Models of Hierarchical Memory TwoLevel Memory Hierarchy Problem
Models of Hierarchical Memory
Two-Level Memory Hierarchy • Problem starts out on disk • Solution is to be written to disk • Cost of an algorithm is the number of input and output operations. • Individual items may be blocked into blocks of size B
Parallel Disk Subsystems Unrestricted Parallel Model [Aggarwal, Vitter 1987] • Items are blocked on the disk, with B items per block • Any D blocks can be read or written simultaneously in one I/O
Parallel Disk Subsystems: Parallel Disk Model [Vitter, Shriver 1990] • D blocks can be read or written simultaneously, but only if they reside on distinct disks • More realistic than the unrestricted parallel model • Still not entirely realistic, since the CPU may now become the bottleneck if D is large enough.
Parallel Memory Hierarchies 1 2 … H • H hierarchies of the same type (with H CPUs) are connected by a “network” • The network can do sorting deterministically in log H time
P Processors/D Disks • The number of disks D can be either more than, the same as, or less than the number of processors.
Multilevel Memory Hierarchies: Hierarchical Memory Model (HMM) [Aggarwal, Alpern, Chandra, Snir 1987] • Access to memory location x takes time f(x) • f is a non-decreasing function such that there exists a constant c such that f(2 x) ≤ cf(x) for all x
Multilevel Memory Hierarchies: Block Transfer Model (BT) [Aggarwal, Chandra, Snir 1987] • Access to memory location x takes time f(x) • Once an access has been made, additional items can be “injected” at a cost of one per item
Multilevel Memory Hierarchies: Uniform Memory Hierarchies (UMH) [Alpern, Carter, Feig 1990] bandwidth b (l ) l l ar blocks each of size r • There is a hierarchy of exponential-sized memory modules • Each bus has a bandwidth associated with it • All the buses can be active simultaneously
Parallel Memory Hierarchies: Results • P-HMM f(x) = log x f(x) = x a Algorithm is uniformly optimal for any cost function • P-BT f(x) = log x a f(x) = x , 0 < a < 1 a f(x) = x , a = 1 a f(x) = x , a > 1 [Vitter, Shriver 1990] These results use a modified Balance Sort for deterministic upper bounds
Parallel Memory Hierarchies: More Results • P-UMH b(l) =1 b(l) = 1/(l+1) b(l) = r-cl • P-RUMH As above except tight lower bound for b(l) = 1/(l+1) • P-SUMH b(l) =1 b(l) = 1/(l+1) b(l) = r-cl These results use a modified Balance Sort for deterministic upper bounds
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