...er search trees in certain situations. Knuth’s method Knuth’s method estimates�, the size of a backtrack tree as �������������� where�� is the branching rate observed at depth � using random probing (=-=Knuth 1975-=-). For binary trees, if�is the depth of a random probe, then�� � � for all � � � and 0 otherwise, so � is ����s�. Knuth’s method simply averages this estimate over multiple random probes, giving����� ...

...ues have been applied to predict the size of search trees. Horvitz and colleagues have used Bayesian methods to predict runtimes of constraint satisfaction algorithms based on wide range of measures (=-=Horvitz et al. 2001-=-; Kautz et al. 2002; Ruan, Horvitz, & Kautz 2002). Such predictions are then used to derive good restart strategies. Leyton-Brown and Nudelman used statistical regression to learn a function to predic...

... to predict the size of search trees. Horvitz and colleagues have used Bayesian methods to predict runtimes of constraint satisfaction algorithms based on wide range of measures (Horvitz et al. 2001; =-=Kautz et al. 2002-=-; Ruan, Horvitz, & Kautz 2002). Such predictions are then used to derive good restart strategies. Leyton-Brown and Nudelman used statistical regression to learn a function to predict runtimes for NP-h...

...(Allen & Minton 1996). They used these estimates to select the most promising algorithm. Lobjois and Lemaitre also used Knuth’s method to select the branch and bound algorithm likely to perform best (=-=Lobjois & Lemaitre 1998-=-). Cornuéjols, Karamanov and Li predict the size of a branch and bound search tree by using the maximum depth, the widest level and the first level at which the tree is no longer complete to build a s...

...s method that uses partial backtracking, a beam-like search procedure (Purdom 1978). Chen has proposed a generalization of Knuth’s method called heuristic sampling that stratifies nodes into classes (=-=Chen 1992-=-). Knuth’s method classifies nodes by depth. Chen applied his method to estimate the size of depth-first, breadth-first, bestfirst and iterative deepening search trees. Allen and Minton adapted Knuth’...

...to search trees explored by procedures like branch and bound since the bounds to be used down a particular branch are not known in advance. Another difficulty for random probing, as argued by Purdom (=-=Purdom 1978-=-), is that it is easily misled if the tree is very unbalanced. More systematic methods are required in such situations. It is sometimes possible to use Knuth’s method in an approximate way by substitu...

...h-first, bestfirst and iterative deepening search trees. Allen and Minton adapted Knuth’s method to constraint satisfaction algorithms by averaging over the last ten branches sampled by backtracking (=-=Allen & Minton 1996-=-). They used these estimates to select the most promising algorithm. Lobjois and Lemaitre also used Knuth’s method to select the branch and bound algorithm likely to perform best (Lobjois & Lemaitre 1...

...s at the phase transition in hardness. This problem set is referred to as 3-unsat. We tested 200 instances, and computed the ratio of estimated tree size over actual. The SAT solver used was satz215 (=-=Li 1999-=-), a DPLL-based solver which uses lookahead in choosing the next variable on which to branch, but which does not learn nogoods. Figure 3 shows the evolution of the estimate as the search proceeds. The...

...rawski, & Sakallah 2002). Musick and Russell abstracted the search space to construct a Markov model for predicting the runtime of heuristic search methods like hill climbing and simulated annealing (=-=Musick & Russell 1992-=-). Slaney, Thiébaux and Kilby have used the search cost to solve easy decision problems away from the phase boundary as a means to predict the cost of solving the corresponding optimization problem (S...

...2). Slaney, Thiébaux and Kilby have used the search cost to solve easy decision problems away from the phase boundary as a means to predict the cost of solving the corresponding optimization problem (=-=Slaney & Thiébaux 1998-=-; Thiébaux, Slaney, & Kilby 2000). They showed that good estimates could be found using a small fraction of the time taken to prove optimality. Finally, statistical techniques have been applied to pre...

...ar model of the branching rate. (Cornuéjols, Karamanov, & Li 2006). Kokotov and Shlyakhter measure the progress of Davis Putnam solvers using an estimator somewhat similar to our recursive estimator (=-=Kokotov & Shlyakhter 2000-=-). Finally, Aloul, Sierawski and Sakallah use techniques based on decision diagrams to estimate progress of a satisfiability solver (Aloul, Sierawski, & Sakallah 2002). Musick and Russell abstracted t...