2 research outputs found
Lattice Linear Problems vs Algorithms
Modelling problems using predicates that induce a partial order among global
states was introduced as a way to permit asynchronous execution in
multiprocessor systems. A key property of such problems is that the predicate
induces one lattice in the state space which guarantees that the execution is
correct even if nodes execute with old information about their neighbours.
Thus, a compiler that is aware of this property can ignore data dependencies
and allow the application to continue its execution with the available data
rather than waiting for the most recent one. Unfortunately, many interesting
problems do not exhibit lattice linearity. This issue was alleviated with the
introduction of eventually lattice linear algorithms. Such algorithms induce a
partial order in a subset of the state space even though the problem cannot be
defined by a predicate under which the states form a partial order.
This paper focuses on analyzing and differentiating between lattice linear
problems and algorithms. It also introduces a new class of algorithms called
(fully) lattice linear algorithms. A characteristic of these algorithms is that
the entire reachable state space is partitioned into one or more lattices and
the initial state locks into one of these lattices. Thus, under a few
additional constraints, the initial state can uniquely determine the final
state. For demonstration, we present lattice linear self-stabilizing algorithms
for minimal dominating set and graph colouring problems, and a parallel
processing 2-approximation algorithm for vertex cover.
The algorithm for minimal dominating set converges in n moves, and that for
graph colouring converges in n+2m moves. The algorithm for vertex cover is the
first lattice linear approximation algorithm for an NP-Hard problem; it
converges in n moves.
Some part is cut due to 1920 character limit. Please see the pdf for full
abstract.Comment: arXiv admin note: text overlap with arXiv:2209.1470