Randomized Binary and Tree Search under Pressure

TL;DR

This work analyzes searching for a hidden target under a subbudget of queries on lines and trees by framing the problem as a zero-sum game between a hider and a seeker. For a line with unit profit, the authors derive a fast $O(\log n)$-time algorithm that samples a structured mixture of efficient search strategies, with the optimal value determined by Bezout’s identity via $c=2^k-2$ and $d=\gcd(c,n-1)$, yielding a value $h/w$ and enabling a Nash equilibrium through a dual construction. Extending to general trees, they formulate the problem as an LP and show that the hider’s best response can be computed in $O(n^2 2^{2k})$ time via a dynamic program over valid edge labelings, while the overall equilibrium is obtainable in $2^k\text{poly}(n)$ time; they also prove NP-hardness for best-responses when $k$ is part of the input. The results identify tractable regimes (e.g., $k=O(\log n)$) and connect discrete search under budget to number-theoretic and DP techniques, with potential applications in network surveillance and epidemiological screening where query budgets are tight.

Abstract

We study a generalized binary search problem on the line and general trees. On the line (e.g., a sorted array), binary search finds a target node in $O(\log n)$ queries in the worst case, where $n$ is the number of nodes. In situations with limited budget or time, we might only be able to perform a few queries, possibly sub-logarithmic many. In this case, it is impossible to guarantee that the target will be found regardless of its position. Our main result is the construction of a randomized strategy that maximizes the minimum (over the target position) probability of finding the target. Such a strategy provides a natural solution where there is no apriori (stochastic) information of the target's position. As with regular binary search, we can find and run the strategy in $O(\log n)$ time (and using only $O(\log n)$ random bits). Our construction is obtained by reinterpreting the problem as a two-player (\textit{seeker} and \textit{hider}) zero-sum game and exploiting an underlying number theoretical structure. Furthermore, we generalize the setting to study a search game on trees. In this case, a query returns the edge's endpoint closest to the target. Again, when the number of queries is bounded by some given $k$, we quantify a \emph{the-less-queries-the-better} approach by defining a seeker's profit $p$ depending on the number of queries needed to locate the hider. For the linear programming formulation of the corresponding zero-sum game, we show that computing the best response for the hider (i.e., the separation problem of the underlying dual LP) can be done in time $O(n^2 2^{2k})$, where $n$ is the size of the tree. This result allows to compute a Nash equilibrium in polynomial time whenever $k=O(\log n)$. In contrast, computing the best response for the hider is NP-hard.

Figures (7)

• Figure 1: Example of a search tree $T$ over a tree $G$. Leafs $\nu$ of $T$ are labeled with $V(\nu)$. Covered vertices in $G$ are marked in yellow. Edge labels of $G$ will be explained in Section \ref{['sec:edge-labelings']}
• Figure 2: An example of two maximal covered sets and their respective search trees on a line of length $n=11$ and a budget of $k=3$ queries. Covered vertices are colored, and dashed arrows point to leaves of the search tree with two or more vertices. Figure \ref{['fig:border']} shows the first case of condition (iii) of Proposition \ref{['prop:non-dominated']} and Figure \ref{['fig:noborder']} shows the second case.
• Figure 3: Example of the game on a line of length $n=11$ with $k=3$ queries, where $\text{gcd}(c,n-1)=2$. The searcher choose uniformly one out of 5 strategies. Each strategy is depicted with a distinct color marking the covered cells. Here all cells are covered with probability $3/5$, except the first cell which gets more coverage. By appropriately shifting the strategies the seeker can choose to overcover exactly one of the nodes 0,2,4,6,8 or 10. These are exactly the nodes which the hider avoids, choosing uniformly among all other nodes. The value of the game is $3/5$.
• Figure 4: Example of the game on a line of length $n=12$ with $k=3$ queries. The searcher chooses one out of 9 strategies. Each strategy is depicted with a distinct color marking the covered cells. The seeker covers uniformly all cells. However the hider chooses a non-uniform distribution where segments have length 1 or 2. The value of the game is $5/9$.
• Figure 5: Cumulative distribution of the hider's solution for $n=38$ and $k=4$ ($c=14$) in the first 14 vertices. In this instance, $h=11$ and $w=29$, which means short segments (resp. long) have length 1 (resp. 2). In this figure, connected vertices are in the same segment. The probability assigned to each vertex is illustrated by its shade. Notice that $y([1,v])$ always remains within $g(v)$ and $g(v+1)$.

Theorems & Definitions (41)

• Proposition 3.1
• Corollary 3.1
• Lemma 3.1
• proof
• Corollary 3.2
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof