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In the Range Minimum/Maximum Query (RMQ) and Range Maximum-Sum Segment Query (RMSQ) problems, we are given an array which we can preprocess in order to answer subsequent queries. In the RMQ query, we are given a range on the array and we need to find the maximum/minimum element within that range. On the other hand, in RMSQ query, we need to return the segment within the given query range that gives the maximum sum. In this paper, we present cache oblivious optimal algorithms for both of the above problems. In particular, for both the problems, we have presented linear time data structures having optimal cache miss. The data structures can answer the corresponding queries in constant time with constant cache miss.

In the point set embeddability problem, we are given a plane graph G with n vertices and a point set S with n points. Now the goal is to answer the question whether there exists a straight-line drawing of G such that each vertex is represented as a distinct point of S as well as to provide an embedding if one does exist. Recently, in [15], a complete characterization for this problem on a special class of graphs known as the plane 3-trees was presented along with an efficient algorithm to solve the problem. In this paper, we use the same characterization to devise an improved algorithm for the same problem. Much of the efficiency we achieve comes from clever uses of the triangular range search technique. We also study a generalized version of the problem and present improved algorithms for this version of the problem as well.