Browse Theses

We consider the simulation of PRAMs (Parallel Random Access Machines) by processor networks of bounded degree.

Since sorting can be used to provide a fast communication network, we study the problem of using sorters of $k$ values to form large sorting and merging networks. For $n$ an integral power of $k$, we show how to merge $k$ sorted vectors of length $n/k$ each using 4 log$\sb{k}n-3$ layers of $k$-sorters and 4 log$\sb{k}n-5$ layers of $k$-input binary mergers. We do so by modifying Leighton's ColumnSort. As a result, we show how to sort $n$ values using 2log$\sbsp{k}{2}n-{\rm log}\sb{k}n$ layers of $k$-sorters and 2log$\sbsp{k}{2}n-3{\rm log}\sb{k}n+1$ layers of $k$-input binary mergers.

Among our main results, we present an improved deterministic simulation of CRCW PRAMs on processor networks of bounded degree. We show how to simulate each step of an $n$ processor, $m$ shared variable PRAM by a bounded-degree network of $n$ processors using $O$(log $n$ log ($m/n$) + (log $n)\sp2$/log log $n$) time per simulated PRAM step. This improves on Alt, Hagerup, Mehlhorn, and Preparata's $O$(log $n$ log $m$) and Peleg and Upfal's $O(m/n +$ log $n$) algorithms for $\Omega(n($log $n$ log $n)\sp2$/log log $n) < m < O(n\sp2)$. We also prove an $\Omega$((log($m/n))\sp2$/log log ($m/n$)) lower bound for $m\geq\Omega(n)$ under the assumption that all interprocessor communication is point-to-point. This extends the lower bound of $\Omega$(log $n$ log $m$/log log $m$) discovered independently by Alt, Hagerup, Mehlhorn, and Preparata and Karlin and Upfal and improves upon the diameter bound of $\Omega$(log $n$) for $\Omega(n2\sqrt{{\rm log}\ n\ {\rm log\ log}\ n})$ $< m < O(n\sp{2 + \varepsilon})$. We also present an $\Omega$(log $m$) unrestricted lower bound for $m \geq\Omega(n\sp2)$. The only previous unrestricted lower bound was the $\Omega$(log $n$) diameter bound.

Finally, we show a lower bound of $\Omega(n\sp2$log$\sp2 m + m$ log $m$ log$\sp2 n$) for the AT$\sp2$VLSI complexity of a shared memory for $m$ variables and $n$ processors and that the AT$\sp2$ complexity of Ranade's probabilistic algorithm matches this lower bound. We also show that our deterministic algorithm has an AT$\sp2$ upper bound to within a factor of polylog of optimal for $m = O(n$ log$\sp{k}n)$. The lower bound for the case where $m\geq\Omega(n)$ has not been studied before and the upper bounds improve in both area and number of processors over constructions of Luccio, Pietracaprina, and Pucci.