Contents
What is recursive bisection?
Abstract. The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal-sized pieces, by a “good” bisection algorithm, and then recursively divides the two pieces. Most such heuristics are designed to find the best possible bisection within allowed time.
What is k way partitioning?
The multilevel k-way partitioning algorithm reduces the size of the graph by collapsing vertices and edges (coarsening phase), finds a k-way partition of the smaller graph, and then it constructs a k-way partition for the original graph by projecting and refining the partition to successively finer graphs (uncoarsening …
Is graph partitioning NP complete?
Problem complexity Solutions to these problems are generally derived using heuristics and approximation algorithms. However, uniform graph partitioning or a balanced graph partition problem can be shown to be NP-complete to approximate within any finite factor.
What is edge partition?
An edge partition of a graph G is a partition of its edges E into subsets {Ej}. We can define {Vj} to be the set of endpoints of edges in {Ej}. We then have that [Vj,Ej] is a graph. Any graph can be partitioned into its maximal connected subgraphs, which are called its connected components.
Can a subgraph have no edges?
Yes. If G has a isolated vertex (one with no edges in or out) then the subgraph of G obtained by removing that vertex has the same number of edges but strictly fewer vertices. Suppose G has no isolated vertices.
Can there be two different simple paths between the vertices of a tree?
Theorem: An undirected graph is a tree iff there is exactly one simple path between each pair of vertices. Since T is connected, there must be at least one simple path between each pair of vertices. If there is more than one path between two vertices, then parts of those paths could be joined to form a cycle.
Can a hypergraph be partitioned using direct k-way?
We claim that hypergraph partitioning with multiple constraints and fixed vertices should be implemented using direct K-way refinement, instead of the widely adopted recursive bisection paradigm.
What is the purpose of the partitioning constraint?
Typically, the partitioning constraint is to maintain one or more balance constraints on the part weights.
How is a multi-level bisection hypergraph coarsened?
In multi-level bisection, the original hypergraph is coarsened into a smaller hypergraph af- ter a series of coarsening levels, in which highly coherent ver- tices are grouped into supervertices]