Imagine a boat navigating from port to port through narrow channel markers, or a drone weaving through sequential checkpoints. This scenario crystallizes into a compelling computational geometry problem: Find the shortest path between two points that crosses a series of line segments ("gates") in strict order. While a brute-force graph solution exists, a recently demonstrated approach offers remarkable elegance and efficiency.

The Problem and Naive Approach

Given points $p_0$ and $p_1$ and gates $G = {g_1, g_2, ..., g_n}$ (each a line segment), the goal is the minimal-length path crossing each gate sequentially. A straightforward solution constructs a graph where vertices include $p_0$, $p_1$, and all gate endpoints ($l_i$, $r_i$). Edges exist only if their connecting line intersects all intermediate gates. Dijkstra’s algorithm then finds the path.

Graph representation of gates and valid paths (Source: mht.wtf)

This works but suffers computationally. Checking edge validity requires $O(n)$ intersection tests per potential edge, leading to $O(n^3)$ complexity. For dense gate configurations, this becomes prohibitive.

A Brilliant Insight: Region Propagation

A more efficient method exploits a geometric observation: The shortest path through a gate either passes straight through it or touches one endpoint before changing direction. This enables propagating "coverage regions" across gates:

  1. Initialization: For $g_1$, all points are reachable via a straight line from $p_0$.

  2. Region Extension: For each subsequent gate $g_i$, determine which parts are "covered" by existing regions (via straight-line extensions) and where new regions rooted at prior endpoints ($l_{i-1}$ or $r_{i-1}$) are needed.

  3. Pruning: Regions that don’t contribute to covering $g_i$ are discarded. Coverage is maintained via orientation checks (left/right of lines).

  4. Termination: $p_1$ is checked against the final gate’s regions. If uncovered, a terminal segment from the nearest endpoint is added.

Algorithmic Nuances and Efficiency

The magic lies in data representation. Each gate’s coverage is defined by:
- An ordered list of roots (points anchoring regions)
- Implicit boundary lines between roots and gate endpoints

Extending to $g_i$ requires only: 1. Checking if $l_i$ lies left of the leftmost region boundary (triggering a new left-rooted region).
2. Checking if $r_i$ lies right of a region boundary (triggering a new right-rooted region or clipping an existing one).

Orientation queries use a simple cross-product check:

def is_left_of_line(line_root, line_dir, point):
    to_point = point - line_root
    rot_dir = [-line_dir.y, line_dir.x]  # 90° counterclockwise
    return rot_dir.dot(to_point) > 0

Complexity: Each gate processes in $O(k)$ time (where $k$ is the number of surviving regions, often $<< n$), reducing total complexity to near-$O(n)$.

Backtracking the Path

After processing all gates:
1. Identify which region contains $p_1$. 2. Backtrack through the region’s root to the preceding gate’s root. 3. Repeat until reaching $p_0$.

The result is a piecewise-linear path of minimal length—achieved without explicitly calculating any distances.

Why This Matters

This algorithm exemplifies computational geometry’s power: translating spatial intuition into efficient computation. Beyond navigation, it applies to motion planning, VLSI routing, and GIS systems. Its elegance lies in leveraging geometric properties (triangle inequality, convexity) to avoid brute-force work, showcasing how clever data representation drives performance.

Source: Navigate Gates (Licensed CC BY-SA 4.0)