refer to that object, saving space, although tracing garbage collectors, particularly incremental ones, can require additional space for other purposes. The naive algorithm described above can't handle , an object which refers directly or indirectly to itself. A mechanism relying purely on reference counts will never consider cyclic chains of objects for deletion, since their reference count is guaranteed to stay nonzero (cf. picture). Methods for dealing with this issue exist but can also increase the overhead and complexity of reference counting — on the other hand, these methods need only be applied to data that might form cycles, often a small subset of all data. One such method is the use of weak references, while another involves using a mark-sweep algorithm that gets called infrequently to clean up. In a concurrent setting, all updates of the reference counts and all pointer modifications must be atomic operations, which incurs an additional cost. There are three reasons for the atomicity requirements. First, a reference count field may be updated by multiple threads, and so an adequate atomic instruction, such as a (costly) compare-and-swap, must be used to update the counts. Second, it must be clear which object loses a reference so that its reference count can be adequately decremented. But determining this object is non-trivial in a setting where multiple threads attempt to modify the same reference (i.e., when data races are possible). Finally, there exists a subtle race in which one thread gains a pointer to an object, but before it increments the object's reference count, all other references to this object are deleted concurrently by other threads and the object is reclaimed, causing the said thread to increment a reference count of a reclaimed object. In addition to these, if the memory is allocated from a free list, reference counting suffers from poor locality. Reference counting alone cannot move objects to improve cache performance, so high performance collectors implement a tracing garbage collector as well. Most implementations (such as the ones in PHP and Objective-C) suffer from poor cache performance since they do not implement copying objects. Graph interpretation When dealing with garbage collection schemes, it is often helpful to think of the reference graph, which is a directed graph where the vertices are objects and there is an edge from an object A to an object B if A holds a reference to B. We also have a