Exploring Simple Triangular and Hexagonal Grid Polygons Online

TL;DR

This paper studies online exploration strategies for a mobile robot in unknown hexagonal and triangular grid environments, providing bounds on tour length and competitiveness.

Contribution

It introduces exploration strategies with proven length bounds and competitiveness ratios for unknown cellular environments with hexagonal and triangular grids.

Findings

Tour length bounds: S <= C + 1/4 E - 2.5 (hexagonal), S <= C + E - 4 (triangular)

Strategies are 4/3-competitive for both grid types

Lower bounds of 14/13 (hexagonal) and 7/6 (triangular) on competitiveness

Abstract

We investigate the online exploration problem (aka covering) of a short-sighted mobile robot moving in an unknown cellular environment with hexagons and triangles as types of cells. To explore a cell, the robot must enter it. Once inside, the robot knows which of the 3 or 6 adjacent cells exist and which are boundary edges. The robot's task is to visit every cell in the given environment and to return to the start. Our interest is in a short exploration tour; that is, in keeping the number of multiple cell visits small. For arbitrary environments containing no obstacles we provide a strategy producing tours of length S <= C + 1/4 E - 2.5 for hexagonal grids, and S <= C + E - 4 for triangular grids. C denotes the number of cells-the area-, E denotes the number of boundary edges-the perimeter-of the given environment. Further, we show that our strategy is 4/3-competitive in both types of grids, and we provide lower bounds of 14/13 for hexagonal grids and 7/6 for triangular grids.