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On the Instance Optimality of Bidirectional Dijkstra's Algorithm

Matic Požar

arXiv:2608.26952v1cs.DS

TL;DR

The paper asks whether shortest st-path algorithms are instance optimal in the standard query model, building on prior claims for Dijkstra variants in weighted and unweighted graphs. It constructs counterexamples, proposes a modified bidirectional Dijkstra algorithm, and proves weighted instance optimality while advancing lower bounds and partial results for unweighted and simple graphs. The results include an O(Δ) unweighted barrier and constant-factor optimality for specified simple-graph instance classes.

  • Problem

    Prior analyses claimed instance optimality for several Dijkstra variants, but the paper identifies incomplete arguments and a gap concerning shortest-path certification.

  • Method

    The paper uses shortcut-based graph constructions and lower-bound certificates, then modifies bidirectional Dijkstra and analyzes unweighted and simple-graph cases in the standard query model.

  • Results

    The considered unidirectional and bidirectional implementations are not instance optimal in general, while the modified bidirectional variant is instance optimal in weighted graphs.

  • Takeaways & Limitations

    The paper establishes weighted instance optimality for the modified algorithm, simplifies the O(Δ) unweighted lower bound, and proves constant-factor optimality for broad classes of simple-graph instances.

  • Takeaways & Limitations

    In the unweighted setting, no algorithm can achieve instance optimality up to a factor better than O(Δ) under the stated degree restriction.

Abstract

from arXiv · show

Recent work by Haeupler, Hladík, Rozhon, Tarjan, and Tětek on the instance optimality of shortest-path algorithms established several results concerning Dijkstra's algorithm and bidirectional Dijkstra's algorithm in weighted and unweighted graphs. Motivated by these results, we revisit the question of instance optimality for shortest $st$-path algorithms in the standard query model. We identify several issues in the analysis of the instance optimality of both unidirectional and bidirectional Dijkstra's algorithms and provide corresponding counterexamples. We then propose a minimal simple modification of the bidirectional Dijkstra algorithm and prove that the resulting variant is instance optimal in the weighted setting. Furthermore, we revisit the unweighted case, provide a simplified proof of the lower bound showing that no algorithm can achieve instance optimality up to a factor better than $O(Δ)$, where $Δ$ denotes the maximum degree of the graph, and discuss the implications of this result for approximation algorithms. Finally, we make progress on the open problem of instance optimality in simple graphs. We show that if the problem instance satisfies $n\ge m/16$, where $n$ is the number of nodes and $m$ is the number of edges queried by our algorithm, then it is optimal up to a constant factor. Additionally, we show instance optimality for a broad class of instances, in particular when the largest degree in the graph is at most the square root of the number of explored edges, our algorithm exhibits optimality up to a constant factor.

1 Introduction

The paper revisits instance optimality for shortest st-path algorithms, emphasizing that correctness requires both finding a shortest path and certifying its optimality. It identifies flaws in prior analyses and presents counterexamples and revised results for weighted, unweighted, and simple graphs.

  • Motivation: Instance optimality requires an algorithm to match every correct algorithm’s expected query complexity on each input up to a constant factor.The paper studies this stronger criterion for shortest st-path problems rather than relying only on worst-case or average-case guarantees.
  • Motivation: Correct shortest-path algorithms must both identify a shortest st-path and certify that no shorter path exists.Classical algorithms satisfy both requirements but may not exploit graph-specific structure.
  • Motivation: Shortcut-based algorithms can exploit graph structure to find candidate paths cheaply, creating a challenge for instance optimality when certification remains possible.The paper’s counterexamples use structural shortcuts followed by lower bounds on shortest-path length.
  • Prior work: The authors revisit prior claims that Dijkstra variants and bidirectional breadth-first search are instance optimal, finding arguments that are incomplete and require refinement.The related work includes weighted and unweighted settings.
  • Contributions: The paper gives counterexamples, modifies bidirectional Dijkstra to restore weighted instance optimality, simplifies the unweighted lower-bound proof, and advances the simple-graph case.Its simple-graph results cover instances where n is at least m/16 and broader bounded-degree conditions.

2 Preliminaries

The preliminaries define instance optimality and the shortest st-path problem in a standard query model for weighted directed multigraphs. They also review Dijkstra’s algorithm, its query operations, and the correctness convention used throughout.

  • Query model: The standard query model measures complexity by the number of queries to a weighted directed multigraph that may contain self-loops and parallel edges.Nodes expose incoming and outgoing incidence lists, and queries have constant cost.
  • Query operations: Degree(i) returns a node’s degree, while Edge(v, i) returns the i-th neighboring edge of v.These are the basic constant-cost graph queries used in the model.
  • Shortest paths: The shortest st-path problem asks for a minimum-length path from s to t in a weighted multigraph with strictly positive edge weights.The weights induce a distance function over node pairs.
  • Dijkstra’s algorithm: Dijkstra repeatedly closes the open vertex with smallest computed distance and relaxes each open neighbor’s distance.Once a vertex is closed, its computed distance equals its true distance.
  • Correctness: An algorithm is correct when it returns the correct answer with probability at least 0.9, although the cutoff is context-dependent.The paper notes that the 0.9 threshold is arbitrary and may change.
  • Instance optimality: Instance optimality requires correctness and a constant c = O(1) such that T_A(x) ≤ c · T_A′(x) for every input and correct comparison algorithm.The comparison uses expected complexity under the query measure.

3 Is Dijkstra’s Algorithm Instance Optimal?

This section constructs graph instances where a shortcut-based bidirectional algorithm is much faster than the analyzed Dijkstra variant. The separation shows that the cited Dijkstra instance-optimality claim fails under the considered restrictions.

  • Prior theorem: The section revisits a prior theorem claiming that a restricted Dijkstra algorithm is instance optimal under an adversarial next-neighbor query model.The theorem assumes directed weighted graphs with positive weights and access through seen vertices’ next out-neighbors.
  • Counterexample construction: The counterexample algorithm checks neighborhoods near s and t for a shortcut, then uses a correct shortest-path algorithm while preserving correctness.Its execution is conditioned on information from both endpoint neighborhoods.
  • Counterexample construction: GD contains s, t, m, two D-node cliques attached to s and t, and positive undirected edge weights.The graph has n = 2D + 3 nodes; the clique attached to t is irrelevant for the unidirectional counterexample but matters for bidirectional analysis.
  • Conclusion: Therefore, the restricted Dijkstra implementation is not instance optimal for the shortest st-path problem.The O(n) versus Ω(n^2) separation rules out a universal constant-factor comparison.
  • Runtime separation: Algorithm 2 uses O(n) expected queries on GD because its nested neighbor scan is capped at three iterations per outer-loop iteration.The first loop executes at most n − 1 times, and the later lower-bound check also costs O(n).
  • Runtime separation: Dijkstra requires Ω(D^2) = Ω(n^2) queries on GD because all s-side clique vertices tie at distance d(s,t)/2 and each has degree D + 1.The algorithm must explore all s_i vertices before it can confirm the path through m.

4 Instance Optimality in Weighted Graphs

The section shows that the cited bidirectional Dijkstra implementation is not instance optimal, then introduces a minimal relaxation that yields instance optimality for weighted directed and undirected multigraphs.

  • Counterexample: Algorithm 3 can require Ω(n^2) expected queries on G_D, while Algorithm 2 uses O(n) queries for n = 3 + 2D.The lower bound holds even when expectations range over all priority-queue orderings.
  • Counterexample: Uniform edge weights prevent either search direction from distinguishing the central node m from the other neighbors, forcing repeated exploration before termination.For orderings with k and r nodes closed before m, the query count is 2 min(k, r) · Ω(D).
  • Proof issue: The counterexample exposes a flaw in the claimed proof: edges skipped by Algorithm 2 can still satisfy d(s, s_i) + d(t_j, t) = d(s, t), not a strict lower bound.Thus the proof’s assumed separation between partial forward and backward distances does not hold.
  • Modified algorithm: Relaxing Line 25 to test ˆd(s, u) + ℓ(uv) + ˆd(v, t) < µ without requiring v to be closed in the backward search defines BID.The modification is designed to eliminate the counterexample’s failure mode.
  • Optimality theorem: BID is instance optimal under query complexity for shortest st-paths in directed and undirected positive-weight multigraphs.The proof establishes that every correct algorithm needs Ω(|E_s| + |E_t|) expected queries, matching BID’s O(|E_s| + |E_t|) cost.

5 Approximate Instance Optimality in Unweighted Graphs

The paper gives a simplified lower-bound construction showing that instance optimality cannot generally improve beyond a factor proportional to maximum degree, while approximation can make these instances easier.

  • The construction uses two stars joined by a direct edge (s, t) whose weight is the minimum allowed edge weight ν.The incidence-list ordering places the direct edge at a randomly selected position among the edges incident to s.
  • O(∆) is the best possible instance-optimality factor in the constructed shortest st-path instances.The lower bound applies to both expected query and time complexity.
  • Any correct algorithm must query a constant fraction of s's incident edges with constant probability before finding the direct edge, yielding expected complexity Ω(∆).
  • For some fixed instance, a tailored algorithm queries the selected edge first and solves the instance in O(1) time and queries.If the queried edge is not (s, t), it falls back to any correct shortest-path algorithm.
  • The proof depends on a particular instance and node ordering, whereas random-order instance-optimality-like measures do not require optimality for every labeling.
  • wmax/wmin is a threshold beyond which approximation may avoid identifying the exact shortest path, although the observation establishes no approximation lower bound.A direct edge can yield a wmax/wmin-approximate solution in constant time on the corresponding instance.

6 Instance Optimality in Simple Graphs

The paper advances instance optimality in simple graphs by combining a vertex-reservoir lower bound with a cross-edge counting argument. These arguments establish constant-factor optimality for broad classes of instances, including classes bounded by explored vertices or degrees.

  • Motivation: The simple-graph problem remains open because prior lower-bound constructions add edges that may create parallel edges, valid only for multigraphs.The paper therefore develops constructions that preserve simplicity, including adding nodes only under restricted assumptions.
  • Lower-bound framework: A reservoir theorem shows that algorithms avoiding a sufficiently large part of the vertex set must pay a constant fraction of BID’s search cost.This supports instance optimality for instances with a large vertex reservoir and classes satisfying |Es| ≤ K|Ns| for constant K.
  • Instance-optimal classes: BID is instance optimal up to a constant factor on classes where its explored forward edges satisfy |Es| ≤ K|Ns| for fixed K = O(1).The proof compares the lower bound TA(G, s, t) = ΩK(|Es| + |Et|) with BID’s O(|Es| + |Et|) query cost.
  • Lower-bound framework: For every cross edge between the forward and backward explored regions, any correct randomized algorithm must query it with probability at least 0.25, implying TA(G, s, t) ≥ m/4.Here m is the number of cross-edges between X = V(Gs) and Y = V(Gt).
  • Instance-optimal classes: The cross-edge argument also yields q = Ω(M) when Ms, Mt = Ω(M) and ∆s∆t = O(M), providing the tensor lower bound used for simple-graph optimality.The bound concerns q = e(X, Y), the number of queried cross edges between the two explored regions.
  • Instance-optimal classes: If ∆² ≤ S, every correct algorithm satisfies TA(G, s, t) ≥ S/64, so BID is instance optimal up to a constant factor.S = |Es| + |Et|, and the argument uses the cross-edge theorem with ∆s∆t ≤ ∆² ≤ S.

7 Conclusion

The paper revisits instance optimality for Dijkstra-style shortest-path algorithms, exposing flaws in prior analyses and restoring weighted-case optimality through a simple bidirectional modification. It also advances results for unweighted and simple graphs under stated structural conditions.

  • 7 Conclusion: The proposed shortcut-based algorithm exploits graph structure while preserving correctness through lower-bound certificates.It identifies paths using structural shortcuts and certifies their optimality with suitable lower bounds.
  • 7 Conclusion: The considered unidirectional and bidirectional Dijkstra implementations are not instance optimal in general.The paper establishes this using a family of graph instances designed to expose their shortcomings.
  • 7 Conclusion: A simple modification restores instance optimality for bidirectional Dijkstra’s algorithm in the weighted setting.The paper also gives a simplified proof of the related prior result.
  • 7 Conclusion: The simplified unweighted lower-bound proof highlights the role of shortest-path-length lower bounds in instance-optimal guarantees.This result concerns the lower bound previously stated as Theorem 6.2.
  • 7 Conclusion: For simple graphs, BID is instance optimal when n is at least 1/16 of its query complexity or when maximum degree is at most the square root of explored edges.Whether instance optimality holds for arbitrary simple graphs remains open.
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