Priority-based task reassignments in hierarchical 2D mesh-connected systems using tableaux

Task reassignments in 2D mesh-connected systems (2D-MSs) have been researched for several decades. We propose a hierarchical 2D mesh-connected system (2D-HMS) in order to exploit the regular nature of a 2D-MS. In our approach priority-based task assignments and reassignments in a 2D-HMS are represented by tableaux and their algorithms. We show how task relocations for a priority-based task reassignment in a 2D-HMS are reduced to a jeu de taquin slide.

💡 Research Summary

The paper addresses the long‑standing problem of task reassignment in two‑dimensional mesh‑connected systems (2D‑MS), extending it to heterogeneous environments where nodes have differing execution rates. Traditional approaches assume homogeneous nodes and restrict task allocations to rectangular submeshes, leaving priority‑driven reassignments under‑explored. To overcome these limitations, the authors introduce a hierarchical 2D mesh‑connected system (2D‑HMS) in which rows and columns are sorted in descending order of node priority (or execution speed).

The core contribution is a novel combinatorial representation of the 2D‑HMS using Young diagrams and Young tableaux. Each cell (i, j) of a Young diagram corresponds to a physical node (i, j) in the mesh, preserving the natural partial order: a node at (i‑1, j) or (i, j‑1) has higher priority than the node at (i, j). A “hierarchical 2D‑mesh tableau” (HMT) is defined as a tableau of a fixed shape λ (a partition with equal row lengths) whose entries are task identifiers. Empty cells are allowed, enabling partial or skew tableaux that capture dynamic task placement and idle nodes.

Task assignment is formalized as an injective mapping A : Tₘ → Rₙ (where Tₘ is the set of m tasks, Rₙ the set of n ≥ m nodes). The mapping must satisfy two constraints: (1) task priorities strictly decrease along rows and columns, mirroring the underlying node priority order; and (2) the set of occupied nodes forms a contiguous submesh. Under these constraints the authors propose a greedy relocation policy: when a node becomes idle, it is treated as an “inner corner” of the current tableau. The algorithm repeatedly slides the smallest‑priority neighboring task into the empty cell, moving the empty cell outward until it reaches an “outer corner”. This process is exactly a forward jeu de taquin slide as defined in the theory of Young tableaux.

The paper proves several key theoretical results linking the mesh‑based problem to classical tableau combinatorics. Lemma 2.1 shows that each jeu de taquin slide transforms the tableau’s reading word into a Knuth‑equivalent permutation. Theorem 2.2 guarantees that any sequence of slides yields a unique normal‑shape tableau, while Theorem 2.3 establishes the equivalence between jeu de taquin equivalence of tableaux and Knuth equivalence of their reading words. Consequently, all possible task reassignments under the greedy policy belong to a single jeu de taquin equivalence class, and the final configuration is the unique standard tableau of that class.

Complexity analysis reveals that each slide is O(1) and the total number of slides is bounded by the number of idle cells, giving an overall O(n) worst‑case runtime for an HMT of size n. Moreover, the number of distinct standard tableaux of shape λ (and thus the number of possible final configurations) is given by the Hook‑length formula f_λ = n! / ∏{(i,j)∈λ} h{i,j}. This provides a precise combinatorial count of feasible reassignments.

Experimental illustrations on a 4 × 4 mesh demonstrate the algorithm’s operation: after a task completes, the empty cell propagates via jeu de taquin slides to a corner, yielding a new tableau that respects priority ordering. Compared with traditional graph‑matching or heuristic methods, the tableau‑based approach reduces computational overhead and offers provable correctness.

In conclusion, the authors present a mathematically rigorous framework that unifies hierarchical heterogeneous mesh systems with Young tableau theory. The approach enables priority‑aware, fragmentation‑reducing task relocations using well‑studied combinatorial algorithms. Future work is suggested on handling dynamic priority changes, incorporating communication latency, and extending the model to irregular submesh shapes beyond the canonical rectangular partitions.