An Analysis of Backtracking with Search Rearrangement

The search rearrangement backtracking algorithm of Bitner and Reingold [Comm. ACM, 18 (1975), pp. 651–655] introduces at each level of the backtrack tree a variable with a minimal number of remaining values; search order may differ on different branches. For conjunctive normal form formulas with v variables, s literals per term $(s \geqq 3)$, and $v^\alpha $ terms $((s/2) < \alpha < s)$, the average number of nodes in a search rearrangement backtrack tree is $\exp [\Theta (v^{(s - \alpha - 1)/(s - 2)} )]$ (i.e., for some positive constants $a_1 ,a_2 $, and $v_0 $, when $v \geqq v_0 $ the number of nodes is between $\exp (a_1 v^{(s - \alpha - 1)/(s - 2)} )$ and $\exp (a_2 v^{(s - \alpha - 1)/(s - 2)} )$. For $1 < \alpha \leqq s/2$ the average number of nodes is between $\exp [\Theta (v^{(s - \alpha - 1)/(s - 2)} )]$ and $\exp [\Theta ((\ln v)^{(s - 1)/(s - 2)} v^{(s - \alpha - 1)/(s - 2)} )]$. This compares with $\exp [\Theta (v^{(s - \alpha )/(s - 1)} )]$ for ordinary backtracking. For $1 < \alpha < s$, s...