Heuristic Priority Queues (HPQ) Algorithm

• Heuristic Priority Queues (HPQ) are adaptive algorithms that dynamically partition work between sequential and parallel modes to optimize performance in resource scheduling.
• They employ elimination protocols, combining techniques, and CAS-based synchronization to reduce latency and improve throughput under workload fluctuations.
• HPQs are applied in high-concurrency and real-time systems such as autonomous traffic control, demonstrating improved safety and efficiency in scheduling decisions.

Heuristic Priority Queues (HPQ) algorithms encompass a family of strategies that dynamically adapt priority queue (PQ) behavior based on observed workload and application context, with goals spanning high-throughput concurrent computation and real-time scheduling in cyber-physical systems. Instances of the HPQ paradigm are found in deeply concurrent data structures for resource management (Calciu et al., 2014) as well as in intelligent decision-making systems for mixed-autonomy traffic control (Zhou et al., 2022). Across domains, HPQs employ heuristic state estimation, dynamic partitioning, and local matching to optimize latency, safety, and fairness under system-specific constraints.

1. Structural Foundation of Heuristic Priority Queues

In concurrent computing, the HPQ as formulated in the Adaptive Priority Queue with Elimination and Combining (Calciu et al., 2014) is realized as a skiplist divided into a sequential “head” region and a parallel “tail” region. The sequential portion is served by a dedicated thread handling batched minima removals and delegated small-key insertions, whereas the parallel section enables fully concurrent insertions by multiple threads.

For intersection management in autonomous traffic, the HPQ maintains multiple lane-specific PQs, each prioritized by a first-come-first-served (FCFS) policy anchored at an upstream “entry area.” The design supports partitioned candidate sets per lane and a global arbitration layer evaluating real-time conflict graphs (Zhou et al., 2022).

2. Core Algorithmic Procedures and Heuristics

The concurrent HPQ operationalizes three major modes: elimination, combining, and parallelism. Insertions with large keys bypass the sequential head and enter the parallel skiplist; small keys and removal requests first attempt elimination via a fixed-size array, realizing immediate key-value exchanges or, failing this, delegation to the server thread for processing by combining. A background heuristic dynamically adjusts the sequential/parallel boundary by migrating buckets between regions, employing “double/half” scaling to adapt the sequential region size $S$ based on in-region insertion/removal statistics (bounded by $S_{\min}=8$ and $S_{\max}=65\,536$; thresholds $N_{hi}=1\,000$, $N_{lo}=100$) (Calciu et al., 2014).

In the intersection control framework, the HPQ algorithm operates per control cycle. Vehicles entering the entry area are assigned priorities incrementally. Each cycle, the system inspects the head of each lane-specific PQ, computes the number of active (already granted right-of-way) and prospective (higher-priority still-waiting) conflicts for each candidate, and grants right-of-way under type-dependent conditions: for CHVs (connected human-driven vehicles), only if there are no active or pending conflicts; for CAVs (connected and automated vehicles), if at most one active conflict and none from higher-priority candidates exist. Post-grant, PQ priorities are updated via decrement (Zhou et al., 2022).

3. Elimination Protocols and Conflict Handling Mechanisms

In the adaptive concurrent HPQ, elimination leverages a fixed-size array, each slot representing a packed value or opcode and a unique stamp to prevent ABA problems. Only insertion requests for keys at or below the minimum are eligible for immediate elimination with removal requests. The linearization point for elimination is established by a CAS that atomically swaps opcodes and clears the stamp. If a partner does not appear within bounded attempts, requests are delegated for sequential combining by the server. The explicit coordination of elimination maximizes throughput in symmetric workloads (add/remove mix near 0.5), empirically yielding 70–80% elimination rates at optimal mixes (Calciu et al., 2014).

In intersection management HPQs, the conflict protocol operates through a precomputed adjacency matrix (conflict graph) over the set of all intersection trajectories, supporting $O(1)$ conflict checks. Safety constraints require minimum time headway maintenance and strict passage ordering; vehicles demonstrating abnormal behavior (e.g., excessive delay through the conflict zone) are ejected from the queue, and downstream priority adjustments are enacted to prevent unfair blockages (Zhou et al., 2022).

4. Parameterization, Pseudocode, and Complexity Analysis

Key variables and parameters in the concurrent HPQ include:

• Sequential region size $S$ (adaptively tuned),
• Elimination array size ELIM_SIZE,
• Skiplists for data structuring,
• CAS-based synchronization primitives,
• Locking (readers-writer locks) or hardware transactional memory in advanced variants.

The pseudocode for $\text{PQ::add}(v)$ and $\text{PQ::removeMin}()$ structurally reflects this dual-moded workflow, with fast-path parallel insertion for large keys, bounded elimination scans for small keys, and fallback to server-based combining.

Complexity per operation is as follows:

Operation	Work per op	Span (critical path)
Parallel add (existing bucket)	$O(1)$ atomic incr	$O(\log n)$
Parallel add (new bucket)	$O(\log n)$ CAS	$O(\log n)$
Sequential add/remove	$O(1)$	$O(1 + S/B)$
Elimination	$O(1)$	$O(1)$

For intersection control HPQ (Zhou et al., 2022), the primary per-cycle cost is $O(n)$: with at most four lane heads examined per cycle, and conflict checks performed against up to $n\approx20$–$50$ vehicles. Lane-specific PQs are efficiently implemented, typically via heaps, supporting $O(\log n)$ insert/delete per vehicle. System throughput remains bounded by the real-time requirements of control infrastructure, with measured latencies in the millisecond regime.

5. Concurrency Control, Atomicity, and Hardware Acceleration

Multiple synchronization strategies are supported. In the concurrent HPQ, baseline locking employs a single readers-writer lock to protect region-splitting (moveHead/chopHead) and parallel inserts (as readers), using atomic primitives for pointer and counter updates and CAS for global minValue. Hardware Transactional Memory (HTM) variants (Intel TSX RTM) replace locks with timestamp-based versioning, encapsulating parallel inserts and head manipulations in TSX transactions. After multiple aborts, the protocol reverts to the pessimistic path. HTM variants provide comparable or superior performance up to four cores; beyond that, abort rates constrain further scaling (Calciu et al., 2014).

In intersection applications, atomicity is guaranteed at the algorithmic (digital control) level, as all state transitions and scheduling decisions are executed synchronously in each control cycle.

6. Integration and Application Contexts

In high-concurrency server applications, HPQ structures adaptively balance batch efficiency and parallelism, achieving throughput improvements (over $2\times$) relative to pure flat-combining or classic skiplists at high contention, particularly at balanced add/remove mixes.

In mixed-autonomy intersection control, the HPQ algorithm outputs mode- and conflict-aware grant signals to each CAV. Each vehicle executes a four-mode controller: “car-following,” “cruise,” “waiting,” and “conflict-solving.” Mode selection is driven by the right-of-way status and pairwise conflicts. Longitudinal control is realized with MPC optimized over linearized error states, with lateral control via the Stanley method. The control sequence is subject to safety bounds on acceleration, jerk, and headway. The system ensures strict FCFS fairness, accounts explicitly for human driver constraints (CHVs may not handle more than one real-time conflict), and dynamically demotes the priority of entire lanes in the presence of abnormal vehicles (Zhou et al., 2022).

7. Evaluation and Empirical Outcomes

Extensive evaluation of concurrent HPQs demonstrates substantial elimination rates and workload-adaptive batching. At a 50/50 add/remove mix, 70–80% of operations are satisfied through elimination/combination without mutating the underling skiplist. Throughput scales with core count up to contention limits, and O($\log n$) performance is preserved for large-key insertions even under skewed workloads (Calciu et al., 2014).

Empirical results for the intersection management HPQ indicate reductions in average halts by up to 40% versus established protocols, travel-time reductions between 5% and 65% depending on flow, and the highest stability and average speeds observed among all tested algorithms. No collisions or unsafe behaviors were observed in both macroscopic SUMO simulations (throughputs up to 1600 pcu/h) and microscopic (joint Matlab/PreScan) experiments (Zhou et al., 2022).

---

In summary, Heuristic Priority Queues represent a class of algorithms and data structures that integrate runtime-adaptive heuristics, dynamic partitioning, and elimination/combination for high-throughput PQ management and context-specific scheduling. Their utility is established both in scalable parallel resource scheduling and real-time, safety-critical multi-agent control, exhibiting robust adaptation to workload skews and heterogeneous agent populations.