Lattice Linear Problems vs Algorithms

10/06/2022
by   Arya Tanmay Gupta, et al.
0

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.

READ FULL TEXT
research
09/29/2022

Fully Lattice Linear Algorithms

This paper focuses on analyzing and differentiating between lattice line...
research
09/27/2021

Extending Lattice linearity for Self-Stabilizing Algorithms

In this article, we focus on extending the notion of lattice linearity t...
research
02/28/2023

DAG-Inducing Problems and Algorithms

In this paper, we show that in a parallel processing system, if a direct...
research
02/14/2023

Multiplication and Modulo are Lattice Linear

In this paper, we analyze lattice linearity of multiplication and modulo...
research
08/02/2022

Lattice Linear Predicate Algorithms for the Constrained Stable Marriage Problem with Ties

We apply Lattice-Linear Predicate Detection Technique to derive parallel...
research
09/05/2023

Run for Cover: Dominating Set via Mobile Agents

Research involving computing with mobile agents is a fast-growing field,...

Please sign up or login with your details

Forgot password? Click here to reset