We present optimal parallel solutions to reporting paths between pairs of nodes in an n-node tree. Our algorithms are deterministic and designed to run on an exclusive read exclusive write parallel random-access machine (EREW PRAM). In particular, we provide a, simple optimal parallel algorithm for pre-processing the input tree such that the path queries can be answered efficiently. Our algorithm for preprocessing runs in O(log n) time using O(n/log n) processors. Using the preprocessing, we can report paths between k node pairs in O(log n + log k) time using O(k + (n + S)/log n) processors on an EREW PRAM, where S is the size of the output. In particular, we can report the path between a single pair of distinct nodes in O(log n) time using O(L/log n) processors, where L denotes the length of the path.

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Series Lecture Notes in Computer Science
Maheshwari, A, & Lingas, A. (Andrzej). (1994). A simple optimal parallel algorithm for reporting paths in a tree. In Lecture Notes in Computer Science.