1,568 research outputs found
FLEET: Butterfly Estimation from a Bipartite Graph Stream
We consider space-efficient single-pass estimation of the number of
butterflies, a fundamental bipartite graph motif, from a massive bipartite
graph stream where each edge represents a connection between entities in two
different partitions. We present a space lower bound for any streaming
algorithm that can estimate the number of butterflies accurately, as well as
FLEET, a suite of algorithms for accurately estimating the number of
butterflies in the graph stream. Estimates returned by the algorithms come with
provable guarantees on the approximation error, and experiments show good
tradeoffs between the space used and the accuracy of approximation. We also
present space-efficient algorithms for estimating the number of butterflies
within a sliding window of the most recent elements in the stream. While there
is a significant body of work on counting subgraphs such as triangles in a
unipartite graph stream, our work seems to be one of the few to tackle the case
of bipartite graph streams.Comment: This is the author's version of the work. It is posted here by
permission of ACM for your personal use. Not for redistribution. The
definitive version was published in Seyed-Vahid Sanei-Mehri, Yu Zhang, Ahmet
Erdem Sariyuce and Srikanta Tirthapura. "FLEET: Butterfly Estimation from a
Bipartite Graph Stream". The 28th ACM International Conference on Information
and Knowledge Managemen
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms
Consider the following 2-respecting min-cut problem. Given a weighted graph
and its spanning tree , find the minimum cut among the cuts that contain
at most two edges in . This problem is an important subroutine in Karger's
celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present
a new approach for this problem which can be easily implemented in many
settings, leading to the following randomized min-cut algorithms for weighted
graphs.
* An -time sequential algorithm:
This improves Karger's and bounds when the input graph is not extremely
sparse or dense. Improvements over Karger's bounds were previously known only
under a rather strong assumption that the input graph is simple [Henzinger et
al. SODA'17; Ghaffari et al. SODA'20]. For unweighted graphs with parallel
edges, our bound can be improved to .
* An algorithm requiring cut queries to compute the min-cut of
a weighted graph: This answers an open problem by Rubinstein et al. ITCS'18,
who obtained a similar bound for simple graphs.
* A streaming algorithm that requires space and
passes to compute the min-cut: The only previous non-trivial exact min-cut
algorithm in this setting is the 2-pass -space algorithm on simple
graphs [Rubinstein et al., ITCS'18] (observed by Assadi et al. STOC'19).
In contrast to Karger's 2-respecting min-cut algorithm which deploys
sophisticated dynamic programming techniques, our approach exploits some cute
structural properties so that it only needs to compute the values of cuts corresponding to removing pairs of tree edges, an
operation that can be done quickly in many settings.Comment: Updates on this version: (1) Minor corrections in Section 5.1, 5.2;
(2) Reference to newer results by GMW SOSA21 (arXiv:2008.02060v2), DEMN
STOC21 (arXiv:2004.09129v2) and LMN 21 (arXiv:2102.06565v1
Intercalates and Discrepancy in Random Latin Squares
An intercalate in a Latin square is a Latin subsquare. Let be
the number of intercalates in a uniformly random Latin square. We
prove that asymptotically almost surely
, and that
(therefore
asymptotically almost surely for any ). This
significantly improves the previous best lower and upper bounds. We also give
an upper tail bound for the number of intercalates in two fixed rows of a
random Latin square. In addition, we discuss a problem of Linial and Luria on
low-discrepancy Latin squares
Applications of Graphical Condensation for Enumerating Matchings and Tilings
A technique called graphical condensation is used to prove various
combinatorial identities among numbers of (perfect) matchings of planar
bipartite graphs and tilings of regions. Graphical condensation involves
superimposing matchings of a graph onto matchings of a smaller subgraph, and
then re-partitioning the united matching (actually a multigraph) into matchings
of two other subgraphs, in one of two possible ways. This technique can be used
to enumerate perfect matchings of a wide variety of bipartite planar graphs.
Applications include domino tilings of Aztec diamonds and rectangles, diabolo
tilings of fortresses, plane partitions, and transpose complement plane
partitions.Comment: 25 pages; 21 figures Corrected typos; Updated references; Some text
revised, but content essentially the sam
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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