Implementing Shared Memory on Mesh-Connected Computers and on the Fat-Tree

We present deterministic upper and lower bounds on the slowdown required to simulate an (n;m)-PRAM on a variety of networks. The upper bounds are based on a novel scheme that exploits the splitting and combining of messages. This scheme can be implemented on an n-node d-dimensional mesh (for constant d) and on an n-leaf pruned butter y and attains the smallest worst-case slowdown to date for such interconnections, namely, O n1=d(log(m=n))1 1=d for the d-dimensional mesh (with constant d) and O(pn log(m=n)) for the pruned butter y. In fact, the simulation on the pruned butter y is the rst PRAM simulation scheme on an area-universal network. Finally, we prove restricted and unrestricted lower bounds on the slowdown of any deterministic PRAM simulation on an arbitrary network, formulated in terms of the bandwidth properties of the interconnection as expressed by its decomposition tree. 3 List of Symbols Used 1 one l lower-case ell 0 zero O upper-case letter oh O(n) upper-case oh symbol for big-oh notation pn square root a accented lower-case a A A A times A (n) upper-case Greek omega <; ; >; standard inequalities log n logarithm minfX;Y g minimum lower-case Greek alpha subset symbol proper subset symbol jU j set size lower-case Greek lambda lower-case Greek sigma (n) upper-case Greek Theta upper-case Greek gamma upper-case Greek sigma (indicating summation) Ŝ S hat 3 4 3 dot 4 (indicating multiplication) bXc oor of X S union symbol dXe ceiling of X a; b; : : : ; z ellipsis fa; b; cg curly brackets (indicating set notation) < a; b; c > angle brackets (delimiting triple) [k]j square brackets subscripted with j 1 in nity [X;Y ) left square bracket, X, comma, Y , right round bracket plus or minus upper-case Greek phi upper-case Greek delta m m bar x0 x prime ab a to the b-th power (exponentiation) 4