LABORATORY 21 · COMPUTER SCIENCE

A* Pathfinding & Heuristic Search: Admissibility, Consistency, Priority Queue Open Sets, and State Space Exploration

To investigate informed state-space search algorithms, analyze the mathematical mechanics of the A* evaluation function f(n) = g(n) + h(n), evaluate heuristic admissibility (h ≤ h*) and consistency (|h(u) - h(v)| ≤ c(u,v)), contrast uninformed Dijkstra search (h = 0) against Manhattan and Euclidean heuristics, verify the linear state expansion law E(D) = k_astar · D + E_0 = 3.0000 · D + 0.0000 expansions across path distances D ∈ [10, 70] grid units, and examine memory-bounded architectures including IDA* and Jump Point Search (JPS). Explore interactive search algorithms, compare operation counts and analyse complexity scaling.

COMPUTE · BENCHMARK · ANALYSE

Your virtual algorithm laboratory.

CLOSED SET NODE EXPANSIONS E(D)—

EXECUTION STATUS

Ready to execute D benchmark.

Optimal Path Distance D40 grid stepsSet D (grid steps)

2D uniform-cost orthogonal grid environment (4-connected movement: North, South, East, West; unit step cost c = 1.0).

Adjust D n, then execute benchmark.

Operation counts are generated deterministically by the exact algorithmic execution model. Modern hardware benchmarks additionally reflect cache line prefetching and branch misprediction penalties.

Algorithmic assumptions and limits

The simulated benchmark environment models idealized 4-connected uniform-cost grid terrain where Manhattan heuristic evaluations execute deterministically (E(D) = 3D expansions). In real-world robotic navigation, video game engines, and GIS routing platforms, execution latency is heavily modulated by priority queue min-heap pointer swap overhead, floating-point Euclidean square root latencies, non-uniform terrain traversal costs (mud, water, elevation slopes), dynamic obstacle appearances requiring D* Lite replanning, and memory cache misses across large graph topologies. In the A* heuristic search benchmark across 2D grid path distances D in [10, 70] steps (under 4-connected orthogonal motion with unit edge costs and an admissible, consistent Manhattan distance heuristic h(n) = |x - x_goal| + |y - y_goal|), the search frontier is tightly focused along the direct line of sight between start and goal. At each step along the optimal path, the algorithm extracts the minimum-f vertex from the binary min-heap Open Set, places it into the Closed Set, and relaxes its 4 orthogonal neighbors. Because the Manhattan heuristic perfectly reflects obstacle-free grid costs, only the on-path cell and its immediate adjacent non-goal lateral neighbors are evaluated before the forward cell is chosen for expansion. Across an optimal trajectory of distance D, the cumulative count of closed set state expansions is strictly governed by the exact linear invariant E(D) = 3.0000 · D + 0.0000 expansions. Plotting closed set expansions against path distance D yields an ordinary least-squares regression slope k_astar = 3.0000 expansions/grid-step and baseline intercept E_0 = 0.0000 expansions with an ideal coefficient of determination R² = 1.000000.

Computational tools & environment.

A* Informed State Space Search Engine

Coordinates graph traversal, priority queue extraction, edge relaxation, and parent pointer backtracking to construct optimal shortest paths.

Binary Min-Heap Priority Queue (Open Set)

Maintains candidate frontier vertices ordered by f(n) = g(n) + h(n), supporting O(log |V|) insertion, extraction, and key decrease operations.

Visited State Registry (Closed Set)

Provides O(1) membership testing to record expanded vertices and prevent redundant re-expansion under consistent heuristics.

Geometric Heuristic Distance Evaluator

Calculates Manhattan (L1), Euclidean (L2), Chebyshev (L_inf), and Octile distance metrics between candidate nodes and the target goal.

Parent Pointer Path Reconstructor

Backtracks from the goal node through stored optimal predecessor pointers to synthesize the final shortest path coordinate sequence.

Algorithmic Profiler & Expansion Telemetry Logger

Logs open set insertions, closed set expansions, edge relaxations, and verifies ordinary least-squares regression against the theoretical model E(D) = 3D.

FOLLOW THE METHOD

A reason behind every step.

To investigate informed state-space search algorithms, analyze the mathematical mechanics of the A* evaluation function f(n) = g(n) + h(n), evaluate heuristic admissibility (h ≤ h*) and consistency (|h(u) - h(v)| ≤ c(u,v)), contrast uninformed Dijkstra search (h = 0) against Manhattan and Euclidean heuristics, verify the linear state expansion law E(D) = k_astar · D + E_0 = 3.0000 · D + 0.0000 expansions across path distances D ∈ [10, 70] grid units, and examine memory-bounded architectures including IDA* and Jump Point Search (JPS). Suggested independent settings: 10, 20, 30, 40, 50, 60, 70 grid steps.

Grid Environment & Coordinate Initialization

Initialize the 2D grid map with start vertex s and goal vertex g separated by optimal path distance D ∈ [10, 20, 30, 40, 50, 60, 70] grid units. Clear the Open Set min-heap and Closed Set registry.

Start Vertex Seeding & Initial Heuristic Evaluation

Set g(s) = 0, evaluate h(s) = |x_s - x_g| + |y_s - y_g|, and insert start node s into the Open Set with priority f(s) = g(s) + h(s). Initialize the closed set expansion counter to zero.

Priority Queue Min-f Extraction & Closed Set Transfer

Extract the vertex n with the minimum f(n) from the Open Set. If n matches the goal vertex g, terminate the search and initiate path reconstruction. Otherwise, add n to the Closed Set and increment the node expansion counter E by 1.

Neighbor Generation & Edge Relaxation

Iterate across all valid orthogonal neighbor vertices n' of n. If n' is in the Closed Set or is an impassable obstacle, skip it. Calculate tentative cost tentative_g = g(n) + c(n, n'). If n' is not in the Open Set or tentative_g < g(n'), update parent(n') = n, g(n') = tentative_g, and f(n') = g(n') + h(n'), then insert or update n' in the Open Set.

Path Backtracking & Verification

Upon extracting goal vertex g, traverse predecessor pointers backward from g to s: path = [g, parent(g), parent(parent(g)), ..., s]. Reverse the list to yield the forward optimal trajectory, verifying that path length matches the expected distance D.

Empirical Complexity Regression & Model Validation

Record total Closed Set node expansions E(D) across all benchmark distances D ∈ [10, 70]. Perform ordinary least-squares linear regression of E against D to determine empirical slope k_astar, intercept E_0, and coefficient of determination R².

YOUR NOTEBOOK · 0 OBSERVATIONS

Turn measurements into asymptotic insight.

See a complete example report ↗

Points are your recorded benchmark observations. The regression line validates A* Heuristic Search complexity scaling.

LINEAR ASYMPTOTIC FIT · y = mx + b

E(D) = k_astar · D + E_0 = 3.0000 · D + 0.0000 expansions

#	Optimal Path Distance D (grid steps)	Closed Set Node Expansions E(D) (expansions)	Optimal path distance D (grid steps)	Total node expansions E(D) = 3.0000 · D + 0.0000 (expansions)	Remove

No readings yet. Execute a D benchmark and record it.

Calculate it yourself.

Your estimate of k_astar (expansions/grid-step)

CHECK YOUR UNDERSTANDING

Explain what the operation counts mean.

What condition must a heuristic function h(n) satisfy to be classified as 'admissible' in A* search?

What occurs when the heuristic function h(n) is set to 0 for every vertex in A* search?

On a 4-connected grid with unit orthogonal movement costs, which heuristic is both admissible and consistent while providing tighter bounds than Euclidean distance?