On the Time and Space Complexity of ABA Prevention and Detection

Abstract

We investigate the time and space complexity of detecting and preventing ABAs in shared memory algorithms for systems with n processes and bounded base objects. To that end, we define ABA-detecting registers, which are similar to normal read/write registers, except that they allow a process q to detect with a read operation, whether some process wrote the register since q's last read. ABA-detecting registers can be implemented trivially from a single unbounded register, but we show that they have a high complexity if base objects are bounded: An obstruction-free implementation of an ABA-detecting single bit register cannot be implemented from fewer than n − 1 bounded registers. Moreover, bounded CAS objects (or more generally, conditional read-modify-write primitives) offer little help to implement ABA-detecting single bit registers: We prove a linear time-space tradeoff for such implementations. We show that the same time-space tradeoff holds for implementations of single bit LL/SC primitives from bounded writable CAS objects. This proves that the implementations of LL/SC/VL by Anderson and Moir [2] as well as Jayanti and Petrovic [15] are optimal.We complement our lower bounds with tight upper bounds: We give an implementation of ABA-detecting registers from n + 1 bounded registers, which has step complexity O(1). We also show that (bounded) LL/SC/VL can be implemented from a single bounded CAS object and with O(n) step complexity. Both upper bounds are asymptotically optimal with respect to their time-space product.These results give formal evidence that the ABA problem is inherently difficult, that even writable CAS objects do not provide significant benefits over registers for dealing with the ABA problem itself, and that there is no hope of finding a more efficient implementation of LL/SC/VL from bounded CAS objects and registers than the ones from [2,15].

Strongly Linearizable Implementations of Snapshots and Other Types

“…An ABA-detecting register [3] is a type that stores a single value R from some domain D, and supports the invocation descriptions DW rite q (x) for x ∈ D, and DRead q () with the following sequential specification: Initially, R = ⊥ ∈ D, and a DW rite q (x) invocation changes the value of R to x. Invocation DRead q () returns a pair (x, a) ∈ D × {true, f alse}, where x is the value of R, and a is true if and only if q performed an earlier DRead q () operation, and a DW rite p was performed by some process p since q's last DRead q ().…”

“…If the X.Read() operation on line 37 performed by dr following xr 2 returns (x, p, s), then by the loop-guard on line 41, dr terminates after this iteration of the main loop. This is a contradiction, since dr must restart the repeat-until loop after xr 2 in order to perform xr 3 . Therefore, the X.Read() operation on line 37 performed by dr following xr 2 returns (x ′ , p ′ , s ′ ) = (x, p, s).…”

“…In this section, we presented the first strongly linearizable implementation of an ABA-detecting register by modifying a previous linearizable implementation by Aghazadeh and Woelfel [3,25]. An obvious extension of this work would examine the possibility of a wait-free strongly linearizable implementation of an ABA-detecting register.…”

“…As a building block, our algorithm uses a lock-free strongly linearizable ABA-detecting register. We obtain this object by modifying the wait-free linearizable ABA-detecting register of Aghazadeh and Woelfel [3], which, as we show, is not strongly linearizable.Aspnes and Herlihy [4] identified a wide class types that have wait-free linearizable implementations from atomic registers. These types require that any pair of operations either commute, or one overwrites the other.…”

“…As a building block, our algorithm uses a lock-free strongly linearizable ABA-detecting register. We obtain this object by modifying the wait-free linearizable ABA-detecting register of Aghazadeh and Woelfel [3], which, as we show, is not strongly linearizable.…”