Bounded-Suboptimal Bidirectional Search

• Bounded-suboptimal bidirectional search is a framework that finds paths with a guarantee of cost not exceeding a specified multiple of the optimal path cost.
• The method leverages dual frontiers and consistent heuristics, using tailored priority functions to balance rapid solution discovery with error correction.
• Empirical results show that these techniques can significantly reduce node expansions and runtime compared to traditional unidirectional search algorithms.

Bounded-suboptimal bidirectional search is a family of algorithms for graph search that seeks paths between designated states subject to an explicit suboptimality bound on solution cost, while leveraging bidirectional strategies and consistent heuristics. These methods build upon optimal bidirectional search frameworks, generalizing them to produce paths whose cost is guaranteed not to exceed a specified multiple of the optimal cost, providing a rigorous performance guarantee with potential for significant empirical efficiency gains (Shperberg et al., 13 Nov 2025).

1. Problem Formulation and Preliminaries

Given a graph $G=(V,E)$ (directed or undirected) with non-negative edge costs $c(u,v)$, a start node $s$, and a goal node $g$, the objective is to find a path whose cost $C$ does not exceed $W$ times the optimal cost $C^*$: $C \leq W\cdot C^*$ where the optimal cost is

$$C^* = \min\{\mathrm{cost}(\pi)\mid \pi \text{ is a path } s\to g\}.$$

The parameter $W\ge1$ defines the allowed suboptimality, commonly expressed as $C\le(1+\varepsilon)C^*$, with $W=1+\varepsilon$. This framework generalizes the classical A* and Weighted A* (WA*) paradigms to the bidirectional setting with explicit suboptimality guarantees.

2. Algorithmic Foundations: BAE* and its Bounded-Suboptimal Variants

The Bidirectional A* front-to-end algorithm (BAE*) is defined for consistent heuristics. It maintains two simultaneous frontiers: one forward from $s$ (F) using $h_F(n)\approx c(n,g)$, and one backward from $g$ (B) using $h_B(n)\approx c(s,n)$. For each node, it computes the heuristic error $d_F(n)=g_F(n)-h_B(n)$ in the forward direction, and analogously in the backward direction. The node expansion priorities are given as

$$b_F(n)=g_F(n)+h_F(n)+d_F(n), \qquad b_B(n)=g_B(n)+h_B(n)+d_B(n).$$

Expansion alternates or uses a direction-choice rule, always expanding the node with minimum $b_D$ ($D\in\{F,B\}$). BAE* terminates when a node $m$ appears in both open lists and $$\mathrm{cost}(s\to m\to g)\le\frac{b_{\min}^F + b_{\min}^B}{2}\le C^*$$.

Bounded-suboptimal variants ("WBAE*") modify these priorities to

$$b_{W,F}(n)=g_F(n)+W\,h_F(n)+\lambda\,d_F(n), \qquad b_{W,B}(n)=g_B(n)+W\,h_B(n)+\lambda\,d_B(n),$$

where $\lambda\leq W$ is a parameter controlling the weighting of the error term. $\lambda=0$ yields weighted bidirectional A* (WBiA*); $\lambda=W$ yields full error correction, and intermediate values trade off solution-finding speed and the tightness of the lower bound. The algorithm terminates when

$$\exists n \text{ in both frontiers: } \mathrm{cost}(s\to n\to g)\le LB_{WB}, \quad LB_{WB} = \frac{\min b_{W,F} + \min b_{W,B}}{2}.$$

3. Theoretical Properties: Consistency, Suboptimality, and Completeness

A heuristic is consistent if

$h(u)\le c(u,v)+h(v)$

for all $(u,v)$. Consistency ensures that a node is never reexpanded, provided that ties are broken in favor of higher $g$-values.

Bounded-suboptimality is rigorously guaranteed: $$LB_{WB} = \frac{\min b_{W,F} + \min b_{W,B}}{2} \le W\cdot \frac{\min b_F + \min b_B}{2} \le W\,C^*.$$ Thus, when WBAE* halts, the incumbent solution cost $C$ satisfies $C\le LB_{WB}\le W\,C^*$.

Completeness is assured if the underlying costs are non-negative and the heuristics are consistent. Bidirectional search exhausts both frontiers unless a connecting path is discovered or a negative result is proved.

4. Practical Algorithmic Enhancements

Key algorithmic details include:

• Direction alternation: Expansions alternate between forward and backward frontiers round-robin.
• Node selection: Expand the open node minimizing $b_{W,D}(n)$ in the selected frontier.
• Tie-breaking: Prefer larger $g$ values to avoid re-expansions under consistency.
• Open/closed management: Standard A* bookkeeping is performed for each frontier independently.
• Lower-bound strengthening:
  • GCD rounding: If $\iota$ is the GCD of all edge costs, round up $LB_{WB}$ to the next multiple of $\iota W$ to potentially reduce node expansions.
  • Alternative lower bound (ALB): Track $W\max(f_{\min}^F, f_{\min}^B)$ and take the tighter lower bound; particularly effective for small $W$, with up to 50% fewer expansions for $W\in\{1.1,1.2\}$ but can increase per-node cost for larger $W$.

5. Empirical Evaluation and Comparative Performance

Extensive experiments compared WA*, WBiA*, WMM, WBS, and WBAE (for multiple $\lambda$ choices and a tuned $\lambda^*$) across the following domains:

• Towers of Hanoi (12 disks)
• 15-puzzle (standard and "heavy" variant)
• 18-pancake (GAP variants)
• DAO (grid pathfinding)
• Mazes (grid pathfinding)

The key metrics were node expansions, runtime, and solution cost ratio $C/C^*$. Table 1 summarizes average node expansions (in thousands):

Algorithm	ToH	STP	Pancake	DAO	Mazes
WA*	697	15,585	2,024	524	56
WBiA*	601	14,660	1,694	502	76
WMM	589	8,450	606	400	75
WBS*	644	15,640	1,985	497	67
WBAE* ($1/W^2$)	607	14,470	1,603	465	66
WBAE* ($1/W$)	540	14,470	1,646	450	66
WBAE* ($1$)	534	14,680	1,717	428	65
WBAE* ($W$)	523	14,640	1,667	336	61
WBAE* ($\lambda^*$)	478	14,540	1,709	430	60

For optimal search ($W=1$), WBAE*($W$) ≡ BAE* dominates the alternatives, outperforming WA* and WBiA* by a factor of 5–10. For large $W$ ($\ge3$), the $d$-term’s significance diminishes; WBiA* ($\lambda=0$) often becomes preferable. Tuning $\lambda$ ($\lambda^*$) nearly matches the best fixed choice in all cases. GCD rounding yields 5–20% reductions in node expansions at no runtime cost, while ALB provides further reductions for small $W$ but can slow computation for larger $W$.

6. Parameter Selection, Domain Dependence, and Guidelines

The choice of $\lambda$ is critical for balancing speed and lower-bound tightness:

• For strong heuristics and/or large $W$, rapid solution finding is prioritized; a small $\lambda$ (including $\lambda=0$, i.e. WBiA*) is effective.
• For weak heuristics or $W$ near 1, suboptimality proof becomes dominant, necessitating larger $\lambda$ (even $\lambda=W$, i.e. full error correction).
• Tuning $\lambda$ on a small training set identifies an efficient trade-off, with empirically observed correlation that $\lambda^*$ decreases as heuristic quality or $W$ increases.

A plausible implication is that domain characteristics and suboptimality requirements must both be actively considered in selecting the optimal weighting scheme.

7. Theoretical and Empirical Synthesis

By generalizing the BAE* error-correcting priorities with a heuristic inflation parameter $W$ and a free error-weighting parameter $\lambda\leq W$, the WBAE* framework encapsulates a spectrum of bidirectional bounded-suboptimal algorithms. This approach interpolates between weighted bidirectional A* and fully error-corrected BAE*, maintaining completeness, $C \leq W\,C^*$ bounded-suboptimality, and no node re-expansion under consistent heuristics. Empirical evidence demonstrates that a small collection of $\lambda$ values, including $\lambda=0, 1/W, 1, W$, achieves near-optimal performance across a variety of domains and suboptimality levels, with an additional tuned parameter providing further refinement (Shperberg et al., 13 Nov 2025).