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Phoenix is an implementation of MapReduce on shared memory, aiming at supporting parallel computing based on multi-core/multi-processor efficiently. The extremal graph is a graph with the maximum number of edges without some given subgraphs. In this paper, by allocating the data of tasks appropriately and setting identifiers to distinguish different tasks, a parallel algorithm is proposed and used to construct the extremal graphs without hexagon. The experimental results show that the average speedup is 7.0432 on 8-core CPU and the average efficiency is 88.04% for constructing the extremal graphs of order no more than 28. Finally, three extremal graphs of order 29 without hexagon are obtained by employing the algorithm. Introduction We consider only finite undirected graphs without loops or multiple edges. For a graph G = (V(G), E(G)) of order |V(G)| and size |E(G)|, V(G) denotes the set of vertices and E(G) denotes the set of edges of G respectively. Km is the complete graph of order m, Cm is the cycle of length m. The extremal graph is a graph with the maximum number of edges containing no given subgraphs. The extremal graphs with small order can be constructed by an algorithm efficiently. However, the execution time of the algorithm grows exponentially with the increase of the orders of extremal graphs. So it seems to be difficult to solve this problem by a single-core computer, thus parallel algorithm for constructing extremal graphs based on multi-core chips will be a new trend in the future. MapReduce model, which is introduced by Google Inc., was originally used in the environments of large-scale clusters of nodes and data center, and its high-level API is also being used to develop the data-intensive applications [1]. Phoenix is an implementation of​MapReduce on shared memory, aiming at supporting computations in parallel based on multi-core/multi-processor efficiently. It contains a set of C and C++ programming API and an efficient runtime system, and it is implemented on top of P-threads [2]. In this paper, we will design a parallel algorithm based on Phoenix system to construct extremal graphs without containing hexagons.

The Phoenix-based Parallel Algorithm for Constructing Extremal Graphs