Bifurcation: How to Explore a Tree
Sariel Har-Peled
TL;DR
This work reframes parametric search as a tree exploration problem on an implicit $(n,k)$-tree with $k$ binary forks, where a target node is located using a comparison oracle. The authors introduce a round-based algorithm that maintains a compact explored subtree $\mathcal{C}$ using trimming and halving, balancing exploration with binary search to achieve $O(n \sqrt{k})$ time and $O(\sqrt{k} + \log n)$ oracle calls. They provide matching lower bounds under two models: a linear-time decider and a leaf-only oracle, demonstrating the limits of this approach. The results establish a principled, independent interpretation of bifurcation as a tree exploration technique with provable efficiency and optimality in key regimes. It offers a potential avenue for faster algorithms in parametric search problems where cheap rollback and near-linear deciders hold.
Abstract
Avraham et al. [AFK+15] presented an alternative approach to parametric search, called \emph{bifurcation}, that performs faster under certain circumstances. Intuitively, when the underlying decider execution can be rolled back cheaply and the decider has a near-linear running time. For some problems, this leads to fast algorithms that beat the seemingly natural lower bound arising from distance selection. Bifurcation boils down to a tree exploration problem. You are given a binary (unfortunately implicit) tree of height $n$ and $k$ internal nodes with two children (all other internal nodes have a single child), and assume each node has an associated parameter value. These values are sorted in the inorder traversal of the tree. Assume there is (say) a node (not necessarily a leaf) that is the target node that the exploration needs to discover. The player starts from the root. At each step, the player can move to adjacent nodes to the current location (i.e., one of the children or the parent). Alternatively, the player can call an oracle on the current node, which returns either that it is the target (thus, mission accomplished!) or whether the target value is strictly smaller or larger than the current one. A naive algorithm explores the whole tree, in $O(n k)$ time, then performs $O(\log k n)$ calls to the oracle to find the desired leaf. Avraham \etal showed that this can be improved to $O(n \sqrt{k} )$ time, and $O( \sqrt{k} \log n)$ oracle calls. Here, we improve this to $O(n \sqrt{k} )$ time, with only $ O( \sqrt{k} + \log n)$ oracle calls. We also show matching lower bounds, under certain assumptions. We believe our interpretation of bifurcation as a tree exploration problem, and the associated algorithm, are of independent interest.