ABSTRACT A point $H$ is hidden in a rooted tree $Q$ which is endowed with asymmetric distances (t... more ABSTRACT A point $H$ is hidden in a rooted tree $Q$ which is endowed with asymmetric distances (travel times) between nodes. We determine the randomized search strategy, starting from the root, which minimizes the expected time to reach $H$, in the worst case. This is equivalent to a zero-sum search game $\Gamma\left(Q\right)$, with minimizing Searcher, maximizing Hider, and payoff equal to the capture time. The worst Hiding distribution (over the leaves) from the Searcher's viewpoint is one where at every node $i$ the probability of each branch is proportional to the minimum time required to tour it from $i$. The optimal randomized search is a mixture over depth-first searches. We also consider briefly some other networks and the possibility of a mobile Hider. Our formulation with asymmetric travel times generalizes that of Gal [SIAM J. Control Optim., 17 (1979), pp. 99-122] for symmetric travel times and also the search games of Kikuta [J. Oper. Res., 38 (1995), pp. 70-88] and Kikuta and Ruckle [Naval Res. Logist., 41 (1994), pp. 821-831], who posited search costs $c_i$ at each node $i$ which were added to the travel time to obtain the payoff. We also briefly consider what happens if we allow the Searcher (Hider) to start (hide) at any leaf node. We determine when properties found by Dagan and Gal [Networks, 52 (2008), pp. 156-161] for the symmetric version of such games hold in our asymmetric context.
Let Q be a connected network with a distinguished (starting) point q0, whose are lengths sum to o... more Let Q be a connected network with a distinguished (starting) point q0, whose are lengths sum to one. We associate with Q a “search value” V(Q) representing the expected time needed for a searcher, starting at q0 and moving at unit speed, to find a moving hider. We assume neither sees the other until they meet. We demonstrate that the “figure‐eight” network, consisting of two equal loops joined at a central starting point, has a search value not exceeding 15/16. This contradicts a conjecture of Gal that the search value of any network is at least 1. In the other direction, we show that V(Q) ≦ 6kD for a network with k edges and diameter D.
We apply a new method of analysis to the asymmetric rendezvous search problem on the line (ARSPL)... more We apply a new method of analysis to the asymmetric rendezvous search problem on the line (ARSPL). This problem, previously studied in a paper of Alpern and Gal (1995), asks how two blind, speed one players placed a distance d apart on the line, can find each other in minimum expected time. The distance d is drawn from a known cumulative probability distribution G, and the players are faced in random directions. We show that the ARSPL is strategically equivalent to a new problem we call the double linear search problem (DLSP), where an object is placed equiprobably on one of two lines, and equiprobably at positions ±d. A searcher is placed at the origin of each of these lines. The two searchers move with a combined speed of one, to minimize the expected time before one of them finds the object. Using results from a concurrent paper of the first author and J. V. Howard (1998), we solve the DLSP (and hence the ARSPL) for the case where G is convex on its support, and show that the solution is that conjectured in a paper of Baston and Gal (1998).
Two agents are placed randomly on nodes of a known graph. They are aware of their own position, u... more Two agents are placed randomly on nodes of a known graph. They are aware of their own position, up to certain symmetries of the graph, but not that of the other agent. At each step, each agent may stay where he is or move to an adjacent node. Their common aim is to minimize the expected number of steps required to meet (occupy the same node). We consider two cases determined by whether or not the players are constrained to use identical strategies. This work extends that of Anderson and Weber on ‘discrete locations’ (complete graph) and is related to continuous (time and space) rendezvous as formulated by Alpern. Probabilistic notions arise in the random initial placement, in the random symmetries determining spatial uncertainty of agents, and through the use of mixed strategies.
The paper is devoted to the construction of dangerous disturbances in linear conflict control pro... more The paper is devoted to the construction of dangerous disturbances in linear conflict control problems. Using the technique of sequential linearization, dangerous disturbances can also be constructed for nonlinear systems such as aircraft dynamics equations, including filters, servomechanisms, etc. The procedure proposed is based on a dynamic programming method and consists in the backward integration of ordinary matrix differential equations defining centers, sizes, and orientations of time-dependent parallelotopes forming a repulsive tube in the time-space domain. A feedback disturbance strategy can keep the state vector of the conflict control system outside the repulsive tube for all admissible inputs of the control.
ABSTRACT A point $H$ is hidden in a rooted tree $Q$ which is endowed with asymmetric distances (t... more ABSTRACT A point $H$ is hidden in a rooted tree $Q$ which is endowed with asymmetric distances (travel times) between nodes. We determine the randomized search strategy, starting from the root, which minimizes the expected time to reach $H$, in the worst case. This is equivalent to a zero-sum search game $\Gamma\left(Q\right)$, with minimizing Searcher, maximizing Hider, and payoff equal to the capture time. The worst Hiding distribution (over the leaves) from the Searcher's viewpoint is one where at every node $i$ the probability of each branch is proportional to the minimum time required to tour it from $i$. The optimal randomized search is a mixture over depth-first searches. We also consider briefly some other networks and the possibility of a mobile Hider. Our formulation with asymmetric travel times generalizes that of Gal [SIAM J. Control Optim., 17 (1979), pp. 99-122] for symmetric travel times and also the search games of Kikuta [J. Oper. Res., 38 (1995), pp. 70-88] and Kikuta and Ruckle [Naval Res. Logist., 41 (1994), pp. 821-831], who posited search costs $c_i$ at each node $i$ which were added to the travel time to obtain the payoff. We also briefly consider what happens if we allow the Searcher (Hider) to start (hide) at any leaf node. We determine when properties found by Dagan and Gal [Networks, 52 (2008), pp. 156-161] for the symmetric version of such games hold in our asymmetric context.
Let Q be a connected network with a distinguished (starting) point q0, whose are lengths sum to o... more Let Q be a connected network with a distinguished (starting) point q0, whose are lengths sum to one. We associate with Q a “search value” V(Q) representing the expected time needed for a searcher, starting at q0 and moving at unit speed, to find a moving hider. We assume neither sees the other until they meet. We demonstrate that the “figure‐eight” network, consisting of two equal loops joined at a central starting point, has a search value not exceeding 15/16. This contradicts a conjecture of Gal that the search value of any network is at least 1. In the other direction, we show that V(Q) ≦ 6kD for a network with k edges and diameter D.
We apply a new method of analysis to the asymmetric rendezvous search problem on the line (ARSPL)... more We apply a new method of analysis to the asymmetric rendezvous search problem on the line (ARSPL). This problem, previously studied in a paper of Alpern and Gal (1995), asks how two blind, speed one players placed a distance d apart on the line, can find each other in minimum expected time. The distance d is drawn from a known cumulative probability distribution G, and the players are faced in random directions. We show that the ARSPL is strategically equivalent to a new problem we call the double linear search problem (DLSP), where an object is placed equiprobably on one of two lines, and equiprobably at positions ±d. A searcher is placed at the origin of each of these lines. The two searchers move with a combined speed of one, to minimize the expected time before one of them finds the object. Using results from a concurrent paper of the first author and J. V. Howard (1998), we solve the DLSP (and hence the ARSPL) for the case where G is convex on its support, and show that the solution is that conjectured in a paper of Baston and Gal (1998).
Two agents are placed randomly on nodes of a known graph. They are aware of their own position, u... more Two agents are placed randomly on nodes of a known graph. They are aware of their own position, up to certain symmetries of the graph, but not that of the other agent. At each step, each agent may stay where he is or move to an adjacent node. Their common aim is to minimize the expected number of steps required to meet (occupy the same node). We consider two cases determined by whether or not the players are constrained to use identical strategies. This work extends that of Anderson and Weber on ‘discrete locations’ (complete graph) and is related to continuous (time and space) rendezvous as formulated by Alpern. Probabilistic notions arise in the random initial placement, in the random symmetries determining spatial uncertainty of agents, and through the use of mixed strategies.
The paper is devoted to the construction of dangerous disturbances in linear conflict control pro... more The paper is devoted to the construction of dangerous disturbances in linear conflict control problems. Using the technique of sequential linearization, dangerous disturbances can also be constructed for nonlinear systems such as aircraft dynamics equations, including filters, servomechanisms, etc. The procedure proposed is based on a dynamic programming method and consists in the backward integration of ordinary matrix differential equations defining centers, sizes, and orientations of time-dependent parallelotopes forming a repulsive tube in the time-space domain. A feedback disturbance strategy can keep the state vector of the conflict control system outside the repulsive tube for all admissible inputs of the control.
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Papers by Steve Alpern