Expanding Search on a Network Thomas Lidbetter London School of Economics

1. Expanding Search on a Network Thomas Lidbetter London School of Economics Joint work with Steve Alpern GRASTA 2014 Wedneday 2nd April 2014

2. Pathwise search An pathwise search on a network with root is a continuous unit speed path starting at the root. 5 5 1 1 2 1 For a Hider located at a node, the search time is first time the path reaches . In this example,

3. Expanding search An expanding search on a rooted network is a nested family of connected sets increasing at unit speed, starting with only the root and ending with the whole network. 5 5 1 1 2 1 For a Hider at a node, the search time is first time is contained in some . In this example,

4. If the Hider is located on according to a known distribution , we can calculate the expected search time for an expanding search . We are interested in the Bayesian Problem of how the Searcher should minimise the expected search time for a given Hider distribution. 8/24 7/24 5 5 1/12 2/12 1 1 1/24 1/12 2 1 Expected search time

5. Search density Definition: The search density (or just density) of a region is the probability the Hider is located in divided by the time required to search , which in expanding searchis just the total length of . Example: The density of the highlighted region in the network below is 8/24 7/24 5 5 1/12 2/12 1 1 1/24 1/12 2 1

6. Search Density Lemma Lemma: Suppose and are disjoint regions of a rooted network with Hider distribution . Let be search of that searches immediately before , and let be an identical search except that it searches before . Then if )). I.e. the region with the largest search density should be searched first.

7. Theorem: Let be a tree with root and suppose a Hider is located on the nodes of according to some distribution. Let be the rooted subtree of maximum density. Then every optimal expanding search of begins by searching. Example 8/24 7/24 5 5 1/12 2/12 Max density = /2 = 1/8 1 1 1 1/24 1/12 8/24 7/24 2 1 1 5 5 1/12 1/24 Max density = /1 = 1/12 2 1 1

8. This doesn’t work for non-trees c 12/20 7/20 Max density subtree = abc Optimal search begins with d. b d 1/20 a

9. Search games • Hider picks a point on the network • Searcher picks a search of the network • Payoff is time taken for Searcher to find Hider (Searcher wants to minimise and Hider wants to maximise)

10. Pathwise search games • A Chinese postman tour is a minimal time tour of a network. • A random Chinese postman tour is an equiprobable choice between a given and its reverse. • If the network has total length and is Eulerian, a finds every point of the network in time . • The Hider can hide uniformly to ensure he is found in expected time . Expected search time of RCPT

11. Pathwise search game: Weakly Eulerian networks Q is Weakly Eulerianif it contains a number of disjoint Eulerian networks which, when each is shrunk to a point, leave a tree. • Theorem (Gal, 2000): • Optimal search strategy is a Random Chinese Postman Tour • Value of the game is half the length of a Chinese Postman Tour

12. Pathwise search game: Trees A tree is a special case of a Weakly Eulerian network. 2/5 2/5 The length of a Chinese Postman Tour is 2, twice the measure of the whole network. Value of the game 5 5 1/10 1/10 1 1 2 1 The Hider hides at the nodes according to the Equal Branch Density (EBD) distribution. This is the unique distribution on the leaf nodes such that at every branch node, the density of each branch is the same.

13. Expanding search games • Theorem: In the expanding search game on trees, • The EBD distribution is optimal for the Hider • it is optimal for the Searcher to randomise between all depth first search using a branching strategy • the value of the game is , where average distance from the root to leaf nodes w.r.t. EBD distribution. 2/5 2/5 5 5 1/10 1/10 1 1 So value 2 1

14. General networks: the three-arc network For pathwise search, the three-arc network search game is notoriously difficult to solve (see Pavlovic 1995). The solution involves a complicated Hider distribution, and a Searcher strategy involving backtracking. 1 1 1

15. However, in expanding search the problem is easy! There is an expanding search, on the three-arc network whose reverse, is also an expanding search. Hence for a Hider found by at time , the expected search time is By hiding uniformly on the network, the Hider can ensure a search time of no more than . 1 1 1

16. Not all expanding searches are reversible... 1 1 1 Theorem: If a rooted network has a reversible expanding search then .

17. Theorem: A rooted network has a reversible expanding search if and only if it is 2-edge-connected. Proof (): We successively construct reversible expanding searches on subnetworks, starting with = a cycle containing the root. This is clearly reversible.

18. Next, since the network is 2-arc-connected, we can find a path between two nodes and in the cycle. Let S1 be the expanding search that follows S0 up to x, then follows the path from x to y, then follows the rest of S0. This is reversible.

19. Find another pair of nodes, and on with a path between them, and add this path in to form . Continuing in this fashion produces a reversible expanding search of the whole network.