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Non-wave shaping parallel bidirectional heuristic search algorithms have been reported to sufler of the bidirectional search anomaly. Although wave-shaping is considered as the most natural approach, parallel bidirectional wave-shaping algorithms are extremely scarce. We introduce a wave-shaping algorithm for parallel bidirectional heuristic search in distributed memory environments. The method is inspired by the celebrated uniprocessor bidirectional wave-shaping algorithm of DeChampeaux-Sint. Our performance evaluation shows that the proposed method scales well, maintains good performance for increasing problem sizes, and attains close to linear speedup over the sequential DeChampeaux-Sint algorithm.

We present B**-tree. a data organization method which achieves storage utilization higher than that of the B *-tree. The proposed structure pe$onns especially well in cases where the B*-tree perfoms the worst. B**-tree attains high storage utilization but incurs some I/O overhead during index creation The observed I/O overhead ranges between -20% (i.e. I/O gains) to 35%. In all tested cases, the storage utilization achieved by the proposed structure is at least 90%. 1.Introduction The B-tree data structure, originally described in [l], is the de facto standard organization for indexes in a database system [2]. There are several variations of the B-tree, the most wellknown ones being the B*-tree and the B+-tree [3]. The B-tree guarantees at least 50% storage utilization, that is, at any given time, the tree has each of its nodes at least 50% full. The B*-tree is an improvement to the B-tree and it guarantees 66% storage utilization. The B+-tree is a slightly different data structure which in addition to indexed access, it also allows sequential data processing and stores all data in the lowest level of the tree (leaf nodes). Following [3], a B-tree of order m is a multiway search tree in which the root contains k, keys, 15 k, S m -1, and every other (internal) node contains k, keys, I k, I nt-1 A B*-tree of order m is a multiway search tree in which the root contains k, keys, and every other (internal) node contains k, keys, In both B-tree and B*-tree, an inorder (symmetric) traversal of the tree produces the keys of the tree in ascending order. Since we refer to the insertion operation of B*-trees later in the 271 0-8186-4212-2'93 $03.00 8 1993 IEEE paper, we provide a brief description here. To insert a key K into a B*-tree we first find the (lea0 node N that must accept K. In case that insertion of K into node N causes an overflow, i.e. node N has m-1 keys prior to the insertion of key K, then, before splitting node N we check its adjacent right brother1. If this brother has fewer than m-1 keys, we "loan" a key to the brother rather than splitting node N. Only when a node overflows and its adjacent right brother is full (m-1 keys), do we perfonn a split. In such a case, we split the original node N and its adjacent right brother into a total of three nodes. Note, 2m keys are involved in the splitting operation (m keys in node N that has overflowed, m-1 in the adjacent right brother, and the key in the parent of N that separates N from its brother. Of these 2m keys, 2 will end up in the parent, separating the three nodes that result from the split. The remaining 2m-2 keys will be divided as evenly as possible among the 3 new nodes. Thus, each new node receives at least [2' :-2i -keys. Note, a special problem arises when the root of a B*-tree overflows. The root has no brother to loan a key, so the process described above is not applicable. So, the root of a B*-tree is allowed to I 2 . m -2 I contain as many as 2 . 131 keys2, in order that when the root is split, then the two resulting (internal) nodes each I2.m-...