Expanding Search on a NetworkThomas Lidbetter

London School of EconomicsJoint work with Steve Alpern

GRASTA 2014Wedneday 2nd April 2014

An pathwise search on a network with root is a continuous unit speed path starting at the root.

𝑂For a Hider located at a node, the search time is first time the path reaches .

In this example,

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Pathwise search

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.

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For a Hider at a node, the search time is first time is contained in some .

In this example,

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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.

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Expected search time 112

(1 )

+112

(2 )+124

(4 )+212

(5 )

+824

(10 )+724

(15 )

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 search is just the total length of .

Example: The density of the highlighted region in the network below is

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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.

𝐴 𝐵

Example

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Max density = /1 = 1/12

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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.

This doesn’t work for non-trees

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Max density subtree = abc

Optimal search begins with d.

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)

Pathwise search Expanding search

Bayesian problem Linear Search Problem (Beck)Otherwise not much known

Alpern & L.

Search game Gal theory Alpern & L.

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 .

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𝑥1−𝑥

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Expected search time of RCPT

Pathwise search game: Weakly Eulerian networks

Q is Weakly Eulerian if 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

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Pathwise search game: Trees

A tree is a special case of a Weakly Eulerian network.

The length of a Chinese Postman Tour is 2, twice the measure of the whole network.

Value of the game

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.

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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.

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So value

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.

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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 .

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Not all expanding searches are reversible...

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Theorem: If a rooted network has a reversible expanding search then .

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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.

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.

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𝑥0 𝑦 0

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.

𝑂

𝑥1

𝑦 1