Representations of task assignments in distributed systems using Young tableaux and symmetric groups

Abstract

This paper presents a novel approach to representing task assignments for partitioned agents (respectively, tasks) in distributed systems. A partition of agents (respectively, tasks) is represented by a Young tableau, which is one of the main tools in studying symmetric groups and combinatorics. In this paper we propose a task, agent, and assignment tableau in order to represent a task assignment for partitioned agents (respectively, tasks) in a distributed system. This paper is concerned with representations of task assignments rather than finding approximate or near optimal solutions for task assignments. A Young tableau approach allows us to raise the expressiveness of partitioned agents (respectively, tasks) and their task assignments.1 In this paper we use agent and processor interchangeably if parallel agents in a distributed system are considered as simple computing entities [44]. A task graph T = (V, E) is a directed acyclic graph, where each node in V = {v 1 , v 2 , . . . , v n } denotes a task, and each edge (v i , v j ) ∈ E ⊂ V × V denotes a precedence relationship between tasks, i.e., v j cannot begin before v i completes. The positive weight associated with each task v ∈ V represents a computation requirement. The nonnegative weight associated with each edge (v i , v j ) ∈ E represents a communication requirement.A fully-connected heterogeneous system A is a set A = {a 1 , a 2 , . . . , a m } of m heterogeneous agents whose network topology is fully-connected. A heterogeneous system A is called consistent if agent a i ∈ A executes a task n-times faster than agent a j ∈ A, then it executes all other tasks n-times faster than agent a j . A heterogeneous system A is called communication homogeneous if each communication link in A is identical.Let T = (V, E) be a task graph and A = {a 1 , a 2 , . . . , a m } be a fully-connected heterogeneous system. Assume that a startup cost of initiating a task on an agent is negligible. In a consistent system, the computation cost of taskis the computation requirement of task v i , and e(a j ) is the execution rate of agent a j . Meanwhile, in an inconsistent system, the computation cost of task v i on a j is given by ω(v i , a j ) = w ij , where w ij is the (i, j) th entry in a |V | × |A| cost matrix W . Note that an inconsistent system model is a generalization of a consistent system model. Now, the communication cost model of A is defined as follows. Let d(v i , v j ) be the amount of data to be transferred from task v i to task v j for each (v i , v j ) ∈ E; let t(a s , a t ) be the data transfer rate between the communication link between agent a s and a t in A. Let M be a task assignment between T and A such that M (v i ) = a s and M (v j ) = a t for v i , v j ∈ T and a s , a t ∈ A. Assume that both local communication and communication startup cost are negligible. If the communication of A has the linear cost model [2], the communication cost between task v i on agent a s and task v j on a t is given by c(M (v i ), M (v j )) = d(v i , v j )/t(a s , a t ) if a ...

Task swapping networks in distributed systems

“…Cayley graph approaches to interconnection networks have been researched in [2,21,30,47], but no (bijective) task assignment of tasks to agents is involved in them. Similarly, the group-theoretic approaches for bijective task assignments have been discussed in [20,28,38], but task swappings among agents in a network topology have not been considered. Although minimum-generator sequence and cycle factorization problems have been researched in [24,26,31,35,47], no task swapping or task assignment is considered in them.…”

“…In this type of problems a group theory can be used to represent task reassignments including iterative task swappings. A grouptheoretic approach to representing task assignments or reassignments have already been researched [28,38]. However, little work has been done about task swappings for task reassignments between tasks and agents by using group theory.…”

Priority-Based Task Reassignments in Hierarchical 2d Mesh-Connected Systems Using Tableaux

“…One of the common methods to represent and solve a task assignment problem is a graph-theoretic method using a graph-matching algorithm [15,25,34]. To complement the graph-theoretic method, our previous work [16] presented a Young tableaux [13,22,29,37] approach to representing task assignments.…”