The privacy preserving multiset union (PPMU) protocol allows a set of parties, each with a multiset, to collaboratively compute a multiset union secretly, meaning that any information other than union is not revealed. We propose an efficient PPMU protocol, using multiplicative homomorphic property of ElGamal encryption over Fq[x]/f(x) where q is a prime and f(x) is an irreducible polynomial ove...

We present a highly optimized implementation of tiered vectors, a data structure for maintaining a sequence of n elements supporting access in time O(1) and insertion and deletion in time O(n ) for > 0 while using o(n) extra space. We consider several different implementation optimizations in C++ and compare their performance to that of vector and multiset from the standard library on sequences...

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that ...

In this paper, we survey on reaction automata theory to model and analyze the biochemical behaviors of vital reactions occurring in nature. Inspired by two notions a system initiated Ehrenfeucht Rozenberg 2007 multiset, (RAs) have been proposed as computing models for accepting string languages. Given an input sequence symbols, RA performs its computation process follows: at every time receivin...

In this paper, we study the Stanley depth for the partially ordered set (poset) of nonempty submultisets of a multiset. We find the Stanley depth explicitly for any multiset with at most five distinct elements and provide an upper bound for the general case. On the other hand, the elements of a product of chains corresponds to the submultisets of a multiset. We prove that the Stanley depth of t...

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic number is determined for every complete multipartite graph as...

Multiset orderings are a key ingredient in certain termination techniques like the recursive path ordering and a variant of size-change termination. In order to integrate these techniques in a certifier for termination proofs, we have added them to the Isabelle Formalization of Rewriting. To this end, it was required to extend the existing formalization on multiset orderings towards a generaliz...

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon’s theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof...

This paper answers the following mathematical question: Can multiset permutations be ordered so that each permutation is a prefix shift of the previous permutation? Previously, the answer was known for the permutations of any set, and the permutations of any multiset whose corresponding set contains only two elements. This paper also answers the following algorithmic question: Can multiset perm...