Efficient Generation of Rectangulations via Permutation Languages

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework

Efficient generation of rectangulations via permutation languages

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework

Combinatorial generation via permutation languages

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an $n$-element set by adjacent transpositions; the binary reflected Gray code to generate all $n$-bit strings by flipping a single bit in each step; the Gray code for generating all $n$-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an $n$-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into $n$ rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group~$S_n$. Recently, Pilaud and Santos realized all those lattice congruences as $(n-1)$-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope

Combinatorial generation via permutation languages. I. Fundamentals

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an $n$-element set by adjacent transpositions; the binary reflected Gray code to generate all $n$-bit strings by flipping a single bit in each step; the Gray code for generating all $n$-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an $n$-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into $n$ rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group~$S_n$. Recently, Pilaud and Santos realized all those lattice congruences as $(n-1)$-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Resilience of power grids and other supply networks: structural stability, cascading failures and optimal topologies

The consequences of the climate crisis are already present and can be expected to become more severe in the future. To mitigate long-term consequences, a major part of the world's countries has committed to limit the temperature rise via the Paris Agreement in the year 2015. To achieve this goal, the energy production needs to decarbonise, which results in fundamental changes in many societal aspects. In particular, the electrical power production is shifting from fossil fuels to renewable energy sources to limit greenhouse gas emissions. The electrical power transmission grid plays a crucial role in this transformation. Notably, the storage and long-distance transport of electrical power becomes increasingly important, since variable renewable energy sources (VRES) are subjected to external factors such as weather conditions and their power production is therefore regionally and temporally diverse. As a result, the transmission grid experiences higher loadings and bottlenecks appear. In a highly-loaded grid, a single transmission line or generator outage can trigger overloads on other components via flow rerouting. These may in turn trigger additional rerouting and overloads, until, finally, parts of the grid become disconnected. Such cascading failures can result in large-scale power blackouts, which bear enormous risks, as almost all infrastructures and economic activities depend on a reliable supply of electric power. Thus, it is essential to understand how networks react to local failures, how flow is rerouted after failures and how cascades emerge and spread in different power transmission grids to ensure a stable power grid operation. In this thesis, I examine how the network topology shapes the resilience of power grids and other supply networks. First, I analyse how flow is rerouted after the failure of a single or a few links and derive mathematically rigorous results on the decay of flow changes with different network-based distance measures. Furthermore, I demonstrate that the impact of single link failures follows a universal statistics throughout different topologies and introduce a stochastic model for cascading failures that incorporates crucial aspects of flow redistribution. Based on this improved understanding of link failures, I propose network modifications that attenuate or completely suppress the impact of link failures in parts of the network and thereby significantly reduce the risk of cascading failures. In a next step, I compare the topological characteristics of different kinds of supply networks to analyse how the trade-off between efficiency and resilience determines the structure of optimal supply networks. Finally, I examine what shapes the risk of incurring large scale cascading failures in a realistic power system model to assess the effects of the energy transition in Europe