Solving tree problems on a mesh-connected processor array

Abstract

In this paper we present techniques that result in $O\left(\sqrt{n}\right)$ time algorithms for computing many properties and functions of an n-node forest stored in an $\sqrt{n}\times \sqrt{n}$ mesh of processors. Our algorithms include computing simple properties like the depth, the height, the number of descendents, the preorder (resp. postorder, inorder) number of every node, and a solution to the more complex problem of computing the Minimax value of a game tree. Our algorithms are asymptotically optimal since any nontrivial computation will require $\Omega \left(\sqrt{n}\right)$ time on the mesh. All of our algorithms generalize to higher dimensional meshes.