Some Problems in Compact Message Passing

Abstract

This paper seeks to address the question of designing distributed algorithms for the setting of compact memory i.e. sublinear (in n -the number of nodes) bits working memory for connected networks of arbitrary topologies. The nodes in our networks may have much lower internal (working) memory (say, O(poly log n)) as compared to the number of their possible neighbours (O(n)) implying that a node may not be even able to store the IDs of all of its neighbours. These algorithms may be useful for large networks of small devices such as the Internet of Things, for wireless or ad-hoc networks, and, in general, as memory efficient algorithms.More formally, we introduce the Compact Message Passing (CM P ) model -an extension of the standard message passing model considered at a finer granularity where a node can interleave reads and writes with internal computations, using a port only once in a synchronous round. The interleaving is required for meaningful computations due to the low memory requirement and is akin to a distributed network with nodes executing streaming algorithms. Note that the internal memory size upper bounds the message sizes and hence e.g. for O(log n) memory, the model is weaker than the CONGEST model; for such models our algorithms will work directly too.We present some early results in the CMP model for nodes with O(log 2 n) bits working memory. We introduce the concepts of local compact functions and compact protocols and give solutions for some classic distributed problems (leader election, tree constructions and traversals). We build on these to solve the open problem of compact preprocessing for the compact self-healing routing algorithm CompactFTZ posed in Compact Routing Messages in Self-Healing Trees (Theoretical Computer Science 2017) by designing local compact functions for finding particular subtrees of labeled binary trees. Hence, we introduce the first fully compact self-healing routing algorithm. In the process, we also give independent fully compact versions of the Forgiving Tree [PODC 2008] and Thorup-Zwick's tree based compact routing [SPAA 2001].