The rendezvous dilemma is a logical dilemma, typically formulated in this way:
If they both choose to wait, they will never meet. If they both choose to walk there are chances that they meet and chances that they do not. If one chooses to wait and the other chooses to walk, then there is a theoretical certainty that they will meet eventually; in practice, though, it may take too long for it to be guaranteed. The question posed, then, is: what strategies should they choose to maximize their probability of meeting?
Examples of this class of problems are known as rendezvous problems. These problems were first introduced informally by Steve Alpern in 1976,and he formalised the continuous version of the problem in 1995. This has led to much recent research in rendezvous search. Even the symmetric rendezvous problem played in n discrete locations (sometimes called the Mozart Cafe Rendezvous Problem) has turned out to be very difficult to solve, and in 1990 Richard Weber and Eddie Anderson conjectured the optimal strategy. Only recently has the conjecture been proved for n = 3 by Richard Weber. This was the first non-trivial symmetric rendezvous search problem to be fully solved. Note that the corresponding asymmetric rendezvous problem has a simple optimal solution: one player stays put and the other player visits a random permutation of the locations.
As well as being problems of theoretical interest, rendezvous problems include real-world problems with applications in the fields of synchronization, operating system design, operations research, and even search and rescue operations planning.
The deterministic rendezvous problem is a variant of the rendezvous problem where the players, or robots, must find each other by following a deterministic sequence of instructions. Although each robot follows the same instruction sequence, a unique label assigned to each robot is used for symmetry breaking.
In mathematics and computer science, an algorithm is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always unambiguous and are used as specifications for performing calculations, data processing, automated reasoning, and other tasks.
In graph theory, the shortest path problem is the problem of finding a path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized.
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In computational complexity theory, SL is the complexity class of problems log-space reducible to USTCON, which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same connected component. This problem is also called the undirected reachability problem. It does not matter whether many-one reducibility or Turing reducibility is used. Although originally described in terms of symmetric Turing machines, that equivalent formulation is very complex, and the reducibility definition is what is used in practice.
Václav (Vašek) Chvátal (Czech: [ˈvaːtslaf ˈxvaːtal] is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada. He has published extensively on topics in graph theory, combinatorics, and combinatorial optimization.
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Shmuel Gal is a mathematician and professor of statistics at the University of Haifa in Israel.
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