Self-healing Routing and Other Problems in Compact Memory
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This paper looks at the question of designing distributed algorithms for the setting of compact memory i.e. sublinear (in n - the number of nodes) bits for connected networks of arbitrary topologies. The nodes in our networks may have much lower internal memory (say, O(poly n)) as compared to the number of their possible neighbours. This implies that a node needs to run the algorithm and do computations without being even able to store the states or IDs of all its neighbours. Towards this end, we introduce the Compact Memory Passing model (CMP)- the standard message passing model at a finer granularity where a node can interleave reads and writes with internal computations. This is required for meaningful computations due to the low memory requirement and is somewhat akin to a distributed network with nodes executing streaming algorithms. We believe this model captures features of large networks of small memory devices (e.g. the Internet of Things (IOT)) and initiates research into theoretical foundations in this area. Compact Routing Messages in Self-Healing Trees (Distributed Computing 2017) introduced the compact self-healing routing algorithm CompactFTZ assuming `regular' memory for preprocessing and posed the problem of its compact preprocessing. We solve this problem and, hence, introduce the first fully compact self-healing routing algorithm. In the process, we also give independent fully compact algorithms for the Forgiving Tree [PODC 2008] Thorup-Zwick's tree based compact routing [SPAA 2001], and fundamental problems of leader election, tree constructions and traversals (BFS, DFS, spanning trees and convergecast). Our nodes have only O(^2 n) local memory but the preprocessing can be accomplished using O( n) bits sized messages (as in the CONGEST model). We also give a faster solution for O( n) bits sized messages.
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