# I. Introduction

A priority queue is a fundamental abstract data type that supports two operations: Insert\((k)\) and Extract-Max. Priority queues have many practical applications, especially in areas such as scheduling and event driven simulations. Some examples of applications of priority queues in distributed environments appear in [1]. One example where there are multiple inserters, but only a single extractor, is the priority queue for a shared printer.

Efforts to build faster computers have shifted towards increasing the number of cores on a single computer. As a result, scalable and highly concurrent data structures in the asynchronous shared memory model are becoming increasingly important. In this model, each process can take an arbitrary amount of time between steps and may crash. Processes communicate by reading from and writing to shared registers, and by accessing LL/SC objects. If \(x\) is an LL/SC object, then LL(x) reads \(x\). SC\((x, v)\) by process \(p\) writes \(v\) to \(x\) if \(x\) has not been written to by an SC operation since the last time \(p\) performed LL(x). If SC\((x, v)\) writes \(v\) to \(x\) then it returns True. Otherwise it returns False.

Lock-freedom is a property that guarantees that, while operations are in progress and processes take steps, some operation will finish within a finite number of steps. Wait-freedom is a stronger property that guarantees that every operation performed by a non-faulty process finishes within a finite number of steps by that process.

Israeli and Rappoport [2] present a lock-free priority queue based on a heap, using LL/SC variables, with \(O(n\log{m})\) amortized step complexity for both Insert and Extract-Max, where \(n\) is the number of processes and \(m\) is the maximum number of elements in the priority queue during the operation. They outline how to extend it to a wait-free implementation using 2-word LL/SC, but with step complexity. This is the best wait-free implementation known.

In 2005, Jayanti and Petrovic [3] showed how to implement single-dequeuer queues with step complexity \(\Theta(\log{}n)\) for both Enqueue and Dequeue.

Inspired by this queue implementation, we consider a priority queue where only one process is allowed to perform Extract-Max. Our single-extractor priority queue implementation uses single word LL/SC objects. It is wait-free and has step complexity for both Insert and Extract-Max, where \(n\) is the number of inserters and \(m\) is the maximum number of elements in the priority queue during the operation.

Our implementation uses a single-inserter single-deleter (SISD) ordered multiset, which supports insertion (SISD-Insert) and deletion (SISD-Delete) with \(O(\log{r})\) step complexity, where \(r\) is the maximum number of elements in the multiset during the operation. Finding the maximum element (SISD-FindMax) can be done with constant step complexity. Both the inserter and the deleter can perform SISD-FindMax.

# II. The Construction

Suppose, for simplicity, that the number of inserters, \(n\), is a power of 2. Our single-extractor priority queue implementation uses a complete binary tree with \(n\) leaves. Each leaf corresponds to one inserting process. At each leaf, there is a single-inserter single-deleter (SISD) ordered multiset. Each internal node stores the largest key in its subtree as well as the index of the leaf that the key came from. An Insert operation first inserts into its own SISD multiset and then helps propagate the largest key up the complete binary tree. An Extract-Max operation reads the largest key from the root of the binary tree, deletes it from the SISD multiset of the appropriate leaf, and propagates the new maximum up the binary tree. We use an SISD multiset rather than an SISD priority queue because, by the time an Extract-Max tries to delete a key from the SISD multiset of a leaf, it might no longer be the largest key in the leaf. Our single-extractor priority queue implementation is wait-free, and has step complexity for both Insert and Extract-Max, in addition to the time it takes to perform the SISD multiset insert and delete operations. Note that any process can perform Extract-Max as long as there is some way of guaranteeing that two or more Extract-Max operations are not performed concurrently.

The SISD multiset is implemented using a persistent AVL tree. Each process announces the operation it wishes to perform, and the processes alternate between helping each other when there is contention. We use a novel adaptation of handshaking to ensure all operations are performed without duplication. The step complexity of insertion and deletion is \(O(\log{r})\), where \(r\) is the maximum size of the multiset during the operation, while finding the maximum element can be implemented in constant time.

# III. Future Work

It would be interesting to obtain either a \(\Omega(\log{n} + \log{m})\) lower bound or a faster implmentation of Insert or Extract-Max. Improvements to the step complexities of SISD-Insert or SISD-Delete will reduce the \(\log{}m\) term in the step complexities of Insert or Extract-Max, respectively.

#### Acknowledgements

We would like to thank our supervisor, Professor Faith Ellen, for the advice she gave us while writing this paper, and for the numerous hours she put into editing our work. This work was supported by NSERC Undergraduate Student Research Awards.

#### References

[1] J. Lindén and B. Jonsson, "A skiplist-based concurrent priority queue with minimal memory contention," in *Proceedings of the 17th International Conference on Principles of Distributed Systems (OPODIS)*, 2013, pp. 206–220.

[2] A. Israeli and L. Rappoport, "Efficient wait-free implementation of a concurrent priority queue," in *Proceedings of the 7th International Workshop on Distributed Algorithms (WDAG)*, 1993, pp. 1–17.

[3] P. Jayanti and S. Petrovic, "Logarithmic-time single deleter, multiple inserter wait-free queues and stacks," in *Proceedings of the 25th International Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS)*, 2005, pp. 408–419.