text: We present two algorithms for the Group Mutual Exclusion (GME) Problem that satisfy the properties of Mutual Exclusion, Starvation Freedom, Bounded Exit, Concurrent Entry and First Come First Served. Both our algorithms use only simple read and write instructions, have O(N) Shared Space complexity and O(N) Remote Memory Reference (RMR) complexity in the Cache Coherency (CC) model. Our first algorithm is developed by generalizing the well-known Lamport's Bakery Algorithm for the classical mutual exclusion problem, while preserving its simplicity and elegance. However, it uses unbounded shared registers. Our second algorithm uses only bounded registers and is developed by generalizing Taubenfeld's Black and White Bakery Algorithm to solve the classical mutual exclusion problem using only bounded shared registers. We show that contrary to common perception our algorithms are the first to achieve these properties with these combination of complexities.
  @cite_4: A jack member onto a pair of which members a vehicle can be driven and then raised to a position inclined to the horizontal. The jack members are used more particularly for stowing vehicles in containers on a number of support frames. The jack members raise the vehicles so that they are supported at an angle to the horizontal with their wheels held in wheel support cradles attached to the frames. The jack members each comprises an approach ramp and a lifting ramp pivotally connected at one of their ends on which the vehicle is initially aligned and supported prior to being elevated at one end for inclined support on the vehicle frame.
  @cite_12: In the group mutual exclusion problem [6], which generalizes mutual exclusion [2], a process chooses a session when it requests entry to the Critical Section. A group mutual exclusion algorithm must ensure that the mutual exclusion property holds: If two processes are in the Critical Section at the same time, then they request the same session. In addition to mutual exclusion, lockout freedom, bounded exit and concurrent entering are basic properties that are desirable in any group mutual exclusion algorithm.Hadzilacos in [4] first introduced a fairness condition, called first-come-first-served (FCFS), for group mutual exclusion. The only known FCFS group mutual exclusion algorithm is due to Hadzilacos [4], and requires Θ(N2) bounded shared registers, where N is the number of processes. We present a FCFS group mutual exclusion algorithm that uses only Θ(N) bounded shared registers. (The existence of such an algorithm was posed as an open problem by Hadzilacos.)Next, we demonstrate that the FCFS property does not fully capture our intuitive notion of fairness. We therefore propose an additional fairness property, called first-in-first-enabled (FIFE). Finally, we present a reduction that transforms any "abortable" FCFS mutual exclusion algorithm M into a group mutual exclusion algorithm G. Thus, different group mutual exclusion algorithms can be obtained by instantiating M with different abortable FCFS mutual exclusion algorithms. The group mutual exclusion algorithms so obtained satisfy all of the properties mentioned above: mutual exclusion, lockout freedom, bounded exit, concurrent entering, FCFS, and FIFE.
summary: Subsequently, @cite_12 , presented an algorithm presumed to be of linear time and space, solving the challenge of Hadzilacos. retained the idea of modular composition and also the ME algorithm'' that Hadzilacos used. They came up with a clever modification to the FCFS algorithm'' of Hadzilacos to reduce the space complexity to @math . Unfortunately, though without writing it explicitly, they inherited from Hadzilacos the mistaken impression that the algorithm they use has @math time complexity. For example, the recent paper by Bhatt and Huang @cite_4 explicitly states that the RMR complexity of the algorithm by is @math .