Parallel Minimum Cuts in Near-linear Work and Low Depth

We present the first near-linear work and poly-logarithmic depth algorithm for computing a minimum cut in an undirected graph. Previous parallel algorithms with poly-logarithmic depth required at least quadratic work in the number of vertices. In a graph with n vertices and m edges, our randomized algorithm computes the minimum cut with high probability in O ( m log 4 n ) work and O (log 3 n ) depth. This result is obtained by parallelizing a data structure that aggregates weights along paths in a tree, in addition exploiting the connection between minimum cuts and approximate maximum packings of spanning trees. In addition, our algorithm improves upon bounds on the number of cache misses incurred to compute a minimum cut.

Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cycles

It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order \(v\) if and only if \(v \equiv 0, 1,\text{ or }2 (\operatorname{mod} 9)\). For all positive integers \(\lambda\) and \(v\), we find a maximum packing with loose 3-cycles of the \(\lambda\)-fold complete 3-uniform hypergraph of order \(v\). We show that, if \(v \geq 6\), such a packing has a leave of two or fewer edges.

Squashing maximum packings of Kn with 8-cycles into maximum packings of Kn with 4-cycles

An 8-cycle is said to be squashed if we identify a pair of opposite vertices and name one of them with the other (and thereby turning the 8-cycle into a pair of 4-cycles with exactly one vertex in common). The resulting pair of 4-cycles is called a bowtie. We say that we have squashed the 8-cycle into a bowtie. Evidently an 8-cycle can be squashed into a bowtie in eight different ways. The object of this paper is the construction, for every n > 8, of a maximum packing of Kn with 8-cycles which can be squashed in a maximum packing of Kn with 4-cycles.

Minimal Overlapping Patterns in Colored Permutations

A pattern $P$ of length $j$ has the minimal overlapping property if two consecutive occurrences of the pattern can overlap in at most one place, namely, at the end of the first consecutive occurrence of the pattern and at the start of the second consecutive occurrence of the pattern. For patterns $P$ which have the minimal overlapping property, we derive a general formula for the generating function for the number of consecutive occurrences of $P$ in words, permutations and $k$-colored permutations in terms of the number of maximum packings of $P$ which are patterns of minimal length which has $n$ consecutive occurrences of the pattern $P$. Our results have as special cases several results which have appeared in the literature. Another consequence of our results is to prove a conjecture of Elizalde that two permutations $\alpha$ and $\beta$ of size $j$ which have the minimal overlapping property are strongly $c$-Wilf equivalent if $\alpha$ and $\beta$ have the same first and last elements.