A Faster Algorithm for Computing the Principal Sequence of Partitions of a Graph

Abstract

We consider the following problem: given an undirected weighted graph G = (V , E, c) with nonnegative weights, minimize function c(δ( )) − λ| | for all values of parameter λ. Here is a partition of the set of nodes, the first term is the cost of edges whose endpoints belong to different components of the partition, and | | is the number of components. The current best known algorithm for this problem has complexity O(|V | 2 ) maximum flow computations. We improve it to |V | parametric maximum flow computations. We observe that the complexity can be improved further for families of graphs which admit a good separator, e.g. for planar graphs.

Improving Computational Efficiency of Communication for Omniscience and Successive Omniscience

“…where the rate vector r λ is updated in steps 4 and 5 in each iteration to maintain the monotonicity: r λ,i ≤ r λ ′ ,i for all i ∈ V and λ < λ ′ . It is show in [29,Lemmas 4 and 5] that the minimizer of min{h λ (X) : φ i ∈ X ⊆ V } forms a 'nesting' set sequence in λ, which, for f being the cut function, can be determined by only one call of the parametric MaxFlow algorithm in [46]. We show in Theorem E.1 and Lemma E.2 below that this 'nesting' property is also due to the strong map property, which is conditioned on the monotonicity of r λ .…”

“…Komolgorov's Algorithm[29, Fig. 3] input : f , V and Φ. output:Q λ,V = argmin P∈Π(V ) f α [P] and r λ ∈ B(f λ )for all λ. Initiate r λ,V := (−λ, .…”

Info-clustering: An efficient algorithm by network information flow

“…By [1,Proposition 9], the PSP of the incut function B → c(V \ B, B) of the digraph D coincides with the PSP of the cut function (divided by 2) of the corresponding undirected graph G, which was shown in [15] to be computable by running a parametric max-flow algorithm O(|V |) times. The parametric max-flow algorithm was introduced by [16], which runs in O(|V | 2 |E|) times using the well-known push-relable/preflow algorithm [20,21] implemented with the highest-level selection rule [22].…”

“…In this section, we will adapt and improve the algorithm in [15] to compute the desired PSP for the info-clustering solution. The algorithm will be illustrated using the same example as in the last section, which is chosen to be the same example as in [15] for ease of comparison.…”

“…as they are zero by default. In contrast with [15], the additional node contraction and edge removal in Steps 2 and 3 above reduce the number of vertices from |V | to |U | = j and therefore the complexity in solving (4.7). For each j ∈ V , (4.7) can be solved by the parametric maxflow algorithm in [16] in O(j 3 |E|) time by invoking O(j) times the preflow algorithm [20,21] implemented with highest level selection rule [22], which in turn runs in O(j 2 |E|) times.…”