The expanding search ratio of a graph Spyros Angelopoulos* Christoph Dürr* Thomas Lidbetter**...
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Transcript of The expanding search ratio of a graph Spyros Angelopoulos* Christoph Dürr* Thomas Lidbetter**...
The expanding search ratioof a graph
Spyros Angelopoulos*Christoph Dürr*
Thomas Lidbetter**
*Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6, Paris, France**Department of Mathematics, London School of Economics, UK
Background
• Searching a fixed graph (Koutsoupias, Papadimitriou, Yannakakis, 1996)
• Mining coal or finding terrorists: The expanding search paradigm (Alpern, L., 2013)
Expanding search
3
21
2
31
𝑂
An expanding search of a (weighted, connected) graph with root is a sequence of edges each one of which is incident to a previously searched vertex.
Search time
For a search , and vertex , the search time is the time is first discovered.Eg. The normalized search time, is .Eg.
𝑖
3+2+2¿7
3
21
2
31
𝑂
Search ratio
The search ratio of is .
The search ratio of a graph is .
If minimises the search ratio we say is optimal.
Eg. 3
21
2
31
𝑂
Proposition
For trees or graphs with unit edge weights, it is optimal to search the vertices in order of their distance from O.
3
21
2
31
𝑂
Counterexample for weighted graphs
5 10
76
𝑂
Counterexample for weighted graphs
5 10
76
𝑂
Theorem
It is NP-complete to decide whether .
Proof: Reduction from 3-SAT.
TheoremThere is a polynomial time algorithm that approximates the search ratio within a factor of
Proof sketch:
𝐺
𝑂Min. cost tree containing all vertices at distance from .
Min. cost tree con-taining all vertices at distance from .
Min. cost tree con-taining all vertices at distance from .
Randomized search ratio
For a randomized search and a vertex , the expected search time and expecte normalized search time are denoted by and
The randomized search ratio of a random search is .
The randomized search ratio of a graph is .
Game theoretic interpretationFinding the optimal randomized search is equivalent to finding the optimal strategy in a zero-sum search game between a Searcher and Hider.
21
𝑂
Hider/Searcher 1,2 2,11 1 32 3/2 1
Optimal randomized search: start with short edge with probability and long edge with probability .
Randomized search ratio, .
2-approximate strategyProposition: For trees or graphs with unit length edges, the optimal deterministic strategy is a 2-approximation for the optimal randomized strategy.
Example
11
𝑂
1
𝑛
and
Randomization can be very bad
𝐿≫1
1𝑂
but searching in a random order has search ratio
Randomized star search
𝑑1
𝑂
𝑑2𝑑3
𝑑𝑛
1=𝑑1≤ 𝑑2≤…≤𝑑𝑛
𝑑2 𝑑3𝑑1 𝑑𝑛
Idea: randomize in “stages”
Randomize between all edges with length satisfying for
Unfortunately, it doesn’t work…
1𝑂
𝑛
2− 2
This has search ratio .
But .2−2
−
Better idea: randomize in “random stages”
1 2 4 8 2𝑘− 2 2𝑘−12𝑘
𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘
𝑂
Better idea: randomize in “random stages”
1 2 4 8 2𝑘− 2 2𝑘−12𝑘
𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘
𝑂
Better idea: randomize in “random stages”
1 2 4 8 2𝑘− 2 2𝑘−12𝑘
𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘
𝑂
Better idea: randomize in “random stages”
1 2 4 8 2𝑘− 2 2𝑘−12𝑘
𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘
𝑂
Better idea: randomize in “random stages”
1 2 4 8 2𝑘− 2 2𝑘−12𝑘
𝑥1 𝑥2 𝑥3 𝑥𝑘− 1 𝑥𝑘
𝑂
Theorem: This has an approximation ratio of 5/4.
Idea of proof
Bound from below using a collection of mixed Hider strategies:
Lemma: If the Hider chooses from the edges with probability proportional to the square of the length of the edge, the expected search ratio is at least
.
An exactly optimal randomized search
Theorem: If the lengths of the edges “don’t increase too fast” then the optimal randomized search can be found inductively and
.
Theorem: The graph with edges that has maximum randomized search ratio is the one with equal length edges, i.e. .
Further directions
• Computational complexity of finding randomized search ratio?
• Continuous version…