Parallel Working-Set Search Structures

Abstract

In this paper 1 we present two versions of a parallel working-set map on p processors that supports searches, insertions and deletions. In both versions, the total work of all operations when the map has size at least p is bounded by the working-set bound, i.e., the cost of an item depends on how recently it was accessed (for some linearization): accessing an item in the map with recency r takes O(1+logr ) work. In the simpler version each map operation has O (logp) 2 + logn span (where n is the maximum size of the map). In the pipelined version each map operation on an item with recency r has O (logp) 2 + logr span. (Operations in parallel may have overlapping span; span is additive only for operations in sequence.)Both data structures are designed to be used by a dynamic multithreading parallel program that at each step executes a unittime instruction or makes a data structure call. To achieve the stated bounds, the pipelined version requires a weak-priority scheduler, which supports a limited form of 2-level prioritization. At the end we explain how the results translate to practical implementations using work-stealing schedulers.To the best of our knowledge, this is the first parallel implementation of a self-adjusting search structure where the cost of an operation adapts to the access sequence. A corollary of the working-set bound is that it achieves work static optimality: the total work is bounded by the access costs in an optimal static search tree.

Optimal Parallel Algorithms in the Binary-Forking Model

“…However, they are not work-efficient, requiring O(m log n) work. There is also previous work focusing on I/O efficiency [15] and concurrent operations [37,57] for parallel trees, and parallel data structures supporting batches [4,64,78]. Some previous algorithms achieve optimal work and polylogarithmic span.…”

“…All the above-mentioned algorithms have O(log m log n) span in the binary-forking model. There have also been parallel bulk operations for self-adjusting data structures [4]. As far as we know, there is no parallel algorithm for ordered set functions (Union, Intersection and Difference) with optimal work and O(log n) span in the binary-forking model.…”

Optimal Multithreaded Batch-Parallel 2-3 Trees

“…Searches are easy, and pose no problem for a multithreaded implementation. But performing a sorted batch of b insertions or deletions on the PVW 2-3 tree with n items essentially involves spawning O(log b) synchronous waves of structural changes from the bottom of the 2-3 tree upwards to the root, each wave taking O (1) steps to move up one level. This relies crucially on the lock-step synchronicity of the processors to ensure that these waves never overlap, and naively attempting to use locking to prevent waves from overlapping will cause the worst-case span to increase from O(log b…”

“…We shall work within the QRMW PPM model that was introduced in [1] as a more realistic PPM (parallel pointer machine) model for parallel programming. Unrealistic assumptions of the synchronous PRAM model include the lock-step synchronicity of processors and the lack of collision on concurrent memory accesses to the same locations [9,10,18].…”

“…Furthermore, since T is a multithreaded data structure whose performance bounds are independent of the schedule, it is trivially composable, which means that we can use T as a black-box data structure in any multithreaded algorithm and easily obtain composable performance bounds. Indeed, the parallel working-set map in [1] and the parallel finger structure in [11] both rely such a parallel 2-3 tree as a key building block.…”