Stefan Edelkamp's research while affiliated with Czech Technical University in Prague and other places

Publications (292)

Preprint
Full-text available
Optimization of heuristic functions for the A* algorithm, realized by deep neural networks, is usually done by minimizing square root loss of estimate of the cost to goal values. This paper argues that this does not necessarily lead to a faster search of A* algorithm since its execution relies on relative values instead of absolute ones. As a mitig...
Article
Sampling-based motion planning algorithms such as Rapidly exploring Random Trees (RRTs) have been used in robotic applications for a long time. In this paper, we propose a method that combines deep learning with RRT* method. We use a neural network to learn a sample strategy for RRT*.We evaluate Deep RRT* in a collection of 2D scenarios. The result...
Article
Effective planning while competing for limited resources is crucial in many real-world applications such as on-demand transport companies competing for passengers. Planning techniques therefore have to take into account possible actions of an adversarial agent. Such a challenge that can be tackled by leveraging game-theoretical methods such as Doub...
Article
Effective decision making while competing for limited resources in adversarial environments is important for many real-world applications (e.g. two Taxi companies competing for customers). Decision-making techniques such as Automated planning have to take into account possible actions of adversary (or competing) agents. That said, the agent should...
Article
This paper shows that domain-independent tools from classical planning can be used to model and solve a broad class of game-theoretic problems we call Cost-Adversarial Planning Games (CAPGs). We define CAPGs as 2-player normal-form games specified by a planning task and a finite collection of cost functions. The first player (a planning agent) stri...
Preprint
Full-text available
Learning a well-informed heuristic function for hard task planning domains is an elusive problem. Although there are known neural network architectures to represent such heuristic knowledge, it is not obvious what concrete information is learned and whether techniques aimed at understanding the structure help in improving the quality of the heurist...
Article
Effective plan generation in adversarial environments has to take into account possible actions of adversary agents, i.e., the agent should know what the competitor will likely do. In this paper we propose a novel approach for estimating strategies of the adversary, sampling actions that interfere with the agent's ones. The estimated competitor str...
Article
This paper exploits parallel computing power of graphics cards to accelerate state space search. We illustrate that modern graphics processing units (GPUs) have the potential to speed up breadth-first search significantly. For a bitvector representation of the search frontier, GPU algorithms with one and two bits per state are presented. Efficient...
Preprint
Assessing the skill level of players to predict the outcome and to rank the players in a longer series of games is of critical importance for tournament play. Besides weaknesses, like an observed continuous inflation, through a steadily increasing playing body, the ELO ranking system, named after its creator Arpad Elo, has proven to be a reliable m...
Preprint
This paper proposes \emph{knowledge-based paraonoia search} (KBPS) to find forced wins during trick-taking in the card game Skat; for some one of the most interesting card games for three players. It combines efficient partial information game-tree search with knowledge representation and reasoning. This worst-case analysis, initiated after a small...
Preprint
Skat is a fascinating combinatorial card game, show-casing many of the intrinsic challenges for modern AI systems such as cooperative and adversarial behaviors (among the players), randomness (in the deal), and partial knowledge (due to hidden cards). Given the larger number of tricks and higher degree of uncertainty, reinforcement learning is less...
Chapter
In this paper we look at multi-player trick-taking card games that rely on obeying suits, which include Bridge, Hearts, Tarot, Skat, and many more. We propose mini-game solving in the suit factors of the game, and exemplify its application as a single-dummy or double-dummy analysis tool that restricts game play to either trump or non-trump suit car...
Article
Full-text available
QuickXsort is a highly efficient in-place sequential sorting scheme that mixes Hoare’s Quicksort algorithm with X, where X can be chosen from a wider range of other known sorting algorithms, like Heapsort, Insertionsort and Mergesort. Its major advantage is that QuickXsort can be in-place even if X is not. In this work we provide general transfer t...
Chapter
Logistics operations often require a robot to pickup and deliver objects from multiple locations within certain time frames. This is a challenging task-and-motion planning problem as it intertwines logical and temporal constraints about the operations with geometric and differential constraints related to obstacle avoidance and robot dynamics. To a...
Chapter
Pattern databases (PDBs) are memory-based abstraction heuristics that are constructed prior to the planning process which, if expressed symbolically, yield a very efficient representation. Recent work in the automatic generation of symbolic PDBs has established it as one of the most successful approaches for cost-optimal domain-independent planning...
Chapter
Inspection is a hot topic of robotics recently, and there are many different ways to solve the inspection problem. In this paper, we propose a new framework for a robust and efficient inspection of the entire workspace in a watchman route based on automatically generated waypoints. The framework architecture design includes several relevant technol...
Chapter
This paper addresses multi-robot multi-goal motion planning with temporal and resources constraints. It solves the vehicle routing problem for mobile robots that operate according to their system dynamics, and which have to visit a number of waypoints scattered in a two-dimensional map environment with obstacles, while satisfying time window and ca...
Article
It is well known that Quicksort -- which is commonly considered as one of the fastest in-place sorting algorithms -- suffers in an essential way from branch mispredictions. We present a novel approach to addressing this problem by partially decoupling control from dataflow: in order to perform the partitioning, we split the input into blocks...
Preprint
Full-text available
QuickXsort is a highly efficient in-place sequential sorting scheme that mixes Hoare's Quicksort algorithm with X, where X can be chosen from a wider range of other known sorting algorithms, like Heapsort, Insertionsort and Mergesort. Its major advantage is that QuickXsort can be in-place even if X is not. In this work we provide general transfer t...
Preprint
Full-text available
The two most prominent solutions for the sorting problem are Quicksort and Mergesort. While Quicksort is very fast on average, Mergesort additionally gives worst-case guarantees, but needs extra space for a linear number of elements. Worst-case efficient in-place sorting, however, remains a challenge: the standard solution, Heapsort, suffers from a...
Article
Full-text available
In this work, we propose the application of the SPIN software model checker to a multiagent system that controls the industrial production of goods. The flow of material is buffered on a production line with assembling stations. As the flow of material is asynchronous at each station, queuing is required as long as buffers provide waiting room. Bes...
Article
Robots used for inspection, package deliveries, moving of goods, and other logistics operations are often required to visit certain locations within specified time bounds. This gives rise to a challenging problem as it requires not only planning collision-free and dynamically-feasible motions but also reasoning temporally about when and where the r...
Preprint
Full-text available
We consider the fundamental problem of internally sorting a sequence of $n$ elements. In its best theoretical setting QuickMergesort, a combination Quicksort with Mergesort with a Median-of-$\sqrt{n}$ pivot selection, requires at most $n \log n - 1.3999n + o(n)$ element comparisons on the average. The questions addressed in this paper is how to mak...
Conference Paper
Flood-filling algorithms as used for coloring images and shadow casting show that improved locality greatly increases the cache performance and, in turn, reduces the running time of an algorithm. In this paper we look at Dijkstra’s method to compute the shortest paths for example to generate pattern databases. As cache-improving contributions, we p...
Conference Paper
This paper develops an approach that enables an aerial vehicle to carry out 3D surface inspections. Given a 3D environment with a set of objects that need to be inspected, and an inspection quality \(0\,<\,\alpha \,<\,1\), the objective is to compute a set of waypoints whose joint visibility ratio is at least \(\alpha \) and a dynamically-feasible...
Conference Paper
The success in learning how to play Go at a professional level is based on training a deep neural network on a wider selection of human expert games and raises the question on the availability, the limits, and the possibilities of this technique for other combinatorial games, especially when there is a lack of access to a larger body of additional...
Article
Full-text available
We study the problem of constructing a binary heap in an array using only a small amount of additional space. Let N denote the size of the input, M the capacity of the cache, and B the width of the cache lines of the underlying computer, all measured as numbers of elements. We show that there exists an in-place heap-construction algorithm that runs...
Conference Paper
In this paper we introduce a novel application of model checking to find optimal planning solutions for a flow production system. Originally controlled by a multiagent system, the production system consists of autonomous products and asynchronous production stations with limited space for waiting products. In this work, we present two different app...
Book
This book constitutes the refereed proceedings of the 5th Computer Games Workshop, CGW 2016, and the 5th Workshop on General Intelligence in Game-Playing Agents, GIGA 2016, held in conjunction with the 25th International Conference on Artificial Intelligence, IJCAI 2016, in New York, USA, in July 2016. The 12 revised full papers presented were care...
Article
This paper develops an efficient approach to generate a collision-free and dynamically-feasible trajectory that enables a robotic vehicle to inspect the entire workspace or a subset consisting of one or several regions. The approach makes it possible to specify constraints on the order in which the regions should be inspected by using colors to ens...
Chapter
Many state spaces are so big that even in compressed form they fail to fit into main memory. As a result, during the execution of a search algorithm, only a part of the state space can be processed in main memory at a time; the remainder is stored on a disk. In this paper we survey research efforts in external-memory search for solving state space...
Article
In cost-optimal planning we aim to find a sequence of operators that achieve a set of goals with minimum cost. Symbolic search with Binary Decision Diagrams (BDDs) performs efficient state space exploration in terms of time and memory. This is crucial in optimal settings, in which large parts of the state space must be explored in order to prove op...
Conference Paper
Full-text available
For combinatorial search in single-player games nested Monte-Carlo search is an apparent alternative to algorithms like UCT that are applied in two-player and general games. To trade exploration with exploitation the randomized search procedure intensifies the search with increasing recursion depth. If a concise mapping from states to actions is av...
Conference Paper
In this work, several approaches to feature extraction on sets of time-based events will be developed and evaluated. On the one hand, these sets of events will be extracted from video files and on the other hand it will be manually annotated. By using methods of supervised machine learning the two sets of events will be mapped onto each other. Afte...
Conference Paper
Freespace navigation for autonomous robots is of growing industrial impact, especially in the logistics and warehousing domain. In this work, we describe a multiagent simulation solution to the physical vehicle routing problem, which extends the physical traveling salesman problem —a recent benchmark used in robot motion planning research— by consi...
Preprint
Full-text available
Since the work of Kaligosi and Sanders (2006), it is well-known that Quicksort -- which is commonly considered as one of the fastest in-place sorting algorithms -- suffers in an essential way from branch mispredictions. We present a novel approach to address this problem by partially decoupling control from data flow: in order to perform the partit...
Conference Paper
A discrete event system (DES) is a dynamic system with discrete states the transitions of which are triggered by events. In this paper we propose the application of the Spin software model checker to a discrete event system that controls the industrial production of autonomous products. The flow of material is asynchronous and buffered. The aim of...
Article
Full-text available
In industrial domains such as manufacturing control, a trend away from centralized planning and scheduling towards more flexible distributed agent-based approaches could be observed over recent years. To be of practical relevance, the local control mechanisms of the autonomous agents must be able to dependably adhere and dynamically adjust to compl...
Article
Toward enhancing automation, this paper proposes an efficient approach for multi-group motion planning, where the set of goals is divided into k groups and the objective is to compute a collision-free and dynamically feasible trajectory that enables a virtual vehicle to reach at least one goal from each group. The approach works with ground and aer...
Book
This book constitutes the refereed proceedings of the Fourth Computer Games Workshop, CGW 2015, and the Fourth Workshop on General Intelligence in Game-Playing Agents, GIGA 2015, held in conjunction with the 24th International Conference on Artificial Intelligence, IJCAI 2015, Buenos Aires, Argentina, in July 2015. The 12 revised full papers presen...
Conference Paper
A growing network of technical systems, embedded and autonomous, influence our daily work. Among them, cyber-physical systems establish a close connection between the virtual and the real world. In this paper we show how an existing multiagent system that controls the physical production of goods on a monorail is virtualized by extracting the agent...
Chapter
Full-text available
This chapter presents the concepts, an example implementation, and the evaluation of an autonomous, self-organized, and adaptive multiagent system to optimize industrial processes in dynamic environments. In order to satisfy the rising requirements which result from the Fourth Industrial Revolution and to benefit from the consequent integration of...
Conference Paper
This paper considers solving a problem in combinatorial search: the automated arrangement of irregular-shaped objects for industrial 3D printing. The input is a set of triangulated models; the output is a set of location and orientation vectors for the objects. The proposed algorithm consists of three stages: (1) translation of the models into an o...
Conference Paper
Full-text available
In this paper we review recent advances of randomized AI search in solving industrially relevant optimization problems. The method we focus on is a sampling-based solution mechanism called Monte-Carlo Tree Search (MCTS), which is extended by the concepts of nestedness and policy adaptation to establish a better trade-off between exploitation and ex...
Conference Paper
Full-text available
An in-place priority queue is a data structure that is stored in an array, uses constant extra space in addition to the array elements, and supports the operations \( top \) (\( find \)-\( min \)), \( push \) (\( insert \)), and \( pop \) (\( delete \)-\( min \)). In this paper we introduce an in-place priority queue, for which \( top \) and \( pus...
Article
The cost-optimal track of the international planning competition in 2014 has seen an unexpected outcome. Different to the precursing competition in 2011, where explicit-state heuristic search planning scored best, advances in the state-set exploration with BDDs showed a significant lead. In this paper we review the outcome of the competition, brief...
Article
Full-text available
This paper presents an autonomous multiagent system which optimizes the planning and scheduling of industrial processes using the example of courier and express services. In order to handle the rising demands and to capitalize on the increasing optimization potential in transport logistics, which both result from the consequent integration of indus...
Article
This paper pursues multi-goal motion planning, where the overall set of goals is divided into k groups and the virtual agent needs to visit at least one goal per group. We have developed a combined task and motion-planning approach which can work with ground and aerial vehicles whose motions are simulated by differential equations or by physics-bas...
Conference Paper
Full-text available
In this paper we present a PDDL-based multi-agent planning system for reasoning about key performance indicators (KPIs) in an industrial production planning and control application scenario. On top of PDDL, numeric key figures and associated objectives are configured by the user at run-time and then processed automatically by the system in order to...
Conference Paper
In this paper we look at packing problems that naturally arise in container loading. Given a set of 3D iso-oriented objects and a container, the task is to find a packing sequence of the input objects consisting of the ID, location, and orientation that minimizes the space wasted by the packing. Instead of the decision problem, we look at the packi...
Conference Paper
Toward enhancing automation in video games, this paper proposes an efficient approach for multi-goal motion planning, where a mobile agent needs to visit several regions in a complex environment containing numerous obstacles. The approach works in conjunction with differential equations and physics-based simulations of vehicle dynamics, efficiently...
Conference Paper
The Physical Traveling Salesman Problem (PTSP) is a current research problem which adds a model of velocity to the classic TSP. In this paper we propose algorithms for solving the PTSP which avoid the fragmented allocation of memory and precompute cell-precise single-source shortest paths for each waypoint by using an engineered implementation of D...
Article
Full-text available
These are the proceedings of the Third Workshop on GRAPH Inspection and Traversal Engineering (GRAPHITE 2014), which took place on April 5, 2014 in Grenoble, France, as a satellite event of the 17th European Joint Conferences on Theory and Practice of Software (ETAPS 2014). The aim of GRAPHITE is to foster the convergence on research interests from...
Article
Full-text available
Let $n$ denote the number of elements currently in a data structure. An in-place heap is stored in the first $n$ locations of an array, uses $O(1)$ extra space, and supports the operations: minimum, insert, and extract-min. We introduce an in-place heap, for which minimum and insert take $O(1)$ worst-case time, and extract-min takes $O(\lg{} n)$ wo...
Conference Paper
Full-text available
Planning with numeric state variables and goal systems today still poses a challenging task within the field of computational intelligence. In this paper a two-tier planning system is presented that enables the optimization of continuous numeric action parameters in combinatorially enumerated plans. It allows resorting to a “satisficing” strategy b...
Conference Paper
Full-text available
In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. We show that up to o(n) terms the average number of comparisons incurred by QuickXsort is equal to the average number of compar...
Article
This work combines recent advances in AI planning under memory limitation, namely bitvector and symbolic search. Bitvector search assumes a bijective mapping between state and memory addresses, while symbolic search compactly represents state sets. The memory requirements vary with the structure of the problem to be solved. The integration of the t...
Article
Transporting goods by courier and express services increases the service quality through short transit times and satisfies individual demands of customers. Determining the optimal route for a vehicle to satisfy transport requests while minimizing the total cost refers to the Single Vehicle Pickup and Delivery Problem. Beside time and distance objec...
Article
A weak heap is a priority queue that supports the operations construct, minimum, insert, and extract-minextract-min. To store n elements, it uses an array of n elements and an array of n bits. In this paper we study different possibilities for optimizing construct and insert such that minimum and extract-minextract-min are not made slower. We provi...
Conference Paper
In this research and technology transfer project, the planning and control processes of the industrial partner Hellmann Worldwide Logistics GmbH & Co. KG are analyzed. An agent-based approach is presented to model current processes and to exploit the identified optimization potential. The developed system directly connects the information flow and...
Conference Paper
This paper shows that AI-methods can improve detection of malicious network traffic. A novel method based on Conditional Random Fields combined with Tolerant Pattern Matching is presented. The proposed method uses background knowledge represented in a description logic ontology, user modeled patterns build on-top of this ontology and training examp...
Conference Paper
The development and maintenance of traffic concepts in urban districts is expensive and leads to high investments for altering transport infrastructures or for the acquisition of new resources. We present an agent-based approach for modeling, simulation, evaluation, and optimization of public transport systems by introducing a dynamic microscopic m...
Preprint
Full-text available
In this paper we generalize the idea of QuickHeapsort leading to the notion of QuickXsort. Given some external sorting algorithm X, QuickXsort yields an internal sorting algorithm if X satisfies certain natural conditions. With QuickWeakHeapsort and QuickMergesort we present two examples for the QuickXsort-construction. Both are efficient algorithm...
Conference Paper
Full-text available
A weak heap is a variant of a binary heap where, for each node, the heap ordering is enforced only for one of its two children. In 1993, Dutton showed that this data structure yields a simple worst-case-efficient sorting algorithm. In this paper we review the refinements proposed to the basic data structure that improve the efficiency even further....
Article
Symbolic search with binary decision diagrams (BDDs) often saves huge amounts of memory and computation time. In this paper we propose two general techniques based on transition relation trees to advance BDD search by refining the image operator to compute the set of successors. First, the conjunction tree selects the set of applicable actions thro...
Conference Paper
The complexity and dynamics in group age traffic requires flexible, efficient, and adaptive planning and controlling processes. While the general problem refers to the Vehicle Routing Problem (VRP), additional requirements have to be fulfilled in application. Individual properties and priorities of orders, a heterogeneous fleet of vehicles, dynamic...
Conference Paper
Full-text available
The well-known traveling salesman problem (TSP) is concerned with determining the shortest route for a vehicle while visiting a set of cities exactly once. We consider knowledge and algorithm engineering in combinatorial optimization for improved solving of complex TSPs with Time Windows (TSPTW). To speed-up the exploration of the applied Nested Mo...
Conference Paper
Full-text available
Intelligent behaviour of robot manipulators become important in unknown and changing environments. Emergent behaviour of a machine arises intelligence from the interactions of robots with its environment. Sensorial materials equipped with networks of embedded miniaturized smart sensors can support this behaviour. In this work an integrated autonomo...
Article
The dynamics and complexity of planning and scheduling processes in groupage traffic require efficient, proactive, and reactive system behavior to improve the service quality while ensuring time and cost efficient transportation. We implemented a multiagent-system to emerge an adequate system behavior and focus on the decision making processes of a...
Conference Paper
Full-text available
The power of fractal computation has been mainly exploited for image compression and halftoning. Here, we consider it for finding a fast approximate solution for the fundamental problem of nearest neighbour computation in the image plane. Traditional solutions use Delaunay triangulations or hierarchies (for the case of optimal solutions) or kd-tree...
Article
Full-text available
The complexity and dynamics in groupage traffic require flexible, efficient, and adaptive planning and control processes. The general problem of allocating orders to vehicles can be mapped into the Vehicle Routing Problem (VRP). However, in practical applications additional requirements complicate the dispatching processes and require a proactive a...
Article
Full-text available
This paper addresses improved urban mobility using multiagent simulation. We provide a description of the agent model and the routing infrastructure as a step towards a rich model of the interactions that happen in intermodal transport planning tasks. The multiagent model is generic in the sense that different public and individual transport agents...
Article
This work combines recent advances in combinatorial search under memory limitation, namely bitvector and symbolic search. Bitvector search assumes a bijective mapping between state and memory addresses, while symbolic search compactly represents state sets. The memory requirements vary with the structure of the problem to be solved. The integration...
Chapter
This chapter gives a general introduction to biological pathway and multiple sequence alignment problems. For the latter, a sparse-memory search approach is presented. Computational biology or bioinformatics is a large research field of its own. It is dedicated to the discovery and implementation of algorithms that facilitate the understanding of b...
Chapter
This chapter studies the limits and possibilities of problem abstractions and their relation to heuristic search. As the most important representative of abstraction data structures, it considers pattern databases in great detail. Abstraction is a method to reduce the exploration efforts for large and infinite state spaces. The abstract space is of...
Chapter
This chapter studies the integration of disk space into the search, trying to minimize the number of block accesses. It introduces the theory of external memory algorithms and studies disk-based search with the delayed detection of duplicates and its variants. Often search spaces are so large that even in a compressed form they fail to fit into the...
Chapter
This chapter gives a general introduction to state space search problems in routing applications. Map matching and various speedup techniques are discussed. Navigation is a ubiquitous need to satisfy today's mobility requirements. Current navigation systems assist almost any kind of motion in the physical world including sailing, flying, hiking, dr...
Chapter
This chapter studies heuristic search in game graphs. For two-player games, tree search from the current node is performed and endgame databases are built. The chapter discusses different refinement strategies to improve the exploration for forward search and suggests a symbolic classification algorithm. Adversaries introduce an element of uncertai...
Chapter
This chapter studies different algorithms that explore the middle ground between linear-space algorithms like iterative-deepening A* (IDA*) and memory-sensitive algorithms like A*. It considers transposition tables for full and lossy/lossless sparse state set representations, as well as omission schemes for frontier and visited state sets. By the v...
Chapter
For the enhanced efficiency of A*, this chapter discusses different dictionary data structures. Priority queues are provided for integer- and total-ordered keys. For duplicate elimination, hashing is studied, including provably constant access time. Moreover, maintaining partial states in a form of substrings or subsets is considered. The explorati...
Chapter
This chapter establishes that besides breadth- and depth-first search the (single-source shortest paths) algorithm of Dijkstra is of particular interest, since the heuristic search algorithm A* is a generalization of it. For nonconsistent heuristics, already explored nodes are reopened to preserve the optimality of the first solution. To compute th...
Chapter
This chapter gives an introduction to search problems in model checking, Petri nets, and graph transition systems. It also introduces automated theorem proving and discusses state space search for proof state-based theorem proving and diagnosis problems. The presence of a vast number of computing devices in our environment imposes a challenge for d...
Chapter
This chapter considers synchronous- and asynchronous-distributed versions of A* and iterative-deepening A* (IDA*). Solutions for multicore CPUs/GPUs and workstation clusters are presented for which effective data structures for concurrent access are studied. Moreover, bidirectional search starting at both ends of the search space is considered. Mod...
Chapter
This chapter gives a general introduction to constraint satisfaction. It discusses traditional constraint satisfaction problems, temporal constraint networks, and critical path scheduling and turns to different NP-hard optimization problems. It also discusses recent results on searching backbones and backdoors as well as path and soft constraints....
Chapter
This chapter gives a general introduction to search problems for optimal and suboptimal action planning, including state space search for numerical and temporal planning domains. It also considers plan constraints and soft goals. In domain-independent action planning, a running system must be able to find plans and exploit search knowledge fully au...
Chapter
This chapter discusses learning approaches to prune the successor set(s). It studies the exclusion of forbidden states or move sequences and localizing the search using the notion of relevance. The chapter distinguishes between on-the-fly and offline learning. One of the most effective approaches to tackle large problem spaces is to prune (i.e., cu...

Citations

... Most modern machines are equipped with a graphics processing unit (GPU) with hundreds or thousands of SIMD processors. Researchers have been able to exploit this hardware for breadth-first search problems (Edelkamp, Sulewski, and Yücel 2010) and for domain-independent planning problems (Sulewski, Edelkamp, and Kissmann 2011). ...
... Over the past decade, symbolic search has proven to be a competitive approach to cost-optimal planning (Edelkamp, Kissmann, and Torralba 2015). In contrast to explicit search, in which individual states are generated and expanded, symbolic search operates on whole state sets. ...
... A sampling-based multi-goal motion planning approach with Monte-Carlo Tree Search (MCTS) computes roadmap tours that enable a robot to reach each goal while reducing the overall distance traveled and the number of times it recharges [15]. This discrete planning method accounts for the energy constraint but does not consider uncertainties in the robot's motion and sensing. ...
... The ELO ranking routine is embedded in our Skat AI [12], [13], [14] written in C++, compiled with gcc version 4.9.2 (optimization level -O2). As the formulas are straight-forward, we strongly believe that a Skat web application with a worldwide ranking list similar to 2700chess.com ...