Parking functions on toppling matrices

Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0$, $\Delta_{ij}\leq 0\text{ for }i\neq j,$ and there exists a vector ${\bf r}=(r_1,\ldots,r_n)>0$ such that ${\bf r}\Delta \geq 0$. Here the notation ${\bf r}> 0$ means that $r_i>0$ for all $i$, and ${\bf r}\geq {\bf r}'$ means that $r_i\geq r'_i$ for every $i$. Let $\mathscr{R}(\Delta)$ be the set of vectors ${\bf r}$ such that ${\bf r}>0$ and ${\bf r}\Delta\geq 0$. In this paper, $(\Delta,{\bf r})$-parking functions are defined for any ${\bf r}\in\mathscr{R}(\Delta)$. It is proved that the set of $(\Delta,{\bf r})$-parking functions is independent of ${\bf r}$ for any ${\bf r}\in\mathscr{R}(\Delta)$. For this reason, $(\Delta,{\bf r})$-parking functions are simply called $\Delta$-parking functions. It is shown that the number of $\Delta$-parking functions is less than or equal to the determinant of $\Delta$. Moreover, the definition of $(\Delta,{\bf r})$-recurrent configurations are given for any ${\bf r}\in\mathscr{R}(\Delta)$. It is proved that the set of $(\Delta,{\bf r})$-recurrent configurations is independent of ${\bf r}$ for any ${\bf r}\in\mathscr{R}(\Delta)$. Hence, $(\Delta,{\bf r})$-recurrent configurations are simply called $\Delta$-recurrent configurations. It is obtained that the number of $\Delta$-recurrent configurations is larger than or equal to the determinant of $\Delta$. A simple bijection from $\Delta$-parking functions to $\Delta$-recurrent configurations is established. It follows from this bijection that the number of $\Delta$-parking functions and the number of $\Delta$-recurrent configurations are both equal to the determinant of $\Delta$

Scheduling Coflows for Minimizing the Makespan in Identical Parallel Networks

With the development of technology, parallel computing applications have been commonly executed in large data centers. These parallel computing applications include the computation phase and communication phase, and work is completed by repeatedly executing these two phases. However, due to the ever-increasing computing demands, large data centers are burdened with massive communication demands. Coflow is a recently proposed networking abstraction to capture communication patterns in data-parallel computing frameworks. This paper focuses on the coflow scheduling problem in identical parallel networks, where the goal is to minimize makespan, the maximum completion time of coflows. The coflow scheduling problem in huge data center is considered one of the most significant $NP$-hard problems. In this paper, coflow can be considered as either a divisible or an indivisible case. Distinct flows in a divisible coflow can be transferred through different network cores, while those in an indivisible coflow can only be transferred through the same network core. In the divisible coflow scheduling problem, this paper proposes a $(3-\tfrac{2}{m})$-approximation algorithm, and a $(\tfrac{8}{3}-\tfrac{2}{3m})$-approximation algorithm, where $m$ is the number of network cores. In the indivisible coflow scheduling problem, this paper proposes a $(2m)$-approximation algorithm. Finally, we simulate our proposed algorithm and Weaver's [Huang \textit{et al.}, In 2020 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pages 1071-1081, 2020.] and compare the performance of our algorithms with that of Weaver's