorithms are proposed
many subgraph classes such as trees, and spanning trees. Induced subtrees are
fundamental object thus it should be studied deeply and there possibly exist
some efficient algorithms. Our algorithm utilizes nice properties of
k-degeneracy to state an effective amortized analysis. As a result, the time
complexity is reduced to O(k) time per induced subtree. The problem is solved
in constant time for each in planar graphs, as a corollary
Illustrated key to the genera of free-living marine nematodes of the Order Enoplida
A pictorial key to US genera of free-living marine nematodes
in the order Enoplida is presented. Specific morphological and anatomical features are iUustrated to facilitate use of the key. The purpose of this work is to provide a single key to the genera of enoplid nematodes to facilitate identification of these organisms by nematologists and marine biologists working with meiofauna. (PDF file contains 32 pages.
Amortized Rotation Cost in AVL Trees
An AVL tree is the original type of balanced binary search tree. An insertion
in an -node AVL tree takes at most two rotations, but a deletion in an
-node AVL tree can take . A natural question is whether
deletions can take many rotations not only in the worst case but in the
amortized case as well. A sequence of successive deletions in an -node
tree takes rotations, but what happens when insertions are intermixed
with deletions? Heaupler, Sen, and Tarjan conjectured that alternating
insertions and deletions in an -node AVL tree can cause each deletion to do
rotations, but they provided no construction to justify their
claim. We provide such a construction: we show that, for infinitely many ,
there is a set of {\it expensive} -node AVL trees with the property
that, given any tree in , deleting a certain leaf and then reinserting it
produces a tree in , with the deletion having done
rotations. One can do an arbitrary number of such expensive deletion-insertion
pairs. The difficulty in obtaining such a construction is that in general the
tree produced by an expensive deletion-insertion pair is not the original tree.
Indeed, if the trees in have even height , deletion-insertion
pairs are required to reproduce the original tree
A Back-to-Basics Empirical Study of Priority Queues
The theory community has proposed several new heap variants in the recent
past which have remained largely untested experimentally. We take the field
back to the drawing board, with straightforward implementations of both classic
and novel structures using only standard, well-known optimizations. We study
the behavior of each structure on a variety of inputs, including artificial
workloads, workloads generated by running algorithms on real map data, and
workloads from a discrete event simulator used in recent systems networking
research. We provide observations about which characteristics are most
correlated to performance. For example, we find that the L1 cache miss rate
appears to be strongly correlated with wallclock time. We also provide
observations about how the input sequence affects the relative performance of
the different heap variants. For example, we show (both theoretically and in
practice) that certain random insertion-deletion sequences are degenerate and
can lead to misleading results. Overall, our findings suggest that while the
conventional wisdom holds in some cases, it is sorely mistaken in others
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