Data Independence of Read, Write, and Control Structures in PRAM Computations

Klaus-Jörn Lange, Rolf Niedermeier
2000 Journal of computer and system sciences (Print)  
We introduce the notions of control and communication structures in PRAM computations and relate them to the concept of data independence. Our main result is to characterize differences between unbounded fan-in parallelism AC k , bounded fan-in parallelism NC k , and the sequential classes DSPACE(log n) and LOGDCFL in terms of a PRAM's communication structure and instruction set. Our findings give a concrete indication that in parallel computations writing is more powerful than reading. Further
more » ... characterizations are given for parallel pointer machines and the semiunbounded fan-in circuit classes SAC k . In particular, we obtain the first characterizations of NC k and DSPACE(log n) in terms of PRAMs. Finally, we introduce Index-PRAMs, which in some sense have built-in data independence. We propose Index-PRAMs as a tool for the development of data-independent parallel algorithms. Index-PRAMs serve for studying the essential differences between the above mentioned complexity classes with respect to the underlying instruction set used. general-purpose simulations of PRAMs on real machines, as opposed to special implementations of certain algorithms on certain machine architectures. The direct, general approach, using techniques such as hashing and slackness (see [1, 51, 61, 62] ), is limited by problems such as hardware cost, (non)scalability, and interconnect length (see [9, 19, 31, 45, 61, 62, 67] for discussion). The main alternative has been to base algorithm design on restricted forms of PRAMs (see [12, 32] ) or on models that try to build in more"computational realism" [2, 3, 19, 21, 33, 62, 61] . Our approach is to stay with the generality and simplicity of the basic PRAM models, but without using the classifications offered by parallel complexity theory in terms of P-completeness and NC-membership. Since this dichotomy apparently is not appropriate when trying to speed-up running times by using rather weak parallel machines or networks of workstations, we tried to abstract out general features of algorithms that lead to efficient implementations. The features we emphasize are data independence of the read, write, and control structures of the algorithm. Many of the computation-intensive tasks targeted by the"grand challenges" [44, 57] seem to be simpler than what is needed for a general PRAM simulation. Their communication and control structures are simple enough that an efficient implementation on existing architectures, working with distributed memories and message passing mechanisms, is possible. For example, tasks like FFT, parallel prefix sums ("scans"), and matrix operations are data independent on all counts. For operations on graphs as pointer jumping (list ranking) this may not be the case. In general, parallel graph algorithms depend on how graphs are represented. There are two simple representations of graphs: adjacency matrices and edge lists. Algorithms working on adjacency matrices usually show data-independent behavior (e.g., Warshall algorithm), whereas algorithms using edge lists (e.g., pointer jumping or list ranking problem) show inherent dependence of the communication structure in the underlying data the addresses of global memory cells used strongly depend on the list structure. The importance of data-independent readsÂwritesÂcontrol in distributed memory computing has been pointed out in several papers, e.g., [29, 39, 58, 66] . The main contributions of this paper are as follows: 1. We formally introduce the notions of data-independent reads, writes, and control: Data independence of control means that the statement executed by a processor of a PRAM depends only on time, processor identification number (PIN for short), and length of the input, but not on the input itself. Data independence of communication structure means that in global read accesses (resp., the receipt of messages) or write accesses (resp., the sending of messages) the addresses of shared memory cells depend only on time, PIN, and input length. 2. From the formal notion we obtain surprising results these restrictions lead to complexity classes whose original definitions had seemingly nothing to do with PRAMs: Whereas unbounded fan-in parallelism, represented by the classes AC k , is characterized by a data-dependent control or write structure in combination with a data-independent read structure, bounded fan-in parallelism, represented by 110 LANGE AND NIEDERMEIER
doi:10.1006/jcss.1999.1665 fatcat:mvcun62f35h6fex7wtz7ttej2m