Online-LOCAL Model for Graph Algorithms

• The Online-LOCAL model is a framework where vertices are revealed sequentially and decisions are made based on a bounded-radius local view.
• It enables sharp separations in complexity, such as achieving O(log n) locality for 3-coloring in bipartite graphs compared to higher bounds in other models.
• The model supports transferring lower bounds and classifying deterministic and randomized LCLs across various graph families, highlighting its theoretical significance.

The Online-LOCAL model is a locality model for graph algorithms in which the vertices of an input graph are revealed one by one in an adversarial order, and the algorithm must irrevocably choose the output of each arriving vertex after inspecting only a bounded-radius neighborhood of the already revealed region. In the graph-theoretic literature, it was introduced as part of a unified treatment of locality in distributed, sequential, dynamic, and online computation, and it is used primarily for locally checkable labeling problems (LCLs), separations between models, and transfer of lower bounds across settings (Akbari et al., 2021).

1. Formal definition

Let $G=(V,E)$ be an undirected graph of size $|V|=n$, and let $T=T(n)\ge 0$ be the locality radius. For any node $v\in V$, the radius-$T$ neighborhood is written

$$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$

The adversary chooses an ordering $\sigma=v_1,v_2,\dots,v_n$ of the nodes. When node $v_i$ arrives, the algorithm is told the induced subgraph

$G_i = G\Bigl[\bigcup_{j=1}^i N^T(v_j)\Bigr]$

together with any fixed input labels. The algorithm maintains a partial output labeling $L_i$ and, upon seeing $|V|=n$0 and $|V|=n$1, must irrevocably choose $|V|=n$2 so that after all $|V|=n$3 nodes have arrived, the final labeling $|V|=n$4 is feasible. A problem $|V|=n$5 has Online-LOCAL complexity $|V|=n$6 if there is an Online-LOCAL algorithm of radius $|V|=n$7 that always produces a valid solution, and no smaller radius suffices (Akbari et al., 2021).

A later formulation for lower-bound work emphasizes that the algorithm may use all labels and outputs in $|V|=n$8, i.e., it has global memory over the revealed region, while locality is still measured only by the radius $|V|=n$9 of the exposed neighborhoods (Chang et al., 2023). This combination of adversarial revelation, irrevocable output, and bounded-radius visibility is the distinctive feature of the model.

2. Position in the locality hierarchy

The Online-LOCAL model is studied alongside three other locality models. In the LOCAL model, each node learns its radius-$T=T(n)\ge 0$0 neighborhood in $T=T(n)\ge 0$1 synchronous rounds and outputs based only on that view. In SLOCAL, nodes are processed sequentially in an adversarial order, and when a node is processed it learns the current labels and local memories of all nodes in its radius-$T=T(n)\ge 0$2 neighborhood. In dynamic-LOCAL, the adversary updates the graph one edge or node at a time, and only nodes within distance $T=T(n)\ge 0$3 of the update may revise their outputs. For every problem $T=T(n)\ge 0$4 and radius $T=T(n)\ge 0$5, these models satisfy

$T=T(n)\ge 0$6

and each inclusion is strict (Akbari et al., 2021).

This hierarchy is not merely definitional. The strictness is witnessed by concrete problems: the foundational separation is 3-coloring of bipartite graphs, which has locality $T=T(n)\ge 0$7 in Online-LOCAL, but much larger locality in weaker models. The model is therefore stronger than classical distributed locality, yet it remains constrained enough to support meaningful lower bounds and complexity classifications. A plausible implication is that Online-LOCAL is best viewed not as a replacement for LOCAL, but as a calibrated strengthening that isolates the algorithmic benefit of adversarial sequential revelation.

3. Deterministic complexity theory for LCLs

An LCL problem is locally checkable of radius $T=T(n)\ge 0$8 if feasibility can be verified by checking that, for every node, the labeled radius-$T=T(n)\ge 0$9 ball belongs to some finite set of allowed patterns, with finite input and output alphabets and bounded degree. For deterministic Online-LOCAL, the basic structural results are strongest on graph families where the LOCAL model is already well understood (Akbari et al., 2021).

On paths and cycles, every LCL has complexity in one of three classes in each of the four models LOCAL, SLOCAL, dynamic-LOCAL, and Online-LOCAL:

• $v\in V$0,
• $v\in V$1,
• $v\in V$2.

Moreover, the class is the same in all four models. On rooted regular trees without input labels, the corresponding trichotomy is

• $v\in V$3,
• $v\in V$4,
• $v\in V$5,

again identical across the same four models (Akbari et al., 2021).

These results show that the extra power of Online-LOCAL is not uniform across graph families. On paths, cycles, and rooted regular trees, deterministic Online-LOCAL does not create new broad complexity classes for LCLs. This is precisely why lower bounds from the LOCAL model can often be transferred to Online-LOCAL on such families.

4. Separations on general graphs

The principal positive result for deterministic Online-LOCAL is the $v\in V$6-locality algorithm for proper 3-coloring of bipartite graphs. The algorithm maintains the revealed subgraph as connected “blobs,” each properly 2-colored by colors $v\in V$7 and equipped with a nonnegative border-count $v\in V$8, while color $v\in V$9 is reserved for barrier layers. When a new node arrives, it either starts a new blob, extends one blob, or, if it touches at least two blobs of incompatible parity, the algorithm surrounds the blob with smaller $T$0 by a ring of color-$T$1 neighbors, increments $T$2, flips parity, and merges. A simple induction shows $T$3, and setting $T$4 suffices (Akbari et al., 2021).

This yields a sharp separation from weaker models already in the original framework: 3-coloring bipartite graphs has locality $T$5 in Online-LOCAL, but locality $T$6 in LOCAL and $T$7 in SLOCAL (Akbari et al., 2021). Hence Online-LOCAL is strictly stronger in general graphs, even for a standard LCL.

Subsequent work converted this example into a precise lower-bound theory. For deterministic algorithms, 3-coloring simple $T$8 grids requires locality $T$9, which matches the known $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$0 upper bound inherited from bipartite graphs. The same work proves a tight $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$1 lower bound for cylindrical and toroidal grids, and an $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$2 lower bound for $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$3-coloring general $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$4-partite graphs for fixed $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$5. It also generalizes the $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$6 upper bound to $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$7-coloring of $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$8-partite graphs that admit a locally inferable unique coloring, including triangular grids, $$N^T(v)=B(v,T)=\{u\in V:\operatorname{dist}_G(u,v)\le T\}.$$9-trees, and bipartite graphs (Chang et al., 2023).

A further refinement shows that a globally consistent north/east/south/west orientation of the grid does not reduce the deterministic or randomized Online-LOCAL complexity of 3-coloring: even on oriented grids, locality $\sigma=v_1,v_2,\dots,v_n$0 is necessary (Boudier et al., 26 Sep 2025). A common misconception is therefore that added geometric structure removes the logarithmic barrier. The established lower bounds show that it does not.

5. Randomized variants and stronger hierarchies

A randomized online-LOCAL line of work studies variants with an oblivious adversary and randomized labels. One formulation gives each node an independent private random string and reveals the induced subgraph $\sigma=v_1,v_2,\dots,v_n$1 when $\sigma=v_1,v_2,\dots,v_n$2 arrives; another uses global shared randomness. In this randomized setting, the model is used as an upper envelope for several locality notions:

$\sigma=v_1,v_2,\dots,v_n$3

Moreover, any LCL solvable by a non-signaling outcome with locality $\sigma=v_1,v_2,\dots,v_n$4 and success probability at least $\sigma=v_1,v_2,\dots,v_n$5 can be solved by a randomized online-LOCAL algorithm with locality $\sigma=v_1,v_2,\dots,v_n$6 and failure probability at most $\sigma=v_1,v_2,\dots,v_n$7 (Akbari et al., 2024).

On rooted trees, this randomized framework yields strong collapse results. If an LCL on rooted trees admits a randomized online-LOCAL algorithm of locality $\sigma=v_1,v_2,\dots,v_n$8, then it also admits a deterministic LOCAL algorithm of locality $\sigma=v_1,v_2,\dots,v_n$9. One corollary is that, on rooted trees, there is no problem with complexity between $v_i$0 and $v_i$1 across deterministic and randomized LOCAL, quantum LOCAL, bounded-dependence, non-signaling, SLOCAL, dynamic-LOCAL, and online-LOCAL (Akbari et al., 2024).

A later classification refines this picture for regular trees. For rooted regular trees, every LCL falls into one of four classes:

1. $v_i$2 in LOCAL and randomized online-LOCAL,
2. $v_i$3 in LOCAL and $v_i$4 in randomized online-LOCAL,
3. $v_i$5 in LOCAL and randomized online-LOCAL,
4. $v_i$6 in LOCAL and randomized online-LOCAL for some integer $v_i$7.

For unrooted regular trees, the super-logarithmic region is fully classified, and for general trees, problems that are global in deterministic LOCAL remain global also in randomized online-LOCAL (Dhar et al., 2024). These results delimit the range of any advantage coming from sequential revelation and randomness: it can be large on general graphs, but it disappears or sharply contracts on trees.

6. Conceptual significance and recurring themes

The Online-LOCAL model has become a focal point for understanding which locality barriers are artifacts of simultaneous distributed symmetry breaking and which persist even when the algorithm is given an adversarial reveal order and access to the revealed region as a whole. The original theory shows that all four deterministic models are distinct in general, yet often equivalent on paths, cycles, and rooted regular trees (Akbari et al., 2021). Later results show that even stronger randomized variants remain tightly constrained on trees, while retaining nontrivial additional power on general graph classes (Akbari et al., 2024).

Two technical themes recur throughout the literature. The first is parity management under sequential merging, exemplified by the blob-and-barrier algorithm for bipartite 3-coloring. The second is potential-based lower bounds, especially the $v_i$8-value arguments used for grids and the gadget constructions used for $v_i$9-partite lower bounds. These methods show that Online-LOCAL is neither an unrestricted centralized model nor a minor perturbation of LOCAL; it is a distinct locality regime with its own upper-bound mechanisms and obstruction techniques (Chang et al., 2023).

This suggests a general interpretation of the model. Its additional power is substantial when irrevocable commitments can be organized around the revealed structure of connected components, but much weaker when locality is controlled by tree-like decomposition or by invariants that survive sequential exposure. That combination of strength and limitation is what makes Online-LOCAL central in the current locality theory of graph algorithms.