Binary Search in Graphs Revisited

Abstract

In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by Emamjomeh-Zadeh et al. (in: Proceedings of the 48th annual ACM SIGACT symposium on theory of computing, STOC 2016, Cambridge, pp. 519–532, 2016) to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates’ set for the target, while each query asks an appropriately chosen vertex—the “median”—which minimises a potential $\varPhi$ among the vertices of the candidates’ set. In this paper we address three open questions posed by Emamjomeh-Zadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates’ set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential $\varGamma$ that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential $\varGamma$ that allows querying approximate medians.

1 Introduction

The classical binary search algorithm detects an unknown target (or “treasure”) t on a path with n vertices by asking at most $\log n$ queries to an oracle which always returns the direction from the queried vertex to t. To achieve this upper bound on the number of queries, the algorithm maintains a set of candidates for the place of t; this set is always a sub-path, and initially it is the whole path. Then, at every iteration, the algorithm queries the middle vertex (“median”) of this candidates’ set and, using the response of the query, it excludes either the left or the right half of the set. This way of searching for a target in a path can be naturally extended to the case where t lies on an n-vertex tree, again by asking at most $\log n$ queries that reveal the direction in the (unique) path to t [25]. The principle of the binary search algorithm on trees is based on the same idea as in the case of a path: for every tree there exists a separator vertex such that each of its subtrees contains at most half of the vertices of the tree [17], which can be also efficiently computed.

Due to its prevalent nature in numerous applications, the problem of detecting an unknown target in an arbitrary graph or, more generally in a search space, has attracted many research attempts from different viewpoints. Only recently the binary search algorithm with $\log n$ direction queries has been extended to arbitrary graphs by Emamjomeh-Zadeh et al. [13]. In this case there may exist multiple paths, or even multiple shortest paths form the queried vertex to t. The direction query considered in [13] either returns that the queried vertex q is the sought target t, or it returns an arbitrary direction from q to t, i.e. an arbitrary edge incident to q which lies on a shortest path from q to t. The main idea of this algorithm follows again the same principle as for paths and trees: it always queries a vertex that is the “median” of the current candidates’ set and any response to the query is enough to shrink the size of the candidates’ set by a factor of at least 2. Defining what the “median” is in the case of general graphs now becomes more tricky: Emamjomeh-Zadeh et al. [13] define the median of a set S as the vertex q that minimizes a potential function $\varPhi$, namely the sum of the distances from q to all vertices of S.

Apart from searching for upper bounds on the number of queries needed to detect a target t in graphs, another point of interest is to derive algorithms which, given a graph G, compute the optimal number of queries needed to detect an unknown target in G (in the worst case). This line of research was initiated in [21] where the authors studied directed acyclic graphs (DAGs). Although computing a query-optimal algorithm is known to be NP-hard on general graphs [5, 8, 19], there exist efficient algorithms for trees; after a sequence of papers [1, 16, 20, 22, 29], linear time algorithms were found in [22, 25]. Different models with queries of non-uniform costs or with a probability distribution over the target locations were studied in [6, 7, 9, 18].

A different line of research is to search for upper bounds and information-theoretic bounds on the number of queries needed to detect a target t, assuming that the queries incorporate some degree of “noise”. In one of the variations of this model [2, 13, 14], each query independently returns with probability $p>\frac{1}{2}$ a direction to a shortest path from the queried vertex q to the target, and with probability $1-p$ an arbitrary edge (possibly adversarially chosen) incident to q. The study of this problem was initiated in [14], where $\varOmega (\log n)$ and $O(\log n)$ bounds on the number of queries were established for a path with n vertices. This information-theoretic lower bound of [14] was matched by an improved upper bound in [2]. The same matching bound was extended to general graphs in [13].

In a further “noisy” variation of binary search, every vertex v of the graph is assigned a fixed edge incident to v (also called the “advice” at v). Then, for a fraction $p>\frac{1}{2}$ of the vertices, the advice directs to a shortest path towards t, while for the rest of the vertices the advice is arbitrary, i.e. potentially misleading or adversarially chosen [3, 4]. This problem setting is motivated by the situation of a tourist driving a car in an unknown country that was hit by a hurricane which resulted in some fraction of road-signs being turned in an arbitrary and unrecognizable way. The question now becomes whether it is still possible to navigate through such a disturbed and misleading environment and to detect the unknown target by asking only few queries (i.e. taking advice only from a few road-signs). It turns out that, apart from its obvious relevance to data structure search, this problem also appears in artificial intelligence as it can model searching using unreliable heuristics [4, 23, 26]. Moreover this problem also finds applications outside computer science, such as in navigation issues in the context of collaborative transport by ants [15].

Another way of incorporating some “noise” in the query responses, while trying to detect a target, is to have multiple targets hidden in the graph. Even if there exist only two unknown targets $t_{1}$ and $t_{2}$, the response of each query is potentially confusing even if every query correctly directs to a shortest path from the queried vertex to one of the targets. The reason of confusion is that now a detecting algorithm does not know to which of the hidden targets each query directs. In the context of the above example of a tourist driving a car in an unknown country, imagine there are two main football teams, each having its own stadium. A fraction $0<p_{1}<1$ of the population supports the first team and a fraction $p_{2}=1-p_{1}$ the second one, while the supporters of each team are evenly distributed across the country. The driver can now ask questions of the type “where is the football stadium?” to random local people along the way, in an attempt to visit both stadiums. Although every response will be honest, the driver can never be sure which of the two stadiums the local person meant. Can the tourist still detect both stadiums quickly enough? To the best of our knowledge the problem of detecting multiple targets in graphs has not been studied so far; this is one of the main topics of the present paper.

The problem of detecting a target within a graph can be seen as a special case of a two-player game introduced by Renyi [28] and rediscovered by Ulam [30]. This game does not necessarily involve graphs: the first player seeks to detect an element known to the second player in some search space with n elements. To this end, the first player may ask arbitrary yes/no questions and the second player replies to them honestly or not (according to the details of each specific model). Pelc [27] gives a detailed taxonomy for this kind of games. Group testing is a sub-category of these games, where the aim is to detect all unknown objects in a search space (not necessarily a graph) [10]. Thus, group testing is related to the problem of detecting multiple targets in graphs, which we study in this paper.

It is worth noting that techniques similar to [13] were used to derive frameworks for robust interactive learning [11] and for adaptive hierarchical clustering [12].

1.1 Our Contribution

In this paper we systematically investigate the problem of detecting one or multiple hidden targets in a graph. Our work is driven by the open questions posed by the recent paper of Emamjomeh-Zadeh et al. [13] which dealt with the detection of a single target with and without “noise”. More specifically, Emamjomeh-Zadeh et al. [13] asked for further fundamental generalizations of the model which would be of interest, namely (a) detecting a single target when the query response is a direction to an approximately shortest path, (b) detecting a single target when querying a vertex that is an approximate median of the current candidates’ set S (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far.

We resolve question (a) in Sect. 2.1 by proving that any algorithm requires $\varOmega (n)$ queries to detect a single target t, assuming that a query directs to a path with an approximately shortest length to t. Our results hold essentially for any approximation guarantee, i.e. for 1-additive and for $(1+\varepsilon )$-multiplicative approximations.

Regarding question (b), we first prove in Sect. 2.2 that, for any constant $0<\varepsilon <1$, the algorithm of [13] requires at least $\varOmega (\sqrt{n})$ queries when we query each time an $(1+\varepsilon )$-approximate median (i.e. an $(1+\varepsilon )$-approximate minimizer of the potential $\varPhi$ over the candidates’ set S). Second, to resolve this lower bound, we introduce in Sect. 2.3 a new potential $\varGamma$. This new potential can be efficiently computed and, in addition, guarantees that, for any constant $0 \le \varepsilon <1$, the target t can be detected in $O(\log n)$ queries even when an $(1+\varepsilon )$-approximate median (with respect to $\varGamma$) is queried each time.

Regarding question (c), we initiate in Sect. 3 the study for detecting multiple targets on graphs by focusing mainly to the case of two targets $t_{1}$ and $t_{2}$. We assume throughout that every query provides a correct answer, in the sense that it always returns a direction to a shortest path from the queried vertex either to $t_{1}$ or to $t_{2}$. The “noise” in this case is that the algorithm does not know whether a query is returning a direction to $t_{1}$ or to $t_{2}$. Initially we observe in Sect. 3 that any algorithm requires $\frac{n}{2}-1$ (resp. $n-2$) queries in the worst case to detect one target (resp. both targets) if each query directs adversarially to one of the two targets. Hence, in the remainder of Sect. 3, we consider the case where each query independently directs to the first target $t_{1}$ with a constant probability $p_{1}$ and to the second target $t_{2}$ with probability $p_{2}=1-p_{1}$. For the case of trees, we prove in Sect. 3 that both targets can be detected with high probability within $O(\log ^2 n)$ queries.

For general graphs, we distinguish between biased queries ($p_{1}>p_{2}$) in Sect. 3.1 and unbiased queries ($p_{1}=p_{2}=\frac{1}{2}$) in Sect. 3.2. For biased queries we prove positive results, while for unbiased queries we derive strong negative results. For biased queries, first we observe that we can utilize the algorithm of Emamjomeh-Zadeh et al. [13] to detect the first target $t_{1}$ with high probability in $O(\log n)$ queries; this can be done by considering the queries that direct to $t_{2}$ as “noise”. Thus our objective becomes to detect the target $t_{2}$ in a polylogarithmic number of queries. Notice here that we cannot apply the “noisy” framework of [13] to detect the second target $t_{2}$ , since now the “noise” is larger than $\frac{1}{2}$. We prove our positive results for biased queries by making the additional assumption that, once a query at a vertex v has chosen which target among $\{t_1,t_2\}$ it directs to, it returns any of the possible correct answers (i.e. any of the neighbors u of v such that there exists a shortest path from v to the chosen target using the edge vu) equiprobably and independently from all other queries. We derive a probabilistic algorithm that overcomes this problem and detects the target $t_{2}$ with high probability in $O(\varDelta \log ^{2}n)$ queries, where $\varDelta$ is the maximum degree of a vertex in the graph. Thus, whenever $\varDelta =O(poly\log n)$, a polylogarithmic number of queries suffices to detect $t_{2}$.

In contrast, we prove in Sect. 3.2 that, for unbiased queries, any deterministic (possibly adaptive) algorithm that detects at least one of the targets requires at least $\frac{n}{2}-1$ queries, even in an unweighted cycle. Extending this lower bound for two targets, we prove that, assuming $2c\ge 2$ different targets and unbiased queries, any deterministic (possibly adaptive) algorithm requires at least $\frac{n}{2}-c$ queries to detect one of the targets.

Departing from the fact that our best upper bound on the number of biased queries in Sect. 3.1 is not polylogarithmic when the maximum degree $\varDelta$ is not polylogarithmic, we investigate in Sect. 4 several variations of queries that provide more informative responses. In Sect. 4.1 we turn our attention to “direction-distance” biased queries which return with probability $p_{i}$ both the direction to a shortest path to $t_{i}$ and the distance between the queried vertex and $t_{i}$. In Sect. 4.2 we consider another type of a biased query which combines the classical “direction” query and an edge-variation of it. For both query types of Sects. 4.1 and 4.2 we prove that the second target $t_{2}$ can be detected with high probability in $O(\log ^{3}n)$ queries. Furthermore, in Sects. 4.3 and 4.4 we investigate two further generalizations of the “direction” query which make the target detection problem trivially hard and trivially easy to solve, respectively.

1.2 Our Model and Notation

We consider connected, simple, and undirected graphs. A graph $G=(V,E)$, where $|V|=n$, is given along with a weight function$w:E\rightarrow \mathbb {R}^{+}$ on its edges; if $w(e)=1$ for every $e\in E$ then G is unweighted. An edge between two vertices v and u of G is denoted by vu, and in this case v and u are said to be adjacent. The distance d(v, u) between vertices v and u is the length of a shortest path between v and u with respect to the weight function w. Since the graphs we consider are undirected, $d(u,v)=d(v,u)$ for every pair of vertices v, u. Unless specified otherwise, all logarithms are taken with base 2. Whenever an event happens with probability at least $1-\frac{1}{ n^{\alpha }}$ for some $\alpha >0$, we say that it happens with high probability.

The neighborhood of a vertex $v\in V$ is the set $N(v)=\{u\in V:vu\in E\}$ of its adjacent vertices. The cardinality of N(v) is the degree $\deg (v)$ of v. The maximum degree among all vertices in G is denoted by $\varDelta (G)$, i.e. $\varDelta (G)=\max \{\deg (v):v\in V\}$. For two vertices v and $u\in N(v)$ we denote by $N(v,u)=\{x\in V:d(v,x)=w(vu)+d(u,x)\}$ the set of vertices $x\in V$ for which there exists a shortest path from v to x, starting with the edge vu. Note that, in general, $N(u,v)\ne N(v,u)$. Let $T=\{t_{1},t_{2},\ldots ,t_{|T|}\}\subseteq V$ be a set of (initially unknown) target vertices. A direction query (or simply query) at vertex $v\in V$ returns with probability $p_{i}$ a neighbor $u\in N(v)$ such that $t_{i}\in N(u,v)$, where $\sum _{i=1}^{|T|}p_{i}=1$. If there exist more than one such vertices $u\in N(v)$ leading to $t_{i}$ via a shortest path, the direction query returns an arbitrary one among them, i.e. possibly chosen adversarially, unless specified otherwise. Moreover, if the queried vertex v is equal to one of the targets $t_{i}\in T$, this is revealed by the query with probability $p_{i}$.

2 Detecting a Unique Target

In this section we consider the case where there is only one unknown target $t=t_{1}$, i.e. $T=\{t\}$. In this case the direction query at vertex v always returns a neighbor $u\in N(v)$ such that $t\in N(v,u)$. For this problem setting, Emamjomeh-Zadeh et al. [13] provided a polynomial-time algorithm which detects the target t in at most $\log n$ direction queries. During its execution, the algorithm of [13] maintains a “candidates’ set” $S\subseteq V$ such that always $t\in S$, where initially $S=V$. At every iteration the algorithm computes in polynomial time a vertex v (called the median of S) which minimizes a potential $\varPhi _{S}(v)$ among all vertices of the current set S. Then it queries a median v of S and it reduces the candidates’ set S to $S\cap N(v,u)$, where u is the vertex returned by the direction query at v. The upper bound $\log n$ of the number of queries in this algorithm follows by the fact that always $|S\cap N(v,u)|\le \frac{|S|}{2}$, whenever v is the median of S.

2.1 Bounds for Approximately Shortest Paths

We provide lower bounds for both additive and multiplicative approximation queries. A c-additive approximation query at vertex $v\in V$ returns a neighbor $u\in N(v)$ such that $w(vu) + d(u,t) \le d(v,t) + c$. Similarly, an $(1+\varepsilon )$-multiplicative approximation query at vertex $v\in V$ returns a neighbor $u\in N(v)$ such that $w(vu) + d(u,t) \le (1+\varepsilon ) \cdot d(v,t)$.

It is not hard to see that in the unweighted clique with n vertices any algorithm requires in worst case $n-1$ 1-additive approximation queries to detect the target t. Indeed, in this case $d(v,t)=1$ for every vertex $v \ne t$, while every vertex $u\notin \{v,t\}$ is a valid response of an 1-additive approximation query at v. Since in the case of the unweighted clique an additive 1-approximation is the same as a multiplicative 2-approximation of the shortest path, it remains unclear whether 1-additive approximation queries allow more efficient algorithms for graphs with large diameter. In the next theorem we strengthen this result to graphs with unbounded diameter.

Theorem 1

Assuming 1-additive approximation queries, any algorithm requires at least $n-1$ queries to detect the target t, even in graphs with unbounded diameter.

Proof

To prove the theorem we will construct a graph and a strategy for the adversary such that any algorithm will need $n-1$ queries to locate the target t. Consider a horizontal $2 \times \frac{n}{2}$ grid graph where we add the two diagonals in every cell of the grid. Formally, the graph has $\frac{n}{2}$ “top” vertices $v_1,\ldots ,v_{\frac{n}{2}}$ and $\frac{n}{2}$ “bottom” vertices $u_1,\ldots ,u_{\frac{n}{2}}$. For every $i \in \{1,2,\ldots ,\frac{n}{2}-1\}$ we have the edges $v_{i}v_{i+1}, u_{i}u_{i+1}, v_{i}u_{i}, v_{i+1}u_{i+1}, v_{i}u_{i+1}, v_{i+1}u_{i}$.

The strategy of the adversary is as follows. If the algorithm queries a top vertex $v_{i}$, then the query returns the bottom vertex $u_{i}$. Similarly, if the algorithm queries a bottom vertex $u_{i}$, then the query returns the top vertex $v_{i}$. Observe that, in every case, the query answer lies on a path of length at most one more than a shortest path from the queried vertex and the target t. To see this assume that the algorithm queries a top vertex $v_i$; the case where the queried vertex is a bottom vertex $u_{i}$ is symmetric.

If $t=u_i$, then the edge $v_{i}u_{i}$ clearly lies on the shortest path between $v_{i}$ and t. If $t=u_{j}$, where $j\ne i$, then the shortest path uses one of the diagonal edges incident to $v_{i}$. In this case the edge $v_{i}u_{i}$ leads to a path with length one more than the shortest one. Finally, if $t=v_{j}$, where $j\ne i$, then the shortest path has length $|j-i|$ and uses either the edge $v_iv_{i-1}$ or the edge $v_iv_{i+1}$ . In both cases the edge $v_{i}u_{i}$ lies on the path from $v_{i}$ to T with length $|j-i|+1$ which uses the edge $v_{i}u_{i}$ and one of the diagonal edges $u_{i+1}v_{i-1}$ and $u_{i+1}v_{i+1}$.

Hence, after each query at a vertex different than t, the algorithm can not obtain any information about the position of t (except the fact that it is not the queried node). Thus, in the worst case the algorithm needs to make $n-1$ queries to detect t. $\square$

In the next theorem we extend Theorem 1 by showing a lower bound of $n\cdot \frac{\varepsilon }{4}$ queries when we assume $(1+\varepsilon )$-multiplicative approximation queries.

Theorem 2

Let $\varepsilon >0$. Assuming $(1+\varepsilon )$-multiplicative approximation queries, any algorithm requires at least at least $n\cdot \frac{\varepsilon }{4}$ queries to detect the target t.

Proof

For the proof we use the same construction from Theorem 1, however the adversary we use here is slightly modified. Assume that the distance between the queried vertex and the target t is d. If $d+1 \le (1+\varepsilon )\cdot d$, or equivalently, if $d \ge \frac{1}{\varepsilon }$, the adversary can respond in the same way as in Theorem 1.

Overall, the adversary proceeds as follows. Initially all vertices are unmarked. Whenever the algorithm queries a vertex $v_{i}$ (resp. $u_{i}$), the adversary marks the vertices $\{v_{j},u_{j}:|j-i| < \frac{1}{\varepsilon } \}$ in order to determine the query response. If at least one unmarked vertex remains in the graph, then the query returns (similarly to Theorem  1) vertex $u_{i}$ (resp. $v_{i}$). In this case the adversary can place the target t at any currently unmarked vertex. By doing so, the adversary ensures that the distance between t and any of the previously queried vertices is at least $\frac{1}{\varepsilon }$. If all vertices of the graph have been marked, then the adversary places the target t at one of the last marked vertices and in this case the query returns a vertex on the shortest path between t and the queried vertex.

With the above strategy, any algorithm needs to continue querying vertices until there is no unmarked vertex left. Thus, since at every query the adversary marks at most $2/\varepsilon$ new vertices, any algorithm needs to perform at least $\frac{n/2}{2/\varepsilon }=n\cdot \frac{\varepsilon }{4}$ queries.

$\square$

2.2 Lower Bound for Querying the Approximate Median

The potential $\varPhi _{S}:V\rightarrow \mathbb {R}^{+}$ of [13], where $S\subseteq V$, is defined as follows. For any set $S\subseteq V$ and any vertex $v\in V$, the potential of v is $\varPhi _{S}(v)=\sum _{u\in S}d(v,u)$. A vertex $x\in V$ is an $(1+\varepsilon )$-approximate minimizer for the potential $\varPhi$ over a set S (i.e. an $(1+\varepsilon )$-median of S) if $\varPhi _{S}(x)\le (1+\varepsilon )\min _{v\in V}\varPhi _{S}(v)$, where $\varepsilon >0$. We prove that an algorithm querying at each iteration always an $(1+\varepsilon )$-median of the current candidates’ set S needs $\varOmega (\sqrt{n})$ queries.

Theorem 3

Let $\varepsilon >0$. If the algorithm of [13] queries at each iteration an $(1+\varepsilon )$-median for the potential function $\varPhi$, then at least $\varOmega (\sqrt{n})$ queries are required to detect the target t in a graph G with n vertices, even if the graph G is a tree.

Proof

We will construct a graph $G=(V,E)$ with $n+1$ vertices such that $\varOmega ( \sqrt{n})$ queries are needed to locate the target. The graph G will be a tree with a unique vertex of degree greater than 2, i.e. G is a tree that resembles the structure of a star. Formally, G consists of $\sqrt{n}$ paths of length $\sqrt{n}$ each, where all these paths have a vertex $v_{0}$ as a common endpoint. Let $P_i = (v_0, v_{i,1}, v_{i,2}, \ldots , v_{i,\sqrt{n }-1}, v_{i,\sqrt{n}})$ be the ith path of G. For every $i\le \sqrt{n}$ denote by $Q_{i} = \{v_{i,2},v_{i,3}, \ldots , v_{i,\sqrt{n}}\}$ be the set of vertices of $P_{i}$ without $v_{0}$ and $v_{i,1}$. Furthermore, for every $k \in \{0,1,\ldots , \sqrt{n}\}$ define $V_{-k} = V \setminus (\bigcup _{1\le i\le k}Q_{i})$ to be the set of vertices left in the graph by keeping only the first edge from each path $P_{i}$, where $i\le k$. Note by definition that $V_{-0}=V$.

Now recall that the algorithm of [13] queries at each step an arbitrary median for the potential function $\varPhi$. To prove the theorem, it suffices to show that, in the graph G that we constructed above, the slight modification of the algorithm of [13] in which we query at each step an arbitrary$(1+\varepsilon )$-median for the potential function $\varPhi$, we need at least $\varOmega (\sqrt{n})$ queries to detect the target in worst case. To this end, consider the target being vertex $v_{0}$. The main idea for the remainder of the proof is as follows. At every iteration the central vertex $v_{0}$ and all its neighbors, who have not yet been queried, are $(1+\varepsilon )$-medians, while $v_{0}$ is the exact median for the potential $\varPhi$ of [13]. For every $k\in \{0,1,\ldots ,\sqrt{n}\}$ we have

$$\begin{aligned} \varPhi _{V_{-k}}(v_{0})&=k+(\sqrt{n}-k)\sum _{j=1}^{\sqrt{n}}j \nonumber \\&=k+\frac{1}{2}(\sqrt{n}-k)(n+\sqrt{n}). \end{aligned}$$

(1)

Next we compute $\varPhi _{V_{-k}}(v_{p,1})$ for every $p > k$. Note that $d(v_{p,1},v_{i,1})=2$ for every $i \le k$, and thus $\sum _{i=1}^k d(v_{p,1},v_{i,1}) = 2k$. Furthermore, for the vertices on the path $P_p$ we have

$$\begin{aligned} \sum _{i=2}^{\sqrt{n}} d(v_{p,1},v_{p,i}) = \sum _{i=1}^{\sqrt{n}-1}i = \frac{1 }{2}(n - \sqrt{n}). \end{aligned}$$

Finally, denote by $R=V_{-k} \setminus \{v_{p,1}, v_{p,2}, \ldots , v_{p,\sqrt{n}}\}$ the remaining of the vertices in $V_{-k}$. Then we have

$$\begin{aligned} \sum _{u \in R} d(v_{p,1},u)&= 1 + (\sqrt{n}-k-1) \cdot \sum _{j=2}^{\sqrt{n} +1} j \\&= 1 + \frac{1}{2}(\sqrt{n}-k-1)(n+3\sqrt{n}). \end{aligned}$$

Therefore, it follows that

$$\begin{aligned} \varPhi _{V_{-k}}(v_{p,1}) = 2k + \frac{1}{2}(n - \sqrt{n}) + 1 + \frac{1}{2}( \sqrt{n}-k-1)(n+3\sqrt{n}). \end{aligned}$$

(2)

Now note that, due to symmetry, $v_0$ is the exact median of the vertex set V (with respect to the potential $\varPhi$ of [13]), that is, $\varPhi _{V}(v_0) = min_{x\in V}\{\varPhi _{V}(x)\}$. Furthermore note by (1) and (2) that $\varPhi _{V_{-k}}(v_{p,1}) \ge \varPhi _{V_{-k}}(v_0)$ for every $k<\sqrt{n}$. Moreover, due to symmetry this monotonicity of $\varPhi _{V_{-k}}(\cdot )$ is extended to all vertices $v_{p,2}, v_{p,3}, \ldots , v_{p,\sqrt{n}}$, that is, $\varPhi _{V_{-k}}(v_{p,j}) \ge \varPhi _{V_{-k}}(v_0)$ for every $1\le j\le \sqrt{n}$. Therefore $v_0$ remains the exact median of each of the vertex sets $V_{-k}$, where $0\le k< \sqrt{n}$.

Let $\varepsilon >0$. Then (1) and (2) imply that $\varPhi _{V_{-k}}(v_{p,1}) \le (1+\varepsilon )\varPhi _{V_{-k}}(v_0)$ for every $k < \sqrt{n}$ and for large enough n. Now assume that the algorithm of [13] queries always an $(1+\varepsilon )$-median of the candidates’ set S , where initially $S=V$. Then the algorithm may query always a different neighbor of $v_0$. Due to symmetry, we may assume without loss of generality that the algorithm queries the vertices $v_{1,1}, v_{2,1}, \ldots , v_{\sqrt{n },1}$ in this order. Note that these vertices are $(1+\varepsilon )$-medians of the candidates’ sets $V_{-0}, V_{-1}, \ldots , V_{-(\sqrt{n}-1)}$, respectively. Therefore the algorithm makes at least $\sqrt{n}$ queries, where the total number of vertices in the graph is $n-\sqrt{n}+1$.

$\square$

2.3 Upper Bound for Querying the Approximate Median

In this section we introduce a new potential function $\varGamma _{S}:V\rightarrow \mathbb {N}$ for every $S\subseteq V$, which overcomes the problem occurred in Sect. 2.2. This new potential guarantees efficient detection of t in at most $O(\log n)$ queries, even when we always query an $(1+\varepsilon )$-median of the current candidates’ set S (with respect to the new potential $\varGamma$), for any constant $0<\varepsilon <1$. Our algorithm is based on the approach of [13], however we now query an approximate median of the current set S with respect to $\varGamma$ (instead of an exact median with respect to $\varPhi$ of [13]).

Definition 4

(Potential$\varGamma$) Let $S\subseteq V$ and $v\in V$. Then $\varGamma _{S}(v)=\max \{|N(v,u)\cap S|:u\in N(v)\}$.

Theorem 5

Let $0\le \varepsilon <1$. There exists an efficient adaptive algorithm which detects the target t in at most $\frac{ \log n}{1-\log (1+\varepsilon )}$ queries, by querying at each iteration an $(1+\varepsilon )$-median for the potential function $\varGamma$.

Proof

Our proof closely follows the proof of Theorem 3 of [13]. Let $S\subseteq V$ be an arbitrary set of vertices of G such that $t\in S$. We will show that there exists a vertex $v\in V$ such that $\varGamma _{S}(v)\le \frac{|S|}{2}$. First recall the potential $\varPhi _{S}(v)=\sum _{x\in S}d(v,x)$ . Let now $v_{0}\in V$ be a vertex such that $\varPhi _{S}(v_{0})$ is minimized, i.e. $\varPhi _{S}(v_{0})\le \varPhi _{S}(v)$ for every $v\in V$. Let $u\in N(v_{0})$ be an arbitrary vertex adjacent to $v_{0}$. We will prove that $|N(v_{0},u)\cap S|\le \frac{|S|}{2}$. Denote $S^{+}=N(v_{0},u)\cap S$ and $S^{-}=S\setminus S^{+}$. By definition, for every $x\in S^{+}$, the edge $v_{0}u$ lies on a shortest path from $v_{0}$ to x, and thus $d(u,x)=d(v_{0},x)-w(v_{0}u)$. On the other hand, trivially $d(u,x)\le d(v_{0},x)+w(v_{0}u)$ for every $x\in S$, and thus in particular for every $x\in S^{-}$. Therefore $\varPhi _{S}(v_{0})\le \varPhi _{S}(u)\le \varPhi _{S}(v_{0})+(|S^{-}|-|S^{+}|)\cdot w(v_{0}u)$, and thus $|S^{+}|\le |S^{-}|$. That is, $|N(v_{0},u)\cap S|=|S^{+}|\le \frac{|S|}{2}$, since $S^{-}=S\setminus S^{+}$. Therefore which then implies that $\varGamma _{S}(v_{0})\le \frac{|S|}{2}$ as the choice of the vertex $u\in N(v_{0})$ is arbitrary.

Let $v_{m}\in V$ be an exact median of S with respect to $\varGamma$. That is, $\varGamma _{S}(v_{m})\le \varGamma _{S}(v)$ for every $v\in V$. Note that $\varGamma _{S}(v_{m})\le \varGamma _{S}(v_{0})\le \frac{|S|}{2}$. Now let $0\le \varepsilon <1$ and let $v_{a}\in V$ be an $(1+\varepsilon )$-median of S with respect to $\varGamma$. Then $\varGamma _{S}(v_{a})\le (1+\varepsilon )\varGamma _{S}(v_{m})\le \frac{1+\varepsilon }{2}|S|$. Our adaptive algorithm proceeds as follows. Similarly to the algorithm of [13] (see Theorem 3 of [13]), our adaptive algorithm maintains a candidates’ set S, where initially $S=V$. At every iteration our algorithm queries an arbitrary $(1+\varepsilon )$-median $v_{m}\in V$ of the current set S with respect to the potential $\varGamma$. Let $u\in N(v_{m})$ be the vertex returned by this query; the algorithm updates S with the set $N(v,u)\cap S$. Since $\varGamma _{S}(v_{a})\le \frac{1+\varepsilon }{2}|S|$ as we proved above, it follows that the updated candidates’ set has cardinality at most $\frac{ 1+\varepsilon }{2}|S|$. Thus, since initially $|S|=n$, our algorithm detects the target t after at most $\log _{\left( \frac{2}{1+\varepsilon }\right) }n=\frac{\log n}{1-\log (1+\varepsilon )}$ queries. $\square$

Notice in the statement of Theorem 5 that for $\varepsilon =0$ (i.e. when we always query an exact median) we get an upper bound of $\log n$ queries, as in this case the size of the candidates’ set decreases by a factor of at least 2. Furthermore notice that the reason that the algorithm of [13] is not query-efficient when querying an $(1+\varepsilon )$-median is that the potential $\varPhi _{S}(v)$ of [13] can become quadratic in |S|, while on the other hand the value of our potential $\varGamma _{S}(v)$ can be at most |S| by Definition 4, for every $S\subseteq V$ and every $v\in V$. Furthermore notice that, knowing only the value $\varPhi _{S}(v)$ for some vertex $v\in V$ is not sufficient to provide a guarantee for the proportional reduction of the set S when querying v. In contrast, just knowing the value $\varGamma _{S}(v)$ directly provides a guarantee that, if we query vertex v the set S will be reduced by a proportion of $\frac{ \varGamma _{S}(v)}{|S|}$, regardless of the response of the query. Therefore, in practical applications, we may not need to necessarily compute an (exact or approximate) median of S to make significant progress.

3 Detecting Two Targets

In this section we consider the case where there are two unknown targets $t_{1}$ and $t_{2}$, i.e. $T=\{t_{1},t_{2}\}$. In this case the direction query at vertex v returns with probability $p_{1}$ (resp. with probability $p_{2}=1-p_{1}$) a neighbor $u\in N(v)$ such that $t_{1}\in N(v,u)$ (resp. $t_{2}\in N(v,u)$). Detecting more than one unknown targets has been raised as an open question by Emamjomeh-Zadeh et al. [13], while to the best of our knowledge no progress has been made so far in this direction. Here we deal with both problems of detecting at least one of the targets and detecting both targets. We study several different settings and derive both positive and negative results for them. Each setting differs from the other ones on the “freedom” the adversary has on responding to queries, or on the power of the queries themselves. We will say that the response to a query directs to$t_{i}$, where $i\in \{1,2\}$, if the vertex returned by the query lies on a shortest path between the queried vertex and $t_{i}$.

It is worth mentioning here that, if an adversary would be free to arbitrarily choose which $t_{i}$ each query directs to (i.e. instead of directing to $t_{i}$ with probability $p_{i}$), then any algorithm would require at least $\lfloor \frac{n}{2}\rfloor$ (resp. $n-2$) queries to detect at least one of the targets (resp. both targets), even when the graph is a path. Indeed, consider a path $v_{1},\ldots ,v_{n}$ where $t_{1}\in \{v_{1},\ldots ,v_{\lfloor \frac{n}{2}\rfloor }\}$ and $t_{2}\in \{v_{\lfloor \frac{n}{2}\rfloor +1},\ldots ,v_{n}\}$. Then, for every $i\in \{1,\ldots ,\lfloor \frac{n}{2}\rfloor \}$, the query at $v_{i}$ would return $v_{i+1}$, i.e. it would direct to $t_{2}$. Similarly, for every $i\in \{\lfloor \frac{n}{2}\rfloor +1,\ldots ,n\}$, the query at $v_{i}$ would return $v_{i-1}$, i.e. it would direct to $t_{1}$. It is not hard to verify that in this case the adversary could “hide” the target $t_{1}$ at any of the first $\lfloor \frac{n}{2}\rfloor$ vertices which is not queried by the algorithm and the target $t_{2}$ on any of the last $n-\lfloor \frac{n}{2}\rfloor$ vertices which is not queried. Hence, at least $\lfloor \frac{n}{2}\rfloor$ queries (resp. $n-2$ queries) would be required to detect one of the targets (resp. both targets) in the worst case.

As a warm-up, we provide in the next theorem an efficient algorithm that detects with high probability both targets in a tree using $O(\log ^{2}n)$ queries.

Theorem 6

For any constant $0<p_{1}<1$, we can detect with probability at least $\left( 1-\frac{\log n}{n}\right) ^{2}$ both targets in a tree with n vertices using $O(\log ^{2}n)$ queries.

Proof

Let $G=(V,E)$ be a tree on n vertices and let $T=\{t_{1},t_{2}\}$ be the two targets. The algorithm runs in two phases. In each phase it maintains a candidates’ set $S\subseteq V$ such that, with high probability, S contains at least one of the yet undiscovered targets. At the beginning of each phase $S=V$. Let without loss of generality $p_{1}\ge p_{2}$. Furthermore let $\alpha =-\frac{1}{\log p_{1}}$; note that $\alpha \ge 1$.

The first phase of the algorithm proceeds in $\log n$ iterations, as follows. At the beginning of the ith iteration, where $1\le i\le \log n$ , the candidates’ set is $S_{i}$; note that $S_{1}=V$ at the beginning of the first iteration. Let $v_{i}$ be a median of $S_{i}$ (with respect to the potential $\varGamma$ of Sect. 2.3). In the first iteration we query the median $v_{1}$ of V once; let $u_{1}$ be the response of this query. Then we know that one of the two targets belongs to the set $N(v_{1},u_{1})$, thus we compute the updated candidates’ set $S_{2}=N(v_{1},u_{1})$. Furthermore, since $v_{1}$ was chosen to be a median of $S_{1}$, it follows that $|S_{2}|\le \frac{|S_{1}|}{2}=\frac{n}{2}$.

For each $i\ge 2$, the ith iteration proceeds as follows. We query the median $v_{i}$ of the set $S_{i}$ for $\alpha \log n$ times. First assume that at least one of these $\alpha \log n$ queries at $v_{i}$ directs to a subtree of $v_{i}$ (within $S_{i}$) that does not contain the first median $v_{1}$ of $S_{1}=V$, and let $u_{i}^{\prime }$ be the response of that query. Then we know that the subtree of $v_{i}$ (within $S_{i}$) which is rooted at $u_{i}^{\prime }$ contains at least one of the targets that belong to $S_{i}$. Thus we compute the updated candidates’ set $S_{i+1}=S_{i}\cap N(v_{i},u_{i}^{\prime })$, where again $|S_{i+1}|\le \frac{|S_{i}|}{2}$.

Now assume that all of the $\alpha \log n$ queries at $v_{i}$ direct to the subtree of $v_{i}$ that contains the median $v_{1}$ of the initial candidates’ set $S_{1}=V$. Let $u_{i}^{\prime \prime }$ be the (unique) neighbor of $v_{i}$ in that subtree, that is, all $\alpha \log n$ queries at $v_{i}$ return the vertex $u_{i}^{\prime \prime }$. Then we compute the updated candidates’ set $S_{i+1}=S_{i}\cap N(v_{i},u_{i}^{\prime \prime })$, where again $|S_{i+1}|\le \frac{|S_{i}|}{2}$. In this case, the probability that at least one of the targets of $S_{i}$ does not belong to the subtree of $v_{i}$ (within $S_{i}$) which is rooted at $u_{i}^{\prime \prime }$ is upper bounded by the probability $p_{1}^{\alpha \log n}$ that each of the $\alpha \log n$ queries at $v_{i}$ directs to a target that does not belong to $S_{i}$. That is, with probability at least $1-p_{1}^{\alpha \log n}$, at least one of the targets of $S_{i}$ (which we are looking for) belongs to the subtree of $v_{i}$ (within $S_{i}$) rooted at $u_{i}^{\prime \prime }$. Since at each iteration the size of the candidates’ set decreases by a factor of 2, it follows that $|S_{\log n}|=1$. The probability that at each of the $\log n$ iterations we maintained a target from the previous candidates’ set to the next one is at least $\left( 1-p_{1}^{\alpha \log n}\right) ^{\log n}=\left( 1-\frac{1}{n}\right) ^{\log n}\ge 1-\frac{\log n }{n}$ by Bernoulli’s inequality. That is, with probability at least $1-\frac{ \log n}{n}$ we detect during the first phase one of the two targets in $\log n$ iterations, i.e. in $\alpha \log ^{2}n$ queries in total.

Let $t_{0}$ be the target that we detected during the first phase. In the second phase we are searching for the other target $t_{0}^{\prime }\in T\setminus \{t_{0}\}$. The second phase of the algorithm proceeds again in $\log n$ iterations, as follows. Similarly to the first phase, we maintain at the beginning of the ith iteration, where $1\le i\le \log n$, a candidates’ set $S_{i}$ with median $v_{i}$, where $S_{1}=V$ at the beginning of the first iteration.

For each $i\ge 1$, in the ith iteration of the second phase we query $\alpha \log n$ times the median $v_{i}$ of the set $S_{i}$. First assume that at least one of these $\alpha \log n$ queries at $v_{i}$ directs to a subtree of $v_{i}$ (within $S_{i}$) that does not contain the target $t_{0}$ that we detected in the first phase, and let $u_{i}^{\prime }$ be the response of that query. Then we can conclude that the other target $t_{0}^{\prime }$ belongs to the set $N(v_{i},u_{i}^{\prime })$, thus we compute the updated candidates’ set $S_{i+1}=S_{i}\cap N(v_{i},u_{i}^{\prime })$, where $|S_{i+1}|\le \frac{|S_{i}|}{2}$.

Now assume that all of the $\alpha \log n$ queries at $v_{i}$ direct to the subtree of $v_{i}$ that contains the target $t_{0}$. Let $u_{i}^{\prime \prime }$ be the (unique) neighbor of $v_{i}$ in that subtree, that is, all $\alpha \log n$ queries at $v_{i}$ return the vertex $u_{i}^{\prime \prime }$ . Then we compute the updated candidates’ set $S_{i+1}=S_{i}\cap N(v_{i},u_{i}^{\prime \prime })$, where again $|S_{i+1}|\le \frac{|S_{i}|}{2 }$. In this case, the probability that the undiscovered target $t_{0}^{\prime }$ does not belong to the subtree of $v_{i}$ (within $S_{i}$) which is rooted at $u_{i}^{\prime \prime }$ is upper bounded by the probability $p_{1}^{\alpha \log n}$ that each of the $\alpha \log n$ queries at $v_{i}$ directs to $t_{0}$. That is, with probability at least $1-p_{1}^{\alpha \log n}$, the target $t_{0}^{\prime }$ belongs to the subtree of $v_{i}$ (within $S_{i}$) rooted at $u_{i}^{\prime \prime }$. Since at each iteration the size of the candidates’ set decreases by a factor of at least 2, it follows that $|S_{\log n}|=1$. The probability that at each of the $\log n$ iterations we maintained the target $t_{0}^{\prime }$ in the candidates’ set is at least $\left( 1-p_{1}^{\alpha \log n}\right) ^{\log n}\ge 1-\frac{\log n}{n}$. That is, with probability at least $1-\frac{\log n}{n}$ we detect in $\alpha \log ^{2}n$ queries during the second phase the second target $t_{0}^{\prime }$, given that we detected the other target $t_{0}$ in the first phase.

Summarizing, with probability at least $\left( 1-\frac{\log n}{n}\right) ^{2}$ we detect both targets in $2\alpha \log ^{2}n$ queries. $\square$

Since in a tree both targets $t_{1},t_{2}$ can be detected with high probability in $O(\log ^{2}n)$ queries by Theorem 6, we consider in the remainder of the section arbitrary graphs instead of trees. First we consider in Sect. 3.1biased queries, i.e. queries with $p_{1}>\frac{1}{2}$. Second we consider in Sect. 3.2unbiased queries, i.e. queries with $p_{1}=p_{2}=\frac{1}{2}$.

3.1 Upper Bounds for Biased Queries

In this section we consider biased queries which direct to $t_{1}$ with probability $p_{1}>\frac{1}{2}$ and to $t_{2}$ with probability $p_{2}=1-p_{1}<\frac{1}{2}$. As we can detect in this case the first target $t_{1}$ with high probability in $O(\log n)$ queries by using the “noisy” framework of [13], our aim becomes to detect the second target $t_{2}$ with the fewest possible queries, once we have already detected $t_{1}$.

For every vertex v and every $i\in \{1,2\}$, denote by $E_{t_{i}}(v)=\{u\in N(v):t_{i}\in N(v,u)\}$ the set of neighbors of v such that the edge uv lies on a shortest path from v to $t_{i}$. Note that the sets $E_{t_{1}}(v)$ and $E_{t_{2}}(v)$ can be computed in polynomial time, e.g. using Dijkstra’s algorithm. We assume that, once a query at vertex v has chosen which target $t_{i}$ it directs to, it returns each vertex of $E_{t_{i}}(v)$ equiprobably and independently from all other queries. Therefore, each of the vertices of $E_{t_{1}}(v)\setminus E_{t_{2}}(v)$ is returned by the query at v with probability $\frac{p_{1}}{ |E_{t_{1}}(v)|}$, each vertex of $E_{t_{2}}(v)\setminus E_{t_{1}}(v)$ is returned with probability $\frac{1-p_{1}}{|E_{t_{2}}(v)|}$, and each vertex of $E_{t_{1}}(v)\cap E_{t_{2}}(v)$ is returned with probability $\frac{p_{1} }{|E_{t_{1}}(v)|}+\frac{1-p_{1}}{|E_{t_{2}}(v)|}$. We will show in Theorem 8 that, under these assumptions, we detect the second target $t_{2}$ with high probability in $O(\varDelta \log ^{2}n)$ queries where $\varDelta$ is the maximum degree of the graph.

The high level description of our algorithm (Algorithm 1) is as follows. Throughout the algorithm we maintain a candidates’ set S of vertices in which $t_{2}$ belongs with high probability. Initially $S=V$. In each iteration we first compute an (exact or approximate) median v of S with respect to the potential $\varGamma$ (see Sect. 2.3). Then we compute the set $E_{t_{1}}(v)$ (this can be done as $t_{1}$ has already been detected) and we query $c\varDelta \log n$ times vertex v, where $c=\frac{7(1+p_{1})^{2}}{p_{1}(1-p_{1})^{2}}$ is a constant. Denote by Q(v) the multiset of size $c\varDelta \log n$ that contains the vertices returned by these queries at v. If at least one of these $O(\varDelta \log n)$ queries at v returns a vertex $u\notin E_{t_{1}}(v)$, then we can conclude that $u\in E_{t_{2}}(v)$, and thus we update the set S by $S\cap N(v,u)$. Assume otherwise that all $O(\varDelta \log n)$ queries at v return vertices of $E_{t_{1}}(v)$. Then we pick a vertex $u_{0}\in N(v)$ that has been returned most frequently among the $O(\varDelta \log n)$ queries at v, and we update the set S by $S\cap N(v,u_{0})$. As it turns out, $u_{0}\in E_{t_{2}}(v)$ with high probability. Since we always query an (exact or approximate) median v of the current candidates’ set S with respect to the potential $\varGamma$, the size of S decreases by a constant factor each time. Therefore, after $O(\log n)$ updates we obtain $|S|=1$. It turns out that, with high probability, each update of the candidates’ set was correct, i.e. $S=\{t_{2}\}$. Since for each update of S we perform $O(\varDelta \log n)$ queries, we detect $t_{2}$ with high probability in $O(\varDelta \log ^{2}n)$ queries in total.

Recall that every query at v returns a vertex $u\in E_{t_{1}}(v)$ with probability $p_{1}$ and a vertex $u\in E_{t_{2}}(v)$ with probability $1-p_{1}$. Therefore, for every $v\in V$ the multiset Q(v) contains at least one vertex $u\in E_{t_{2}}(v)$ with probability at least $1-p_{1}^{|Q(v)|}=1-p_{1}^{|c\varDelta \log n|}$. In the next lemma we prove that, every time we update S using Step 8, the updated set contains $t_{2}$ with high probability.

Lemma 7

Let $S\subseteq V$ such that $t_{2}\in S$ and let $S^{\prime }=S\cap N(v,u)$ be the updated set at Step 8 of Algorithm 1. Then $t_{2}\in S^{\prime }$ with probability at least $1-\frac{2}{n}$.

Proof

First we define the vertex subset $\widehat{E}_{t_{2}}(v) = E_{t_{2}}(v) \cap E_{t_{1}}(v)$. Assume that Step 8 of Algorithm 1 is executed; then note that $Q(v)\subseteq E_{t_{1}}(v)$, i.e. the query always returns either a vertex of $E_{t_{1}}(v)\setminus E_{t_{2}}(v)$ or a vertex of $\widehat{E}_{t_{2}}(v)$. Given the fact that Step 8 of Algorithm 1 is executed, note that each of the vertices of $E_{t_{1}}(v)\setminus E_{t_{2}}(v)$ is returned by a query with probability $\frac{p_{1}}{|E_{t_{1}}(v)|}$ and each of the vertices of $\widehat{E}_{t_{2}}(v)$ is returned with probability $\frac{p_{1}}{|E_{t_{1}}(v)|}+\frac{1-p_{1}}{|\widehat{E}_{t_{2}}(v)|}$. Observe that these probabilities are the expected frequencies for these vertices in Q(v), given the fact that $Q(v)\subseteq E_{t_{1}}(v)$. To prove the lemma it suffices to show that, whenever $Q(v)\subseteq E_{t_{1}}(v)$, the most frequent element of Q(v) belongs to $E_{t_{1}}(v)\cap E_{t_{2}}(v)$ with high probability. To this end, let $\delta =\frac{1-p_{1}}{1+p_{1}}$ and $c=\frac{7(1+p_{1})^{2}}{p_{1}(1-p_{1})^{2}}$ be two constants. Note that, for the chosen value of $\delta$, the inequality $|\widehat{E}_{t_{2}}(v)|\le |E_{t_{}}(v)|$ is equivalent to

$$\begin{aligned} (1+\delta )\frac{p_{1}}{|E_{t_{1}}(v)|}\le (1-\delta )\left( \frac{p_{1}}{ |E_{t_{1}}(v)|}+\frac{1-p_{1}}{|\widehat{E}_{t_{2}}(v)|}\right) \end{aligned}$$

(3)

Let $u\in E_{t_{1}}(v)\setminus E_{t_{2}}(v)$, i.e. the query at v directs to $t_{1}$ but not to $t_{2}$. We define the random variable $Z_{i}(u)$, such that $Z_{i}(u)=1$ if u is returned by the i-th query at v and $Z_{i}(u)=0$ otherwise. Furthermore define $Z(u)=\sum _{i=1}^{c\varDelta \log n}Z_{i}(u)$. Since $\Pr (Z_{i}(u)=1)=\frac{p_{1}}{|E_{t_{1}}(v)|}$, it follows that $E(Z(u))=c\varDelta \log n\frac{p_{1}}{|E_{t_{1}}(v)|}$ by the linearity of expectation. Then, using Chernoff’s bounds it follows that

$$\begin{aligned} \Pr (Z(u)\ge (1+\delta )E(Z(u)))&\le \exp \left( -\frac{\delta ^{2}}{3} \frac{p_{1}}{|E_{t_{1}}(v)|}c\varDelta \log n\right) \nonumber \\&\le \exp \left( -2\delta ^{2}\frac{(1+p_{1})^{2}}{(1-p_{1})^{2}}\log n\right) \nonumber \\&=\exp \left( -2\log n\right) =\frac{1}{n^{2}}. \end{aligned}$$

(4)

Thus (4) implies that the probability that there exists at least one $u\in E_{t_{1}}(v)\setminus E_{t_{2}}(v)$ such that $Z(u)\ge (1+\delta )E(Z(u))$ is

$$\begin{aligned} \Pr \left( \exists u\in E_{t_{1}}(v)\setminus E_{t_{2}}(v):Z(u)\ge (1+\delta )\frac{p_{1}}{|E_{t_{1}}(v)|}\right) \le (\varDelta -1)\frac{1}{n^{2}}\le \frac{1}{n}. \end{aligned}$$

(5)

Now let $u^{\prime }\in \widehat{E}_{t_{2}}(v)$. Similarly to the above we define the random variable $Z_{i}^{\prime }(u^{\prime })$, such that $Z_{i}^{\prime }(u^{\prime })=1$ if $u^{\prime }$ is returned by the ith query at v and $Z_{i}^{\prime }(u^{\prime })=0$ otherwise. Furthermore define $Z^{\prime }(u^{\prime })=\sum _{i=1}^{c\varDelta \log n}Z_{i}^{\prime }(u^{\prime })$. Since $\Pr (Z_{i}^{\prime }(u^{\prime })=1)=\frac{p_{1}}{ |E_{t_{1}}(v)|}+\frac{1-p_{1}}{|\widehat{E}_{t_{2}}(v)|}$, by the linearity of expectation it follows that $E(Z(u))=c\varDelta \log n\left( \frac{p_{1}}{|E_{t_{1}}(v)|}+\frac{1-p_{1}}{ |\widehat{E}_{t_{2}}(v)|}\right)$. Then we obtain similarly to (4) that

$$\begin{aligned} \Pr (Z^{\prime }(u^{\prime })\le (1-\delta )E(Z^{\prime }(u^{\prime })))&\le \exp \left( -\frac{\delta ^{2}}{2}\left( \frac{p_{1}}{|E_{t_{1}}(v)|}+ \frac{1-p_{1}}{|\widehat{E}_{t_{2}}(v)|}\right) c\varDelta \log n\right) \nonumber \\&\le \exp \left( -3\delta ^{2}\frac{(1+p_{1})^{2}}{p_{1}(1-p_{1})^{2}}\log n\right) \nonumber \\&\le \exp \left( -3\log n\right) \le \frac{1}{n^{2}}. \end{aligned}$$

(6)

Thus, it follows by the union bound and by (3), (5), and (6) that

$$\begin{aligned} \Pr (\exists u\in E_{t_{1}}(v)\setminus E_{t_{2}}(v):Z(u)\ge Z^{\prime }(u^{\prime }))\le \frac{2}{n}. \end{aligned}$$

(7)

That is, the most frequent element of Q(v) belongs to $\widehat{E}_{t_{2}}(v)$ with probability at least $1-\frac{2}{n}$. This completes the proof of the lemma. $\square$

With Lemma 7 in hand we can now prove the main theorem of the section.

Theorem 8

Assume that every query at a vertex v directs to $t_1$ and to $t_2$ with probability $p_1>\frac{1}{2}$ and $p_2=1-p_1$, respectively. Furthermore, once a query at a vertex v has chosen which target it directs to, it returns any of the possible correct answers equiprobably and independently from all other queries. Then, given $t_{1}$, Algorithm 1 detects $t_{2}$ in $O(\varDelta \log ^{2}n)$ queries with probability at least $(1- \frac{2}{n})^{O(\log n)}$.

Proof

Since we query at each iteration an $(1+\varepsilon )$-median for the potential function $\varGamma$, recall by Theorem 5 that after at most $\frac{\log n}{1-\log (1+\varepsilon )}=O(\log n)$ iterations we will obtain $|S|=1$. Furthermore, in every iteration the algorithm queries $c\varDelta \log n$ times the $(1+\varepsilon )$-median of the current set, and thus the algorithm makes $O(\varDelta \log ^{2}n)$ queries in total. Whenever the algorithm updates S in Step 6 the target $t_{2}$ belongs to the updated set with probability 1. Moreover, whenever the algorithm updates S in Step 8, Lemma  7 implies that the target $t_{2}$ belongs to the updated set with probability at least $(1-\frac{2}{n})$ . Thus, the probability all the $O(\log n)$ updates of S were correct, i.e. $t_{2}$ belongs to S after each of the $O(\log n)$ updates, is at least $(1-\frac{2}{n})^{O(\log n)}$. $\square$

Note by Theorem 8 that, whenever $\varDelta =O(poly\log n)$ we can detect both targets $t_{1}$ and $t_{2}$ in $O(poly\log n)$ queries. However, for graphs with larger maximum degree $\varDelta$, the value of the maximum degree dominates any polylogarithmic factor in the number of queries. The intuitive reason behind this is that, for an (exact or approximate) median v of the current set S, whenever $\deg (v)$ and $E_{t_{1}}(v)$ are large and $E_{t_{2}}(v)\subseteq E_{t_{1}}(v)$, we can not discriminate with a polylogarithmic number of queries between the vertices of $E_{t_{2}}(v)$ and the vertices of $E_{t_{1}}(v)\setminus E_{t_{2}}(v)$ with large enough probability. Although this argument does not give any lower bound for the number of queries in the general case (i.e. when $\varDelta$ is unbounded), it seems that more informative queries are needed to detect both targets with polylogarithmic queries in general graphs. We explore such more informative queries in Sect. 4.

3.2 Lower Bounds for Unbiased Queries

In this section we consider unbiased queries, i.e. queries which direct to each of the targets $t_{1},t_{2}$ with equal probability $p_{1}=p_{2}=\frac{1}{2}$. In this setting every query is indifferent between the two targets, and thus the “noisy” framework of [13] cannot be applied for detecting any of the two targets. In particular, in this section we generalize our study to the case of $2c\ge 2$ different targets $T=\{t_{1},t_{2},\ldots ,t_{2c}\}$, where the query to any vertex $v\notin T$ is unbiased. That is, $p_{i}=\frac{1}{2c}$ for every $i\in \{1,2,\ldots ,2c\}$. In the next theorem we prove that any deterministic (possibly adaptive) algorithm needs at least $\frac{n}{2}-c$ queries to detect one of the 2c targets.

Theorem 9

Suppose that there are 2c targets in the graph and let $p_{i}=\frac{1}{2c}$ for every $i\in \{1,2,\ldots ,2c\}$. Then, any deterministic (possibly adaptive) algorithm requires at least $\frac{n}{2}-c$ queries to locate at least one target, even in an unweighted cycle.

Proof

Let $T=\{t_{1},t_{2},\ldots ,t_{2c}\}$ be the set of targets. Again, let G be the unweighted cycle with $n=2k$ vertices $v_{0},v_{1},\ldots ,v_{2k-1}$. For each $i\in \{1,2,\ldots ,c\}$ the targets $\{t_{i},t_{i+c}\}$ are placed by the adversary on two anti-diametrical vertices of the cycle, i.e. $t_{i}=v_{j}$ and $t_{i+c}=v_{j+k}$, for some $j\in \{0,1,\ldots ,2k-1\}$. Thus, for any vertex $v_{x}\notin T$, the unbiased query at $v_{x}$ returns $v_{x-1}$ with probability $\frac{1}{2}$ and $v_{x+1}$ with probability $\frac{1}{2}$. That is, for each vertex $v_{x}\notin T$ the response of the query at $v_{x}$ is exactly the same. Let $\mathcal {A}$ be a deterministic algorithm that queries at most $k-c-1$ different vertices. Then there exist at least $c+1$ pairs $\{v_{i_{1}},v_{i_{1}+k}\},\{v_{i_{2}},v_{i_{2}+k}\},\ldots ,\{v_{i_{c}},v_{i_{c}+k}\}$ of anti-diametrical vertices such that none of these vertices is queried by the algorithm. Then the adversary can place the 2c targets any c of these $c+1$ pairs of anti-diametrical vertices, without affecting the validity of the previous answers. Thus the algorithm $\mathcal {A}$ needs to query at least $k-c=\frac{n}{2}-c$ different vertices to detect a target. $\square$

Corollary 10

Let $p_{1}=p_{2}=\frac{1}{2}$. Then any deterministic (possibly adaptive) algorithm needs at least $\frac{n}{2}-1$ queries to detect one of the two targets, even in an unweighted cycle.

4 More Informative Queries for Two Targets

A natural alternative to obtain query-efficient algorithms for multiple targets, instead of restricting the maximum degree $\varDelta$ of the graph (see Sect. 3.1), is to consider queries that provide more informative responses in general graphs. As we have already observed in Sect. 3.1, it is not clear whether it is possible to detect multiple targets with $O(poly\log n)$direction queries in an arbitrary graph. In this section we investigate natural variations and extensions of the direction query for multiple targets which we studied in Sect. 3.

4.1 Direction-Distance Biased Queries

In this section we strengthen the direction query in a way that it also returns the value of the distance between the queried vertex and one of the targets. More formally, a direction-distance query at vertex $v\in V$ returns with probability $p_{i}$ a pair $(u,\ell )$, where $u\in N(v)$ such that $t_{i}\in N(u,v)$ and $d(v,t_{i})=\ell$. Note that here we impose again that all $p_{i}$’s are constant and that $\sum _{i=1}^{|T|}p_{i}=1$, where $T=\{t_{1},t_{2},\ldots ,t_{|T|}\}$ is the set of targets. We will say that the response $(u,\ell )$ to a direction-distance query at vertex vdirects to$t_{i}$ if $t_{i}\in N(v,u)$ and $\ell =d(v,t_{i})$. Similarly to our assumptions on the direction query, whenever there exist more than one such vertices $u\in N(v)$ leading to $t_{i}$ via a shortest path, the direction-distance query returns an arbitrary vertex u among them (possibly chosen adversarially). Moreover, if the queried vertex v is equal to one of the targets $t_{i}\in T$, this is revealed by the query with probability $p_{i}$. These direction-distance queries have also been used in  [13] for detecting one single target in directed graphs.

Here we consider the case of two targets and biased queries, i.e. $T=\{t_{1},t_{2}\}$ where $p_{1}>p_{2}$. Similarly to Sect. 3.1, initially we can detect the first target $t_{1}$ with high probability in $O(\log n)$ queries using the “noisy” model of [13]. Thus, in what follows we assume that $t_{1}$ has already been detected. We will show that the second target $t_{2}$ can be detected with high probability with $O(\log ^{3}n)$ additional direction-distance queries using Algorithm 2. The high level description of our algorithm is the following. We maintain a candidates’ set S such that at every iteration $t_{2}\in S$ with high probability. Each time we update the set S, its size decreases by a constant factor. Thus we need to shrink the set S at most $\log n$ times. In order to shrink S one time, we first compute an $(1+\varepsilon )$ -median v of the current set S and we query $\log n$ times this vertex v. Denote by Q(v) the set of all different responses of these $\log n$ direction-distance queries at v. As it turns out, the responses in Q(v) might not always be enough to shrink S such that it still contains $t_{2}$ with high probability. For this reason we also query $\log n$ times each of the $\log n$ neighbors $u\in N(v)$, such that $(u,\ell )\in Q(v)$ for some $\ell \in \mathbb {N}$. After these $\log ^{2}n$ queries at v and its neighbors, we can safely shrink S by a constant factor, thus detecting the target $t_{2}$ with high probability in $\log ^{3}n$ queries.

For the description of our algorithm (see Algorithm 2) recall that, for every vertex v, the set $E_{t_{1}}(v)=\{u\in N(v):t_{1}\in N(v,u)\}$ contains all neighbors of v such that the edge uv lies on a shortest path from v to $t_{1}$.

In the next theorem we prove the correctness and the running time of Algorithm 2.

Theorem 11

Given $t_{1}$, Algorithm 2 detects $t_{2}$ in at most $O(\log ^{3}n)$ queries with probability at least $1-O\left( \log n\cdot p_{1}^{\log n}\right)$.

Proof

Throughout its execution, Algorithm 2 maintains a vertex set S that contains the second target $t_{2}$ with high probability. Initially $S=V$. Let v be an $(1+\varepsilon )$-median of the set S (with respect to the potential $\varGamma$ of Sect. 2.3) at some iteration of the algorithm, and assume that $t_{2}\in S$. We query $\log n$ times vertex v; let Q(v) be the set of all different query responses. Since each query directs to $t_{1}$ with probability $p_{1}$ and to $t_{2}$ with probability $p_{2}$, it follows that at least one of the queries at v directs to $t_{2}$ with probability at least $1-p_{1}^{\log n}$.

Consider a response-pair $(u,\ell )\in Q(v)$. If this query directs to $t_{1}$, then $u\in E_{t_{1}}(v)$ and $\ell =d(v,t_{1})$. Hence, if we detect at least one response pair $(u,\ell )\in Q(v)$ such that $u\notin E_{t_{1}}(v)$ or $\ell \ne d(v,t_{1})$, we can safely conclude that this query directs to $t_{2}$ (lines 5, 6 of Algorithm 2). Therefore, in this case, $u\in E_{t_{2}}(v)=\{u\in N(v):t_{2}\in N(v,u)\}$, and thus we safely compute the updated set $S\cap N(v,u)$ at line 6.

Assume now that $u\in E_{t_{1}}(v)$ and $\ell =d(v,t_{1})$ for every response-pair $(u,\ell )\in Q(v)$ (see lines 8–11 of the algorithm). Then every query at v directs to $t_{1}$. However, as we proved above, at least one of these queries $(u,\ell )\in Q(v)$ also directs to $t_{2}$ (i.e. $u\in E_{t_{2}}(v)$) with probability at least $1-p_{1}^{\log n}$. Therefore $\ell =d(v,t_{1})=d(v,t_{2})$ with probability at least $1-p_{1}^{\log n}$. Note that, in this case, we can not use only the response-pairs of Q(v) to distinguish which query directs to $t_{2}$.

In our attempt to detect at least one vertex $u\in E_{t_{2}}(v)$, we query $\log n$ times each of vertices u such that $(u,\ell )\in Q(v)$. For each such vertex u denote by Q(u) the set of all different response-pairs from these $\log n$ queries at u. Similarly to the above, at least one of these $\log n$ queries at u directs to $t_{2}$ with probability at least $1-p_{1}^{\log n}$. Recall that $d(v,t_{2})=\ell$ and let $(z,\ell ^{\prime })\in Q(u)$. If $u\in E_{t_{2}}(v)$ then $d(u,t_{2})=\ell -w(vu)$, otherwise $d(u,t_{2})>\ell -w(vu)$. Furthermore note that $d(u,t_{1})=\ell -w(vu)$, since $u\in E_{t_{1}}(v)$. Therefore, if we detect at least one response-pair $(z,\ell ^{\prime })\in Q(u)$ such that $\ell ^{\prime }>\ell -w(vu)$, then we can safely conclude that $u\notin E_{t_{2}}(v)$. Otherwise, if for every response-pair $(z,\ell ^{\prime })\in Q(u)$ we have that $\ell ^{\prime }=\ell -w(vu)$, then $u\in E_{t_{2}}(v)$ (i.e. $t_{2}\in N(v,u)$) with probability at least $1-p_{1}^{\log n}$.

Recall that there exists at least one query at v that directs to $t_{2}$ with probability at least $1-p_{1}^{\log n}$, as we proved above. That is, among all response-pairs $(u,\ell )\in Q(v)$ there exists at least one vertex $u\in E_{t_{2}}(v)$ with probability at least $1-p_{1}^{\log n}$. Therefore, we will correctly detect a vertex $u\in E_{t_{2}}(v)$ at lines 10, 11 of the algorithm with probability at least $\left( 1-p_{1}^{\log n}\right) ^{2}$, i.e. with at least this probability the updated candidates’ set at line 11 still contains $t_{2}$. Thus, since we shrink the candidates’ set $\frac{\log n}{1-\log (1+\varepsilon )}=O(\log n)$ times, we eventually detect $t_{2}$ as the unique vertex in the final candidates’ set with probability at least $\left( 1-p_{1}^{\log n}\right) ^{O(\log n)}\ge 1-O(\log n\cdot p_{1}^{\log n})$ by Bernoulli’s inequality. Finally, it is easy to verify from the above that the algorithm will terminate after at most $O(\log ^{3}n)$ queries with probability at least $1-O(\log n\cdot p_{1}^{\log n})$. $\square$

4.2 Vertex-Direction and Edge-Direction Biased Queries

An alternative natural variation of the direction query is to query an edge instead of querying a vertex. More specifically, the direction query (as defined in Sect. 1.2) queries a vertex $v\in V$ and returns with probability $p_{i}$ a neighbor $u\in N(v)$ such that $t_{i}\in N(u,v)$. Thus, as this query always queries a vertex, it can be also referred to as a vertex-direction query. Now we define the edge-direction query as follows: it queries an ordered pair of adjacent vertices (v, u) and it returns with probability $p_{i}$YES (resp. NO) if $t_{i}\in N(v,u)$ (resp. if $t_{i}\notin N(v,u)$). Similarly to our notation in the case of vertex-direction queries, we will say that the response YES (resp. NO) to an edge-direction query at the vertex pair (v, u) refers to $t_{i}$ if $t_{i}\in N(v,u)$ (resp. if $t_{i}\notin N(v,u)$). Similar but different edge queries for detecting one single target on trees have been investigated in [13, 16, 24, 29].

Here we consider the case where both vertex-direction and edge-direction queries are available to the algorithm, and we focus again to the case of two targets and biased queries, i.e. $T=\{t_{1},t_{2}\}$ where $p_{1}>p_{2}$. Similarly to Sects. 3.1 and 4.1, we initially detect $t_{1}$ with high probability in $O(\log n)$ vertex-direction queries using the “noisy” model of [13]. Thus, in the following we assume that $t_{1}$ has already been detected. We will show that Algorithm 3 detects the second target $t_{2}$ with high probability using $O(\log ^{2}n)$ additional vertex-direction queries and $O(\log ^{3}n)$ edge-direction queries, i.e. in total $O(\log ^{3}n)$ queries.

In the next theorem we prove the correctness and the running time of Algorithm 3.

Theorem 12

Given $t_{1}$, Algorithm 3 detects $t_{2}$ in at most $O(\log ^{2}n)$ vertex-direction queries and $O(\log ^{3}n)$ edge-direction queries with probability at least $1-O(\log n\cdot p_{1}^{\log n})$.

Proof

The proof follows a similar approach as the proof of Theorem . Throughout its execution, Algorithm  maintains a vertex set S that contains the second target $t_{2}$ with high probability. Initially $S=V$. Let v be an $(1+\varepsilon )$-median of the set S (with respect to the potential $\varGamma$ of Sect. 2.3) at some iteration of the algorithm, and assume that $t_{2}\in S$. We query $\log n$ times vertex v; let Q(v) be the set of all different query responses. Similarly to the analysis of Algorithm 2 in the proof of Theorem , at least one of the queries at v directs to $t_{2}$ with probability at least $1-p_{1}^{\log n}$.

Consider a response-vertex $u\in Q(v)$. If this query directs to $t_{1}$, then $u\in E_{t_{1}}(v)$. Hence, if we detect at least one $u\in Q(v)$ such that $u\notin E_{t_{1}}(v)$, we can safely conclude that this query directs to $t_{2}$ (lines 5, 6 of Algorithm ). Therefore, in this case, $u\in E_{t_{2}}(v)=\{u\in N(v):t_{2}\in N(v,u)\}$, and thus we safely compute the updated set $S\cap N(v,u)$ at line 6.

Assume now that $u\in E_{t_{1}}(v)$ for every response $u\in Q(v)$ (see lines 8–11 of the algorithm). Then every query at v directs to $t_{1}$, although at least one of them also directs to $t_{2}$ (i.e. $Q(v)\cap E_{t_{2}}(v)\ne \emptyset$) with probability at least $1-p_{1}^{\log n}$, as we proved above. Note that, in this case, we can not use only the vertices of Q(v) to distinguish which query directs to $t_{2}$.

In our attempt to detect at least one vertex $u\in E_{t_{2}}(v)$, we apply $\log n$ edge-direction queries at each of the ordered pairs (v, u), where $u\in Q(v)$. For each such pair (v, u) denote by Q(v, u) the set of all different YES/NO responses from these $\log n$ queries at (v, u). Similarly to the above, at least one of these $\log n$ queries at (v, u) refers to $t_{2}$ with probability at least $1-p_{1}^{\log n}$. Therefore, if NO$\in Q(v,u)$, then we can safely conclude that $u\notin E_{t_{2}}(v)$. Otherwise, if $Q(v,u)=\{\mathtt {YES}\}$, then $u\in E_{t_{2}}(v)$ (i.e. $t_{2}\in N(v,u)$) with probability at least $1-p_{1}^{\log n}$.

Recall that there exists at least one query at v that directs to $t_{2}$ with probability at least $1-p_{1}^{\log n}$. That is, among all responses in Q(v) there exists at least one vertex $u\in E_{t_{2}}(v)$ with probability at least $1-p_{1}^{\log n}$. Therefore, we will correctly detect a vertex $u\in E_{t_{2}}(v)$ at lines 10, 11 of the algorithm with probability at least $\left( 1-p_{1}^{\log n}\right) ^{2}$, i.e. with at least this probability the updated candidates’ set at line 11 still contains $t_{2}$. Thus, similarly to the proof of Theorem 11, we eventually detect $t_{2}$ as the unique vertex in the final candidates’ set with probability at least $1-O(\log n\cdot p_{1}^{\log n})$. Finally, it is easy to verify from the above that the algorithm will terminate after at most $O(\log ^{2}n)$ vertex-direction queries and $\log ^{3}n$ edge-direction queries with probability at least $1-O(\log n\cdot p_{1}^{\log n})$. $\square$

4.3 Two-Direction Queries

In this section we consider another variation of the direction query that was defined in Sect. 1.2 (or “vertex-direction query” in the terminology of Sect. 4.2), which we call two-direction query. Formally, a two-direction query at vertex v returns an unordered pair of (not necessarily distinct) vertices $\{u,u^{\prime }\}$ such that $t_{1}\in N(v,u)$ and $t_{2}\in N(v,u^{\prime })$. Note here that, as $\{u,u^{\prime }\}$ is an unordered pair, the response of the two-direction query does not clarify which of the two targets belongs to N(v, u) and which to $N(v,u^{\prime })$.

Although this type of query may seem at first to be more informative than the standard direction query studied in Sect. 3, we show that this is not the case. Intuitively, this type of query resembles the unbiased direction query of Sect. 3.2. To see this, consider e.g. the unweighted cycle where the two targets are placed at two anti-diametrical vertices; then, applying many times the unbiased direction query of Sect. 3.2 at any specific vertex v reveals with high probability the same information as applying a single two-direction query at v. Based on this intuition the next theorem can be proved with exactly the same arguments as Theorem 9 of Sect. 3.2.

Theorem 13

Any deterministic (possibly adaptive) algorithm needs at least $\frac{n}{2}-1$ two-direction queries to detect one of the two targets, even in an unweighted cycle.

4.4 Restricted Set Queries

The last type of queries we consider is when the query is applied not only to a vertex v of the graph, but also to a subset $S\subseteq V$ of the vertices, and the response of the query is a vertex $u\in N(v)$ such that $t\in N(v,u)$ for at least one of the targets t that belong to the set S. Formally, let T be the set of targets. The restricted-set query at the pair (v, S), where $v\in V$ and $S\subseteq V$ such that $T\cap S\ne \emptyset$, returns a vertex $u\in N(v)$ such that $t\in N(v,u)$ for at least one target $t\in T\cap S$. If there exist multiple such vertices $u\in N(v)$, the query returns one of them adversarially. Finally, if we query a pair (v, S) such that $T\cap S=\emptyset$, then the query returns adversarially an arbitrary vertex $u\in N(v)$, regardless of whether the edge vu leads to a shortest path from v to any target in T. That is, the response of the query can be considered in this case as “noise”.

In the next theorem we prove that this query is very powerful, as $|T|\cdot \log n$ restricted-set queries suffice to detect all targets of the set T.

Theorem 14

Let T be the set of targets. There exists an adaptive deterministic algorithm that detects all targets of T with at most $|T|\cdot \log n$ restricted-set queries.

Proof

To detect the first target we simply apply binary search on graphs. At every iteration we maintain a candidates’ set S (initially $S=V$). We compute a median v of S (with respect to the potential $\varGamma$ of Sect. 2.3) and we query the pair (v, S). If the response of the query at (v, S) is vertex $u\in N(v)$ then we update the candidates’ set as $S\cap N(v,u)$. We know that there is at least one target in the updated set S and that the size of the candidates’ set decreased by a factor of at least 2 (cf. Theorem 5). Thus, after at most $\log n$ restricted-set queries we end up with a candidates’ set of size 1 that contains one target.

We repeat this procedure for another $|T|-1$ times to detect all remaining targets of T,as follows. Assume that we have already detected the targets $t_{1},t_{2},\ldots ,t_{i}\in T$. To detect the next target of T we initially set $S=V\setminus \{t_{1},t_{2},\ldots ,t_{i}\}$ and we apply the above procedure. Then, after at most $\log n$ restricted-set queries we detect the next target $t_{i+1}$. Thus, after at most $|T|\cdot \log n$ restricted-set queries in total we detect all targets of T. $\square$

5 Conclusions

This paper resolves some of the open questions raised by Emamjomeh-Zadeh et al. [13] and makes a first step towards understanding the query complexity of detecting two targets on graphs. Our results provide evidence that different types of queries can significantly change the difficulty of the problem and make it from almost trivial impossible to solve.

The potential $\varGamma$ we introduced in this paper has several interesting properties that have not yet been fully explored. As we mentioned in the paper, just knowing the value $\varGamma _S(v)$ for a vertex v directly provides enough information to quantify the “progress” a direction query can make by querying vertex v, without the need to know the values $\varGamma _S(u)$ for any other vertex $u\ne v$. This property of $\varGamma$ may be exploited to provide computationally more efficient algorithms for detecting one target; an algorithm might only need to compute $\varGamma _S(v)$ for all vertices v lying within a wisely chosen subset such that one of these vertices is an approximate median. Of course, this approach cannot break the $\log n$ lower bound on the number of queries needed to detect the target (e.g. in the path of n vertices), but it could potentially improve the computational complexity of the detection algorithm. Furthermore, the potential $\varGamma$ might be a useful tool for deriving an optimal number of queries for classes of graphs other than trees, since every exact median of $\varGamma$ separates the graph into roughly equal subgraphs. By resolving an open question of [13] we proved that, assuming that a query directs to a path with an approximately shortest path to the (single) target t, any algorithm requires $\varOmega (n)$ queries to detect t. It remains open to specify appropriate special graph classes (or other special conditions) that allow the detection of t using a polylogarithmic number of such approximate-path queries.

For the setting where two, or more, targets need to be detected there is a plethora of interesting questions. We believe that the most prominent one is to derive lower bounds on the number of queries needed to detect both targets in the biased setting. Can the number of queries be improved to $O(\log n)$, or $O(\log n \cdot poly \log \log (n))$? We have preliminary results that suggest a lower bound of $\log n \log \log n$ bound for a special type of algorithms, however a general lower bound seems to require new techniques. Another intriguing question is to find the minimal requirements a query has to satisfy in order to detect even one target in the unbiased setting. Furthermore, in the biased setting, it is not completely clear whether all our assumptions in the statement of Theorem 8 are necessary to prove its correctness; however we believe they are. In particular, can we get in Theorem 8 an upper bound of $O(\varDelta \log ^2 n)$ biased queries for detecting the second target, if we assume that, whenever a query has chosen to direct to a specific target (with a biased probability), it directs to an adversarially chosen correct answer? Is the dependence on $\varDelta$ necessary, even if we assume (as in Theorem 8) that a query randomly chooses among the correct answers?