STOCHASTIC ENUMERATION METHOD

• Published in: Simulation and the Monte Carlo Method Vol. 10; pp. 351 - 376
• Main Authors: Rubinstein, Reuven Y, Kroese, Dirk P
• Format: Book Chapter
• Language: English
• Published: United States John Wiley & Sons, Incorporated 2016; John Wiley & Sons, Inc
• Subjects: breadth‐first search (BFS) methods; depth‐first search (DFS); Monte Carlo technique; stochastic enumeration (SE); stochastic enumeration (SE) algorithm
• ISBN: 9781118632161; 1118632168
• DOI: 10.1002/9781118631980.ch10

This chapter introduces a new generic Monte Carlo technique, called stochastic enumeration (SE), to solve the counting problems that can be formulated in terms of estimating the cost of a tree. The SE method is a sequential importance sampling technique in which random paths are generated through the tree, in a parallel fashion. Two of the most useful algorithms for searching trees or more generally, graphs or networks, are the depth‐first search (DFS) and breadth‐first search (BFS) methods. Starting from a root of the tree, the algorithm enumerates all the vertices of the tree in a certain order. A systematic way to determine this is to sequentially construct a tree of paths of various lengths, all of which start from the top vertex and end either at the bottom‐left vertex or at a vertex whose neighbors. The stochastic enumeration (SE) algorithm generalizes Knuth's algorithm by simulating multiple paths through the tree simultaneously. These paths are also allowed to interact, in the sense that paths avoid each other and therefore explore a larger part of the tree.