We consider the classic setting in which a mobile Searcher must locate a stationary hidden object, called the Hider, in a network with given arc lengths. This general problem goes back to early work in Isaacs (1965) and Gal (1979), who introduced it in the context of the standard, pathwise search; namely, in this usual setting, the Searcher moves at unit speed starting from a given point of the network that we call the root, and the search time is defined as the first time at which the Searcher reaches the Hider. A different approach was recently introduced in Alpern and Lidbetter (2013), and allows the Searcher to move at infinite speed within any region of the network that it has already visited; see Section 2.1 for a formal definition. This paradigm captures several situations in which the cost of re-exploration is negligible, compared to the cost of first-time exploration, and thus can model settings such as mining for coal, hunting a fugitive, or searching for a missing person.
The above works take the approach of seeking mixed, i.e., randomized search strategies, with the objective of minimizing the expected search time, in the worst case; that is, the maximum expected search time over all hiding points in the network. This is accomplished by studying a zero-sum game with payoff the search time, between a minimizing Searcher and a maximizing Hider. In this paper, instead, we study a normalized variant of the search time, in which the search time for reaching a point in is divided by the shortest path from to in ; we call this the normalized search time of . The objective thus becomes to find strategies that minimize the worst-case (normalized) search time, by considering all points in the network .
This normalized formulation was first applied in search games over unbounded domains, such as the linear search Beck and Newman (1970) and star search Gal (1972) problems. Normalization is essential in unbounded domains, since otherwise the Hider can induce unbounded search times, by hiding arbitrarily far from . Further motivation behind the study of normalized objectives is provided by competitive analysis of online algorithms in which the algorithm operates in a status of total uncertainty about the input, and the normalized objective describes how much close the algorithm’s output is, in comparison to an ideal solution with complete information on the input. For this reason, Jaillet and Stafford (1993) refer to searching under the competitive ratio as online searching. Competitive analysis has been applied even in search games over a bounded domain, as in Koutsoupias et al. (1996); Fleischer et al. (2008); Angelopoulos et al. (2019). We will refer to the competitive ratio of a strategy as the worst-case normalized search time among all points of the network222In Koutsoupias et al. (1996); Angelopoulos et al. (2019) the term search ratio is used in order to refer to the competitive ratio. In this work we choose the latter, since it is more prevalent, and since it has been adopted both by the Operations Research and the Computer Science communities; see e.g., the discussion in Alpern and Gal (2003) and Jaillet and Stafford (1993).. Lastly, we define the competitive ratio of a network (with a given root ) as the minimum competitive ratio of any search strategy for . We will further distinguish between the deterministic and the randomized competitive ratios, depending on whether we consider deterministic or randomized search strategies, respectively.
1.1 Main results
In this work we study the competitive ratio of general networks, both in the expanding and the pathwise search paradigms, which are defined precisely in Section 2. For expanding search, we first show in Section 3 that the deterministic competitive ratio is achieved by a simple strategy. This strategy can be visualized as the frontier that is obtained by “flooding” the network starting at , assuming that the arcs represent pipes of corresponding lengths. We then move to randomized strategies for expanding search in Section 4. Here, we show that, unlike the deterministic case, optimal search strategies have a complex statement even on a very simple network that consists of three arcs. Motivated by this observation, we give approximations to the value of the game. First, we show that the randomized competitive ratio of a network is within a factor of 2 of its deterministic competitive ratio, and this bound is tight. More importantly, we give a class of randomized strategies that approximate the randomized competitive ratio of a network within a factor of . This class of strategies is based on iterative applications of Randomized Depth-First-Search, in randomly chosen and increasingly large subsets of the network. This strategy is inspired by a randomized strategy used for tree graphs in the discrete setting, namely when the Hider can only hide over vertices of a given, finite tree Angelopoulos et al. (2019). We emphasize that, unlike Angelopoulos et al. (2019), in this work, the search domain may be substantially more complex than a tree, and it may also be unbounded.
Moreover, we give further approximations of the value of the game by relating the payoff of the search strategy to the function , which informally gives the measure of the set of points within a certain given radius from the root. As a corollary, we show that if the function is concave, the randomized competitive ratio is identical to the deterministic one. This finding may have practical implications in the context of searching in a big city, since the road network is naturally much more dense in its center than it its outskirts, and one expects this density to decrease the further we move from the city center.
Our approach in studying expanding search, and more specifically, our lower bounds on the randomized competitive ratio, have implications for pathwise search as well. More precisely, in Section 5 we give a randomized pathwise search strategy, inspired by the one for expanding search, which is a 5-approximation of the randomized competitive ratio. This is an improvement over the -approximation that can be derived from techniques in Koutsoupias et al. (1996).
To illustrate the significance of the results and the approaches, consider the star-search problem in which the search domain consists of infinite, concurrent rays (Figure (a)a). Star search has a long history of research, and several of variants of this problem have been studied under the competitive ratio (see Chapters 7 and 9 in Alpern and Gal (2003)). It is known that the deterministic competitive ratio is equal to Gal (1972). In contrast, the randomized competitive ratio is not known (in Kao et al. (1998) optimality is shown under the fairly restrictive assumption of periodic strategies). The strategy we obtain in this work has randomized competitive ratio which is at most a factor of 5 from the optimal one. Furthermore, the result applies to much more complicated unbounded domains, for instance such as the one depicted in Figure (b)b, under the mild (and necessary) assumption that for any , the number of points at distance from the root of the network is bounded.
1.2 Related work
Expanding search on a network was introduced in Alpern and Lidbetter (2013), with the focus on the Bayesian problem of minimizing the expected search time against a known Hider distribution. In a followup paper Alpern and Lidbetter (2019), the same authors studied expanding search on general networks and gave two strategy classes that have expected search times that are within a factor close to 1 of the value of the game. Both these works apply to the unnormalized search time. For normalized objectives Angelopoulos et al. (2019) recently studied expanding search in a fixed (finite) graph in which the Hider can only hide on vertices. In terms of finding a strategy of optimal deterministic competitive ratio Angelopoulos et al. (2019) showed that the problem is NP-hard, and gave a approximation. Concerning the randomized competitive ratio, the same work presented a strategy that is a -approximation in the special case of tree graphs.
The competitive ratio of pathwise search was first studied by Beck and Newman in the context of the linear search problem Beck and Newman (1970) and later by Gal Gal (1972, 1974) for star search. For fixed graphs, assuming that the Hider can only hide on vertices, it is NP-hard to approximate the deterministic competitive ratio Koutsoupias et al. (1996). The same paper also gave constant-factor approximations for both the deterministic and the randomized competitive ratio, assuming the graph is undirected. Extensions to edge-weighted graphs were studied in Ausiello et al. (2000), which also showed connections between graph searching and classic optimization problems such as the Traveling Salesman problem and the Minimum Latency problem. The setting in which the search graph is not known to the Searcher, but is rather revealed as the search progresses was studied in Fleischer et al. (2008).
The exact and approximate competitive ratio of pathwise search has been studied in many settings, mostly assuming a star-like search domain. Examples include multi-Searcher strategies López-Ortiz and Schuierer (2004); Angelopoulos et al. (2016a), searching with turn cost Demaine et al. (2006); Angelopoulos et al. (2017), searching with probabilistic information Jaillet and Stafford (1993), searching with upper/lower bounds on the distance of the Hider from the root Hipke et al. (1999); López-Ortiz and Schuierer (2001); Bose et al. (2015), and searching for multiple hiders Angelopoulos et al. (2014); McGregor et al. (2009); Kirkpatrick (2009). All these works assume that the search domain is either the unbounded line or the unbounded star.
We consider a search domain that is represented by a connected network which consists of vertices and arcs, and which has a certain vertex designated as its root. Moreover, is endowed with Lebesgue measure corresponding to length. The measure of a subset of is denoted by , and in the case that has finite measure, we will denote by the total measure of . This defines a metric on , where is the length of the shortest path from to . We write for the distance from to . We denote by the degree of in , namely the number of arcs incident to .
We do not limit ourselves to bounded networks, but make the standing assumption that the network satisfies the condition that there exists some integer such that for any ,
That is, there are at most points at distance from . Any network with a finite number of arcs automatically satisfies this condition. We will see that this condition ensures that the competitive ratio exists. As an example of an unbounded network, if is an -ray star, we have , for all , hence this quantity is bounded. In contrast, if is an unbounded full binary tree in which there are vertices at distance from the root, for all , then this quantity is unbounded, and this implies the competitive ratio is also unbounded. Indeed, any deterministic search strategy (pathwise or expanding) on this network must take at least time to reach all points at distance from the root, so the deterministic competitive ratio would be at least , which is unbounded. We will see later (Proposition 7) that this implies that the randomized competitive ratio of this tree network is also unbounded.
Given a network , and any , we denote the closed disc of radius around by . Let be the distance of the furthest point in from , where if is unbounded. We define the real function given by , so is the measure of the set of points at distance no more than from the root.
We begin with preliminary definitions and results concerning expanding search, since it is a more recent paradigm, and somewhat more subtle to define. We then explain how these definitions change in what concerns pathwise search.
2.1 Expanding search
In expanding search, we allow the search to move at no cost over any part of the network that it has previously explored. This is formalized in the following definition.
Definition 1 (Alpern and Lidbetter (2013)).
An expanding search on a network with root is a family of connected subsets (for ) satisfying: (i) ; (ii) for all ; and (iii) for all .
If the context is clear, we will refer to an expanding search as a search search strategy. For a given expanding search of and a point , let be the (expanding) search time of under . This was shown to be well defined in Alpern and Lidbetter (2013). For , let be the ratio of the search time of to the distance of from the root. We refer to as the normalized search time. It is convenient to define to be equal to .
The deterministic competitive ratio of a deterministic expanding search of a network is given by
The (deterministic, expanding) competitive ratio, of is given by
where the infinum is taken over all search strategies . If we say that is optimal.
Note that the competitive ratio of a strategy may be infinite. For example, suppose that consists of two unit-length arcs and meeting at the root and suppose searches first and then . If lies on the arc at distance from the root then as . It is not immediately obvious whether or not the competitive ratio of a network is finite in general, but we will show in Section 3 that this is indeed the case, by explicitly giving the optimal search strategy for any network.
In addition, we consider randomized search strategies: that is, search strategies that are chosen according to some probability distribution.
We denote randomized strategies by lower case letters, and for randomized strategies
search strategies: that is, search strategies that are chosen according to some probability distribution. We denote randomized strategies by lower case letters, and for randomized strategiesand for the Searcher and the Hider, respectively, we denote the expected search time by and the expected normalized search time by .
The randomized competitive ratio of a randomized expanding search of a network is given by
The randomized competitive ratio, of is given by
where the infimum is taken over all possible randomized search strategies . If we say that is optimal.
When clear from context, we omit for simplicity, e.g., we will use instead of .
We can view the randomized competitive ratio of a network as the value of the following zero-sum game . A strategy for the Searcher is a search strategy as described above and a strategy for the Hider is a point on . The payoff of the game is the normalized search time . For randomized strategies and of the Searcher and Hider, respectively, the expected payoff is denoted by .
In Alpern and Lidbetter (2013) the authors considered a similar zero-sum game on finite networks in which the players’ strategy sets are the same but the payoff is the unnormalized search time . They showed that the strategy sets are compact with respect to the uniform Hausdorff metric and that is lower semicontinuous in for fixed . Since is a constant for fixed , it follows that is also lower semicontinuous in for fixed , and by the Minimax Theorem Alpern and Gal (1988), we have the following theorem.
Let be a finite network with root . The game has a value, which is equal to the randomized competitive ratio . The Searcher has an optimal mixed strategy (with competitive ratio ) and the Hider has -optimal mixed strategies.
It is not so straightforward to show that the game has a value if is unbounded. Nonetheless, this is not important for our analysis, and we will rely on the following general result for zero-sum games that for any mixed Hider strategy ,
where the supremum is taken over all search strategies .
2.2 Pathwise search
For pathwise search, which is the usual search paradigm, the Searcher follows a continuous, unit-speed path: that is a trajectory with and for all . For such a pathwise search and a point on , the (pathwise) search time of under is the first time that is reached by the Searcher, i.e., . The concepts of deterministic and randomized search times, as well as the deterministic and randomized competitive ratios are defined analogously to Definitions 2 and 3.
As in the case of expanding search, we may view the randomized competitive ratio of a network as the value of a game played between a minimizing Searcher and a maximizing Hider where the payoff is the search time. In the case of finite networks, it is easy to show that the value exists, whereas for unbounded networks, it is again the inequality (2) which will be most essential in our analysis.
3 Deterministic, expanding competitive ratio
In this section we show how to obtain an expanding search of optimal deterministic ratio, using a “water filling” principle. Informally, the network is searched in such a way that the set of points that have been searched at any given time form an expanding disc around . Recall the definition of from Section 2. is piecewise linear and strictly increasing so has an inverse . The interpretation is that is the unique radius for which has measure .
For a network with root , consider the expanding search defined by for .
Thus, is an expanding disc of radius . It is easy to verify that is indeed an expanding search. First, we note that is connected, since is always connected. It also trivially satisfies (i) and (ii) from Definition 1, and (iii) is also satisfied since
We will show that attains the optimal competitive ratio. First, note that the search time of a point under is the unique time such that , so . Hence, the competitive ratio of is
This has an intuitive interpretation as follows: if we draw the graph of then the competitive ratio is the slope of the steepest straight line through the root that intersects with the graph of . Condition (1) ensures that is finite for unbounded networks, since for all .
Theorem 6.333This theorem appeared without proof as Theorem 6 of Angelopoulos et al. (2016b).
The expanding search is optimal and the competitive ratio of a network with root is given by
let be an optimal search, and let be the first time that contains . Then the maximum search time of any point at some fixed distance from is , and it follows that is given by
Clearly, , so , by (3). The optimality of and the expression for follows. ∎
4 Randomized, expanding competitive ratio
In this section we study the randomized competitive ratio of expanding search, which is significantly more challenging to analyze than the deterministic one. We begin by showing that the randomized competitive ratio is at most half the deterministic competitive ratio and that there exist networks for which this bound is tight (Section 4.1). In Section 4.2 we give a Hider strategy that allows us to get useful lower bounds on the randomized competitive ratio. We also obtain bounds on that are parameterized by the function , from which we can deduce the randomized competitive ratio for networks with concave . In Section 4.3 we show that the randomized strategy may have a quite complex statement, even for very simple networks that consist only of three arcs. We address this difficulty in Section 4.4, in which we give a strategy that is within a factor at most of the optimal randomized competitive ratio, for all networks.
4.1 A simple approximation of the randomized competitive ratio
Recall that denotes the optimal deterministic search strategy of Section 3.
Proposition 7.444This proposition appeared without proof as Proposition 7 of Angelopoulos et al. (2016b).
For a network with root , the randomized competitive ratio satisfies
Furthermore, the bounds are tight, in the sense that they are the best possible.
The right-hand inequality is clear, since every deterministic search strategy is also a randomized search strategy. To prove the left-hand inequality, we first observe that since is an optimal deterministic search, for any , we can find some point on such that . Let so that . Let be the Hider strategy that hides on uniformly: that is, it chooses a subset of with probability proportional to the measure of that subset. For any search strategy , the expected search time is at least , so
Since can be arbitrarily small, it follows that .
We will now argue that both bounds are tight. This is trivially true for the right-hand inequality since the network consisting of one arc with the root at its end has the same deterministic and randomized competitive ratio.
For the left-hand inequality, consider the network depicted in Figure 2. The normalized search time is maximized at leaf nodes , so that .
Consider now the randomized strategy that searches the arc of length first before searching the remainder of the arcs in a uniformly random order. Then all points at distance no greater than have expected normalized search time ; a point at distance has
so . Since , we must have that , as .
A corollary of Proposition 7 is that the “water-filling” search approximates the optimal randomized search by a factor of .
4.2 A Hider strategy, and lower bounds on the randomized competitive ratio
For a general network , let be a connected subset. Let be the distance from to and let be the Hider strategy (probability measure) that hides uniformly on , so that for a measurable subset . Then denote the average distance from to points in by .
Consider the Hider strategy given by
By adopting the strategy , the Hider ensures that the randomized competitive ratio satisfies
Let be any search strategy, and note that . We have
To illustrate the applicability of Theorem 8, we show how to obtain, in a different way, the corollary of Proposition 7 that the optimal deterministic search strategy approximates the optimal randomized strategy by a factor of .
The optimal deterministic search approximates the randomized competitive ratio by a factor of .
Let and let be a point of such that , where is the set of all points at distance at most . By Theorem 8, , so
Since can be arbitrarily small, the corollary follows. ∎
More importantly, Theorem 8 allows us to obtain the following lower bound on the randomized competitive ratio of .
For any network with root , it holds that .
Let denote the set of arcs in that are incident with . Fix such that ; clearly, such an must exist. Let be the ball of points in that are at distance at most from , and let be the Hider strategy associated with , and defined as in the statement of Theorem 8. We calculate the average distance from to points in by writing
Moreover, from the definition of , we have that . By Theorem 8, we have
The above lemma implies a tight bound on the randomized competitive ratio for all networks for which the function is concave, as shown in the following corollary.
For any network for which is concave, we have that , and strategy is an optimal randomized strategy.
An example of a network for which is concave is depicted in Figure 3, along with a plot of its function .
More generally, we have established the following approximation.
Suppose that for the network it holds that , for some . Then approximates the optimal randomized ratio of within a factor of at most .
4.3 Optimal randomized strategies are complex: -networks
We now consider a class of the simplest networks for which the function is not concave, and thus Corollary 11 does not apply. In particular, we consider the -network depicted in Figure 4 consisting of a node which is incident to three arcs of lengths , and . The root node is the other endpoint of the arc of length . We refer to the arc of length as the “left arc” and the arc of length as the “right arc”.
Clearly the optimal Hider strategy on the -network will hide on the arc incident to the root with probability . Let be the subset of consisting of all the points on the left arc at distance at most from and all the points on the right arc at distance at most from . From Theorem 8, by using the strategy , the Hider ensures that the competitive ratio is at least
By elementary calculus, this bound is maximized for and , giving
We show that the expression given in (5) is indeed the randomized competitive ratio by giving an optimal Searcher strategy. The optimal Searcher strategy mixes between four different strategies which we list below. (Each strategy begins by searching the arc incident to the root, so we do not mention this part of the search.)
Search the left arc first then search the right arc.
Search the right arc up to length first then the left arc then search the remainder of the right arc.
Search the left arc and the right arc at the same time, at speeds proportional to and respectively, until the whole of the left arc has been searched, then search the remainder of the right arc. In other words, in the time interval , search of the left arc and of the right arc, for , then search the remainder of the right arc.
Begin by searching the right arc, but at some time chosen uniformly at random between and , search the whole of the left arc before completing the search of the right arc.
In Table 1 we list probabilities that the Searcher should choose each of these four strategies along with the expected search time of a point at distance from on the left arc and a point at distance from on the right arc.
|Search strategy||Probability||Expected search time on left||Expected search time on right|
A simple calculation shows that for points on the left arc at distance from , the expected search time is and for points on the right arc at distance from , the expected search time is . For points on the right arc at distance from (if such points exist), the expected search time is , and it is easy to show that this is strictly less than . Hence the randomized competitive ratio is .
4.4 A -approximation of the randomized competitive ratio
In this section we give a search strategy that is a -approximation of the optimal randomized search. This is inspired by the strategy of Angelopoulos et al. (2019) for the discrete case, namely for searching in a given graph when the Hider can only hide at a vertex.
We first define the concept of a Randomized Depth-First Search (RDFS) of a tree . Let be any depth-first search of and let be the depth-first search that visits the leaf nodes of in the reverse order from . Then the randomized search that chooses between and equiprobably is a RDFS of .
Let be a RDFS of a tree . Then the expected time at which a point is found by satisfies
Suppose is an equiprobable mixture of the depth-first search and its reverse . Let and . Then
It is easy to see that is the path from to , so
The lemma follows. ∎
Now we can define the randomized search that is a -approximation. For an arbitrary network , let be its shortest path tree. We will define a search of which naturally translates to a search of . First we partition into infinitely many randomly chosen subsets , . To define the sets , we choose numbers uniformly at random from the interval , and set . We call the the levels of the search.
The randomized doubling strategy is defined as follows. At the start of the th iteration, has already been searched, and we shrink it to the root, so that now is a subtree of the resulting network. The th iteration is then a RDFS of . Note that this means that begins with infinitely small RDFS’s, similarly to optimal strategies for the linear search problem, as studied in Gal (1974).
Before proving that the randomized doubling strategy is a -approximation for the optimal randomized search, we first establish two technical lemmas. Let , for , and let .
For any ,
Applying Theorem 8 to ,
Regarding the right-hand side of the expression above as a quadratic in , it is maximized when , and the lemma follows. ∎
The expected measure of is .
A point is contained in if and only if . This occurs with probability . Therefore, the expected measure of is
The randomized doubling strategy is a -approximation of the optimal randomized search. In particular, .
Suppose that the randomized competitive ratio of is maximized at some point which is contained in , for some . Let
be a random variable that takes the valueor depending on whether is contained in or , respectively. Let be the random variable equal to the sum of half the measure of and the measure of all levels preceding . Then, by Lemma 13, the expected search time of is at most . Hence
We just have to show that . Let and be the contributions to from and , respectively, so that .
We first compute . Note that if , which happens with probability , then so that is disjoint from . In this case . Otherwise, with probability , we have that , and is equal to the sum of half the expected measure of and the measure of , or equivalently, the sum of and half the expected measure of . Applying Lemma 15, with , this is equal to
Putting this together,
Next, we consider . With probability , we have that , so that , and is disjoint from . In this case, is zero. Otherwise, , and is equal to half the expected measure of . Applying Lemma 15 again, this time with , gives
Lastly, we consider . Denote by the set of points in at distance at most from . If , then and . If then and . Integrating over all possible value of , we obtain
Now, the second integral above is equal to the expected measure of , and using Lemma 15 with gives
The first term in the expression on the right-hand side above is non positive, since and , so, dividing by , we obtain