Polylogarithmic time algorithms for shortest path forests in programmable matter <jats:title>Abstract</jats:title> <jats:p> In this paper, we study the computation of shortest paths within the <jats:italic>geometric amoebot model</jats:italic> , a commonly used model for programmable matter. Shortest paths are essential for various tasks and therefore have been heavily investigated in many different contexts. We consider the <jats:italic>reconfigurable circuit extension</jats:italic> of the model where the amoebot structure is able to interconnect amoebots by so-called circuits. These circuits permit the instantaneous transmission of simple signals between connected amoebots. We propose distributed algorithms for the <jats:italic>shortest path forest problem</jats:italic> where, given a set of <jats:italic>k</jats:italic> sources and a set of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\ell $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> destinations, the amoebot structure has to compute a forest that connects each destination to its closest source on a shortest path. Our main results are two algorithms for hole-free structures. The first algorithm constructs a shortest path tree for a single source within <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$O(\log \ell )$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> rounds, and the second algorithm a shortest path forest for an arbitrary number of sources within <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$O(\log n \log ^2 k)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mo>log</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> rounds. The former algorithm also provides an <jats:italic>O</jats:italic> (1) rounds solution for the <jats:italic>single pair shortest path problem</jats:italic> (SPSP) and an <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$O(\log n)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> rounds solution for the <jats:italic>single source shortest path problem</jats:italic> (SSSP) since these problems are special cases of the considered problem. Then, we adapt the latter algorithm to an offset version of the problem. This allows us to solve the problem for amoebot structures with holes within <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$O(h \log ^3 n)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>h</mml:mi> <mml:msup> <mml:mo>log</mml:mo> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> rounds w.h.p. where <jats:italic>h</jats:italic> denotes the number of holes. </jats:p> 39 2 Springer Science and Business Media LLC