Spatial problem-solving algorithms

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Formula

Explanation

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Worked solution

1. The problem requires finding the shortest path from hub A to hub D, passing through hub E exactly once, without revisiting any hub. The hubs are positioned on a 2D grid with given coordinates.
2. The path must go from A to E to D, and must not revisit any hub. Since only six hubs exist and the path must not revisit, the only valid path is A → E → D, as any detour would either revisit a hub or not satisfy the 'exactly once' condition for E.
3. The Euclidean distance formula is used to compute the distance between two points (x1, y1) and (x2, y2): distance = sqrt((x2 - x1)^2 + (y2 - y1)^2). This formula applies because the hubs are on a 2D plane and distances are straight-line (as per the problem's complete weighted graph with Euclidean distances).
4. Calculate the distance from A (0, 0) to E (2, 1). Apply the Euclidean formula: sqrt((2 - 0)^2 + (1 - 0)^2) = sqrt(2^2 + 1^2) = sqrt(4 + 1) = sqrt(5).
5. Calculate the distance from E (2, 1) to D (0, 3). Apply the formula: sqrt((0 - 2)^2 + (3 - 1)^2) = sqrt((-2)^2 + 2^2) = sqrt(4 + 4) = sqrt(8) = 2*sqrt(2).
6. Sum the two distances: sqrt(5) + 2*sqrt(2). This is the total path length from A to D via E, with no revisits and only one visit to E.
7. The path A → E → D is the only valid path satisfying the constraints: it visits E exactly once, does not revisit any hub, and uses only direct connections (as required by the shortest-path algorithm in a complete graph).
8. No alternative path (e.g., A → B → E → D) is shorter because any detour would increase distance, and the problem specifies the shortest path using a digital tool, so the direct A-E-D path must be optimal.
9. The total path length is sqrt(5) + 2*sqrt(2) metres. This is an exact expression and does not require decimal approximation unless specified.
10. Verify that no other path satisfies the constraints. For example, A → C → E → D would visit E only once but would be longer due to longer segments. Similarly, A → E → C → D would revisit E if not allowed, but E is only allowed once. Thus, A-E-D is the only viable shortest path.
11. The path A → E → D is valid, uses only allowed hubs, passes through E exactly once, and is the shortest possible under Euclidean distance. Therefore, the total length is correct.
12. The exact length of the shortest path from A to D passing through E exactly once, without revisiting any hub, is sqrt(5) + 2*sqrt(2) metres.

Reference answer: sqrt(5) + 2*sqrt(2)

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