Destination-Oriented DODAG for IoT & Quantum

• Destination-Oriented DODAG is a directed acyclic graph with unique, destination-oriented paths that guarantee loop-free connectivity.
• It underpins robust routing protocols like RPL in IoT and enhances efficient entanglement distribution in quantum networks.
• Advanced algorithms such as SB-RPL and DODAG-X optimize load balancing and reduce skewness to improve network performance.

A Destination-Oriented Directed Acyclic Graph (DODAG) is a specialized directed acyclic graph structure commonly implemented in network protocols and quantum communication algorithms to facilitate loop-free, destination-driven connectivity. In a DODAG, connectivity is explicitly oriented toward one or more distinguished root (“destination”) nodes, such that every other node has at least one outgoing edge leading toward the root(s), supporting unique-path, acyclic traversal. This structure underpins robust routing in lossy wireless networks and underlies scalable entanglement distribution in dynamic quantum networks, with key applications in Internet of Things (IoT), geographic forwarding, and multipartite quantum communication.

1. Formal Definition and Mathematical Structure

Let $G=(V,E)$ denote a physical or logical network, where nodes $V$ may represent devices, routers, or quantum stations, and edges $E$ denote communication links or quantum channels. A DODAG $D=(V_D,E_D)\subseteq G$ is a spanning, acyclic directed graph characterized by these properties (Negrin et al., 13 Aug 2024, Nguyen et al., 2019, Ramachandran et al., 2011):

• Rooted orientation: There exists a distinguished root node $r\in V$; every $v\neq r$ has a directed path to $r$.
• Out-degree constraint (in many protocols): For $v\neq r$, $|\{w:(v\to w)\in E_D\}|=1$ (single parent), though some implementations permit backup parents for reliability (Nguyen et al., 2019).
• Acyclicity: No directed cycles occur.
• Unique-path property: Each non-root node has a unique directed path to the root.
• Spanning property: $V_D=V$; $E_D$ consists of only those edges consistent with destination orientation.

In routing contexts, node “rank” is assigned via an objective function over local and link metrics, ensuring monotonic decrease along directed edges toward the root: for $(n\to p)\in E_D$, $R_p<R_n$ (Nguyen et al., 2019). In quantum network contexts, parent pointers encode the DODAG tree structure to facilitate efficient entanglement swapping and routing (Negrin et al., 13 Aug 2024).

2. Constructing DODAGs: Algorithms and Protocols

IoT/LLN Routing (RPL)

RPL (Routing Protocol for Low Power and Lossy Networks) constructs a DODAG using rank-based parent selection. Two objective functions are commonly implemented:

• Objective Function Zero (OF0): Rank is incremented by a fixed constant per hop: $R_n = R_{p^\star} + \mathit{MinHopRankIncrease}$.
• MRHOF/ETX: Rank is updated based on path quality (Expected Transmission Count), minimizing $$R_n(t) = \min_{p\in\mathcal{P}_n}\{R_p(t) + ETX_{n,p}(t)\}$$ (Nguyen et al., 2019).

Parent selection is governed by minimizing these metrics, with hysteresis to stabilize routing: parent change occurs only if $R(p′) < R(\text{current-parent}) - \sigma$.

Neighbor-Oblivious Link Reversal

Link-reversal algorithms transform a destination-disoriented DAG into a DODAG:

• Full reversal: Stuck nodes increment a local counter $t_i$, update height $h_i(t_i)$, and reverse all incoming links.
• Partial reversal: Stuck nodes increment $t_i$, updating only unresolved incoming links (Ramachandran et al., 2011).

These algorithms require either minimal state (1-bit or 2-bit per node in finite-state variants), and operate asynchronously without explicit neighbor state exchange.

Quantum Networks (DODAG-X Protocol)

In DODAG-X, a spanning arborescence is formed via breadth-first search rooted at the chosen destination node $r$, with parent pointers for unique paths. The construction algorithm selects $r$ to minimize maximum eccentricity, ensures acyclicity and unique-path via BFS, and encodes the tree for efficient routing (Negrin et al., 13 Aug 2024).

3. Fundamental Properties and Operational Significance

DODAGs possess critical properties underpinning their widespread use:

• Loop-freedom: By enforcing monotonicity (rank or state order), cycles are excluded (Nguyen et al., 2019, Ramachandran et al., 2011).
• Destination-reachability: From every non-root node, a directed path to the root guarantees message or quantum state delivery (Negrin et al., 13 Aug 2024, Ramachandran et al., 2011).
• Multi-parent resilience (selective implementations): Backup parents are maintained for failover.
• Low protocol overhead: Minimal state, absence of neighbor table maintenance, and scalable construction make DODAGs suitable for resource-constrained and dynamic environments.

4. Skewness, Balancing, and Performance Analysis

DODAGs constructed by naïve rank-based protocols (e.g., OF0 or MRHOF/ETX) tend to exhibit “skewness,” manifesting as severe load imbalance among subtrees at equal rank levels (Nguyen et al., 2019). The skewness for level-$k$ is quantified by metrics: $$\mathcal{M}_1 = \frac{ST^{\max} - ST^{\min}}{ST^{avr}},\quad \mathcal{M}_2 = \frac{\sum_{i=1}^m |ST_i - ST^{avr}|}{mST^{avr}},\quad \mathcal{M}_3 = \frac{ST^{\max}}{ST^{\min}},\quad \mathcal{M}_4 = \frac{ST^{\max} - ST^{\min}}{ST^{\min}}$$ where $ST_i$ is the size of subtree $i$ at rank-$k$ (Nguyen et al., 2019). Balanced DODAGs yield $\mathcal{M}_1 = \mathcal{M}_2 = \mathcal{M}_4 = 0$, $\mathcal{M}_3=1$.

SB-RPL addresses skewness by introducing a “Node Influence” metric to parent selection: $NI_{n,p}(t) = \alpha ST_p(t) + \beta ETX_{n,p}(t)$ leading to rank update

$R_n(t) = R_p(t) + NI_{n,p}(t)$

This biases new joins away from overloaded parents, balancing the subtree sizes.

Experimental and simulation results (Nguyen et al., 2019):

Traffic	OF0 M₁	MH_ETX M₁	SB-RPL M₁	PDR OF0	PDR SB-RPL
60 pps	3.2	2.1	1.0	62.3%	79.6%
40 pps	—	—	—	87.1%	94.7%
20 pps	—	—	—	91.2%	96.6%

SB-RPL achieves ≈65% skewness reduction and significant reliability improvements.

5. Application Domains: IoT, Geographic Forwarding, Quantum Networks

IoT and LLN Routing: RPL DODAGs underpin scalable, loop-free routing and support flexibility in rank and parent selection. The SB-RPL extension mitigates skewness, optimizing load distribution for up to 800-node deployments without growth in imbalance metrics (Nguyen et al., 2019).

Geographic Forwarding: DODAG link-reversal algorithms tackle local minima and voids by guaranteeing reachability via asynchronous neighbor-oblivious reversal (Ramachandran et al., 2011). Finite-state variants inhibit integer overflow with 1-bit or 2-bit per node state.

Quantum Networks: The DODAG-X protocol leverages the DODAG structure to minimize classical and quantum overhead in multipartite entanglement distribution. In grid and small-world topologies, DODAG-X achieves 17–35% reduction in measurement cost and enables efficient $n$-party GHZ generation (Negrin et al., 13 Aug 2024).

6. Computational Complexity and Scalability

Construction and verification of DODAGs via BFS incur $O(|V|+|E|)$ time; single-source queries are answered in $O(\mathrm{depth}(D))$ (Negrin et al., 13 Aug 2024). DODAG-X provides linear cost scaling with the number of parties and logarithmic or square-root scaling with network size in relevant topologies. Conventional repeated path-searches incur superlinear cost; DODAG parent pointers reduce this to upfront BFS plus constant-time queries. In SB-RPL, subtree size dissemination and Node Influence ranking add negligible communication and computation overhead.

7. Practical Limitations and Implementation Considerations

While DODAGs guarantee destination-oriented, loop-free paths, induced imbalance (skewness) may degrade performance in naïve deployments. Advanced balancing schemes such as SB-RPL (Node Influence metrics) are required for high-traffic or large-scale scenarios (Nguyen et al., 2019). Link-reversal algorithms necessitate asynchronous stuckness detection but eschew neighbor table maintenance (Ramachandran et al., 2011). In quantum network applications, robustness of the DODAG under link failures and the ability to split/fuse subtrees for multipartite entanglement are crucial for scalability (Negrin et al., 13 Aug 2024).

This suggests that ongoing research into load-aware parent selection, finite-state control, and DODAG layering will further extend the reach of DODAGs across heterogeneous network environments.