Dynamic Multi-Region Division Algorithm

• The paper introduces a dynamic partitioning method that couples region division, adaptive routing, and multi-agent task offloading to minimize delay and balance resources.
• It leverages a two-phase genetic algorithm for initial equal-sized partitioning and subsequent adaptive refinement, integrating downstream execution feedback to adjust region boundaries.
• Experimental evaluations demonstrate higher task completion rates, reduced delays, and energy savings compared to grid-based or non-partitioned approaches.

Searching arXiv for the named algorithm and closely related LEO satellite task-management papers. Tool invocation placeholder: arXiv search for "Intelligent Task Management via Dynamic Multi-region Division in LEO Satellite Networks" and related LEO satellite task offloading papers. The dynamic multi-region division algorithm denotes a workload-adaptive partitioning method for intelligent task management in Low Earth Orbit (LEO) satellite networks. In the formulation introduced in "Intelligent Task Management via Dynamic Multi-region Division in LEO Satellite Networks" (Song et al., 14 Jul 2025), the constellation is partitioned into multiple rectangular regions whose sizes are adjusted according to the workload status of individual satellites. The partitioning stage is not an isolated geometric procedure: it is coupled to intra- and inter-region routing, task splitting, and resource allocation, with the explicit objective of minimizing task delay while balancing computational and communication resources. The algorithm is implemented through a Genetic Algorithm (GA) for region division, a second GA for adaptive routing, and a Multi-Agent Deep Deterministic Policy Gradient (MA-DDPG) scheme for task splitting and offloading.

1. Definition and system scope

In the underlying network model, a LEO constellation contains $N$ satellites organized on $P$ polar orbital planes. The satellite set is $V=\{v_1,\dots,v_N\}$, while $S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$ denotes the source satellites that receive tasks from ground stations. Each source satellite $s_i$ receives $M$ tasks with sizes $\{q_m\}_{m=1}^M$, and each task $m$ can be split into $K$ sub-tasks $q_{m,1},\dots,q_{m,K}$ such that $P$0 (Song et al., 14 Jul 2025).

The region-division component partitions the constellation into $P$1 rectangular regions of width $P$2 and height $P$3. The set of regions is denoted $P$4, and the decision vector is

$P$5

where $P$6 maps each satellite to exactly one region. The adjective "dynamic" refers to the fact that the region sizes are not fixed once for all; they are adjusted when workload conditions indicate congestion or imbalance. This places the algorithm in a network-control setting rather than in image segmentation or geometric partitioning.

A recurrent misconception is to interpret "multi-region division" as a generic spatial subdivision procedure. In the present usage, the division is only one layer of a larger task-management stack. Its purpose is to organize routing and offloading decisions under limited onboard resources and uneven workload distributions, both of which can produce congestion on inter-satellite links and degrade task processing efficiency (Song et al., 14 Jul 2025).

2. Optimization formulation

For each sub-task $P$7, offloading proceeds along a route

$P$8

with computational resource $P$9 and communication rate $V=\{v_1,\dots,v_N\}$0 allocated along the route. The delay model contains a computational term and a communication term, and task completion is governed by the slowest sub-task: $V=\{v_1,\dots,v_N\}$1 Energy is defined analogously through computational and communication components, and total task energy is

$V=\{v_1,\dots,v_N\}$2

The task completion indicator is $V=\{v_1,\dots,v_N\}$3 if task $V=\{v_1,\dots,v_N\}$4 on source $V=\{v_1,\dots,v_N\}$5 finishes and $V=\{v_1,\dots,v_N\}$6 otherwise. The overall completion rate is

$V=\{v_1,\dots,v_N\}$7

Average delay is

$V=\{v_1,\dots,v_N\}$8

The joint optimization problem $V=\{v_1,\dots,v_N\}$9 is

$S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$0

subject to computational-resource, communication-rate, and energy constraints at each satellite; the task-splitting conservation constraint $S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$1; and the requirement that each satellite be assigned to exactly one region (Song et al., 14 Jul 2025).

This formulation is notable because region geometry, route selection, and per-task resource decisions are optimized in a coupled manner. The division of the constellation is therefore evaluated through downstream execution quality rather than by a purely geometric criterion. This suggests that region boundaries are treated as operational control variables rather than merely descriptive partitions.

3. Genetic algorithm for dynamic multi-region division

The region-division algorithm employs a two-phase GA. The first phase obtains an initial equal-sized partition; the second adaptively re-partitions congested regions. Two chromosome representations are used. For initial division, the chromosome is

$S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$2

which encodes region height and width. For re-division, the chromosome becomes

$S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$3

where $S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$4 indicates an incremental adjustment direction (Song et al., 14 Jul 2025).

The fitness of a candidate chromosome is not computed from region shape alone. Instead, the evaluation procedure: (i) divides the constellation into regions of size $S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$5, (ii) simulates routing and MA-DDPG offloading for a batch of $S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$6 predicted tasks, and (iii) computes

$S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$7

The GA then maximizes

$S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$8

The genetic operators are standard but specialized to the partition variables. Selection is tournament or roulette-wheel on the fitness score. Crossover is single-point crossover swapping height and width genes. Mutation, with small probability, increments or decrements $S=\{s_1,\dots,s_{N_{\rm src}}\}\subset V$9 or $s_i$0 by one unit within $s_i$1; in re-division, mutation may also flip $s_i$2 (Song et al., 14 Jul 2025).

Dynamic adjustment is triggered when a region $s_i$3 satisfies either of two conditions: its task-failure rate obeys $s_i$4, or its mean delay $s_i$5 exceeds neighboring regions by more than a threshold. The algorithm then runs the GA over $s_i$6 to search for a revised partition that improves fitness under joint simulation of overlapping regions.

The complexity analysis isolates simulation cost. With population size $s_i$7, maximum generations $s_i$8, and fitness-evaluation cost $s_i$9, each GA iteration costs $M$0, and total complexity is $M$1. Empirically, the GA converges in tens of generations (Song et al., 14 Jul 2025).

4. Coupling with adaptive routing and MA-DDPG offloading

The framework couples region division to a second GA for routing. For a source-destination pair $M$2, a route chromosome is a variable-length ordered list of satellite IDs,

$M$3

Its fitness is based on average resource richness: $M$4 where $M$5 and $M$6 are the residual computational and communication resources of satellite $M$7. After planning, the route is executed hop-by-hop; if a hop's predicted transmission delay exceeds $M$8, a sub-GA replans from the current satellite to $M$9. The routing complexity is $\{q_m\}_{m=1}^M$0 per source-destination pair, with occasional re-planning overhead (Song et al., 14 Jul 2025).

Task splitting and offloading are handled by MA-DDPG. Each source satellite $\{q_m\}_{m=1}^M$1 is modeled as an agent in a multi-agent Markov Decision Process $\{q_m\}_{m=1}^M$2. The state at time $\{q_m\}_{m=1}^M$3 is

$\{q_m\}_{m=1}^M$4

where $\{q_m\}_{m=1}^M$5 and $\{q_m\}_{m=1}^M$6 are residual compute and communication resources on the $\{q_m\}_{m=1}^M$7-th hop of the GA-planned route, $\{q_m\}_{m=1}^M$8 is the number of other agents currently planning to use satellite $\{q_m\}_{m=1}^M$9, and $m$0 is the total task size. The action is

$m$1

with $m$2, while $m$3 and $m$4 allocate compute and bandwidth fractions to sub-task $m$5.

The reward is

$m$6

where $m$7 shifts the reward baseline, $m$8 is the instantaneous objective, and $m$9 is the variance of resource usage along the route. The variance term explicitly encourages load balance. Training follows the centralized-training decentralized-execution paradigm, with actor and critic networks updated using standard DDPG policy-gradient and temporal-difference formulas with Polyak averaging (Song et al., 14 Jul 2025).

The architectural significance lies in the division of labor across three optimization layers: coarse-grained region layout, path selection, and fine-grained task-resource allocation. A plausible implication is that the region-division stage reduces the search burden faced by routing and offloading, while routing and MA-DDPG provide the feedback through which partition quality is actually judged.

5. Performance characterization

The evaluation uses three principal metrics: task completion rate $K$0, average delay $K$1, and average energy per task $K$2. Simulation settings include constellation sizes $K$3 ($K$4) and $K$5 ($K$6) satellites, $K$7, $K$8 tasks per batch, $K$9 batches, task splits $q_{m,1},\dots,q_{m,K}$0, GA population $q_{m,1},\dots,q_{m,K}$1, GA generations $q_{m,1},\dots,q_{m,K}$2, and MA-DDPG parameters replay buffer $q_{m,1},\dots,q_{m,K}$3, batch size $q_{m,1},\dots,q_{m,K}$4, $q_{m,1},\dots,q_{m,K}$5, $q_{m,1},\dots,q_{m,K}$6, and learning rates $q_{m,1},\dots,q_{m,K}$7 (Song et al., 14 Jul 2025).

For region division, the dynamic GA partition is compared with Quadtree (QTRD), grid (GRD), and no-partition (NRD) baselines. The reported gains are up to $q_{m,1},\dots,q_{m,K}$8 higher completion rate in the $q_{m,1},\dots,q_{m,K}$9 case, $P$00 to $P$01 lower average delay, and $P$02 to $P$03 energy savings per task. The $P$04 network benefits further from finer GA-tuned regions, improving delay and completion rate (Song et al., 14 Jul 2025).

For routing, the adaptive GA route planner is evaluated against centralized SDN+Dijkstra (CRP), greedy resource-priority (ORP), and random-hop (RSH). The reported improvements are $P$05 to $P$06 higher task completion and $P$07 to $P$08 lower delay in the 1600-satellite configuration.

For task offloading, MA-DDPG is compared with single-agent DDPG, DQN, Transmission-Loss-Reduction Allocation (TRLA), and random offload (ROLA). The reported results are $P$09 to $P$10 improvement in completion rate, $P$11 to $P$12 delay reduction, $P$13 to $P$14 lower energy consumption, and the lowest variance in resource usage, which is interpreted as superior load balancing (Song et al., 14 Jul 2025).

Taken together, these results support the central design claim of the framework: region division is most effective when evaluated jointly with routing and offloading rather than as a standalone partitioning heuristic.

6. Relation to other uses of “region division”

The phrase "region division" appears in several research areas, but those usages refer to different computational objects and objectives. In "CrowdCam: Dynamic Region Segmentation" (Zarrabi et al., 2018), the problem is the segmentation of dynamic regions in image collections of the same event, where a dynamic region is the projection of a moving 3D object on the image plane. That method combines geometry, appearance, and proximity in a discrete-label MRF, and its output is a pixel-wise dynamic/static segmentation rather than a workload-aware partition of a satellite network.

In "Uniform Partitioning of a Bounded Region using Opaque ASYNC Luminous Mobile Robots" (Pramanick et al., 2023), the objective is to partition a known bounded region such as a rectangle, square, or circle into equal-area subregions, each containing exactly one robot, under an opaque ASYNC luminous oblivious robot model. The relevant concerns are visibility obstruction, color states, collision-freedom, and epoch complexity, not inter-satellite congestion, routing, or multi-agent offloading.

These distinctions matter because the LEO dynamic multi-region division algorithm is not primarily a geometric partitioning method. Its essential feature is the use of partition boundaries as adaptive control variables inside a larger optimization loop involving predicted tasks, residual resources, and decentralized execution. This suggests that the defining criterion of a "good" region division in this setting is operational performance—delay, completion, energy, and resource balance—rather than geometric regularity alone.