The expanding search ratio
of 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
An expanding search of a (weighted, connected) graph with root
𝑂 is a sequence of edges each one of which is incident to a
previously searched vertex.
1
3
3
2
2
𝑂
1
Search time
3
1
2
2
3
1
𝑖
𝑂
For a search 𝑆, and vertex 𝑖, the search time 𝑇(𝑆, 𝑖) is the time 𝑖 is
first discovered.
Eg. 𝑇(𝑆, 𝑖) = 3+2 +2 = 7
The normalized search time, 𝑇(𝑆, 𝑖) is 𝑇(𝑆, 𝑖)/𝑑(𝑂, 𝑖).
Eg. 𝑇 𝑆, 𝑖 = 7/2 = 3.5
Search ratio
1
3
3
2
1
2
Eg. 𝜎𝑆 = 3.5
𝑂
The search ratio 𝜎𝑆 of 𝑆 is max 𝑇(𝑆, 𝑖) .
𝑖
The search ratio 𝜎 of a graph 𝐺 is min 𝜎𝑆 = min max 𝑇(𝑆, 𝑖).
𝑆
𝑆
𝑖
If 𝑆 minimises the search ratio we say 𝑆 is optimal.
Proposition
For trees or graphs with unit edge weights, it is
optimal to search the vertices in order of their
distance from O.
1
3
3
2
2
𝑂
1
Counterexample for weighted graphs
7
6
10
5
𝑂
Counterexample for weighted graphs
7
6
10
5
𝑂
Theorem
It is NP-complete to decide whether 𝜎 ≤ 𝑅.
Proof: Reduction from 3-SAT.
Theorem
There is a polynomial time algorithm that approximates
the search ratio within a factor of 4log 4 + 𝜖 < 5.55 .
Proof sketch:
𝐺
Min. cost tree containing all vertices at
distance ≤ 8 from 𝑂.
Min. cost tree containing all vertices at
distance ≤ 4 from 𝑂.
Min. cost tree containing all
vertices at distance ≤ 2 from 𝑂.
𝑂
Randomized search ratio
For a randomized search 𝑠 and a vertex 𝑖, the
expected search time and expecte normalized
search time are denoted by 𝑇(𝑠, 𝑖) and 𝑇 𝑠, 𝑖 .
The randomized search ratio 𝜌𝑠 of a random
search 𝑠 is max 𝑇(𝑠, 𝑖).
𝑖
The randomized search ratio 𝜌 of a graph is
min 𝜎𝑠 = min max 𝑇(𝑠, 𝑖).
𝑠
𝑠
𝑖
Game theoretic interpretation
Finding the optimal randomized search is equivalent to finding
the optimal strategy in a zero-sum search game between a
Searcher and Hider.
2
1
Hider/Searcher 1,2 2,1
1
1
3
2
3/2 1
𝑂
Optimal randomized search: start with short edge with probability
4/5 and long edge with probability 1/5.
Randomized search ratio, 𝜌 = 7/5.
2-approximate strategy
Proposition: For trees or graphs with unit length edges, the
optimal deterministic strategy is a 2-approximation for the
optimal randomized strategy.
Example
1
𝑛
1
𝑂
1
𝜎 = 𝑛 and 𝜌 ≈ 𝑛/2
Randomization can be very bad
𝜎 ≈ 1 but searching in
a random order has
search ratio ≈ 𝐿/2
𝐿≫1
1
𝑂
Randomized star search
𝑑2
𝑑𝑛
𝑑3
1 = 𝑑1 ≤ 𝑑2 ≤ ⋯ ≤ 𝑑𝑛
𝑑1
𝑂
𝑑1
𝑑2
𝑑3
𝑑𝑛
Idea: randomize in “stages”
Randomize between all edges with length 𝑙
satisfying 2𝑗 ≤ 𝑙 < 2𝑗+1 for 𝑗 = 0,1,2, …
Unfortunately, it doesn’t work…
𝑛
This has search ratio
2𝑛
≈ = 𝑛.
2
2 − 2−
2−
1
𝑂
2
But 𝜌 ≈
𝑛
.
2
Better idea: randomize in
“random stages”
𝑥1
1
𝑂
𝑥2
2
𝑥3
4
𝑥𝑘−1
8
2𝑘−2
𝑥𝑘
2𝑘−1
2𝑘
Better idea: randomize in
“random stages”
𝑥1
1
𝑂
𝑥2
2
𝑥3
4
𝑥𝑘−1
8
2𝑘−2
𝑥𝑘
2𝑘−1
2𝑘
Better idea: randomize in
“random stages”
𝑥1
1
𝑂
𝑥2
2
𝑥3
4
𝑥𝑘−1
8
2𝑘−2
𝑥𝑘
2𝑘−1
2𝑘
Better idea: randomize in
“random stages”
𝑥1
1
𝑂
𝑥2
2
𝑥3
4
𝑥𝑘−1
8
2𝑘−2
𝑥𝑘
2𝑘−1
2𝑘
Better idea: randomize in
“random stages”
𝑥1
1
𝑥2
2
𝑥3
4
𝑥𝑘−1
8
2𝑘−2
𝑥𝑘
2𝑘−1
2𝑘
Theorem: This has an
approximation ratio of
5/4.
𝑂
Idea of proof
Bound 𝜌 from below using a collection of mixed
Hider strategies:
Lemma: If the Hider chooses from the edges 𝑖 =
1,2, … 𝑗 with probability proportional to the square
of the length 𝑑𝑖 of the edge, the expected search
ratio is at least
1
2
1+
2
𝑗
𝑖=1 𝑑𝑖
𝑗
2
𝑑
𝑖
𝑖=1
.
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
1
2
1+
2
𝑛
𝑑
𝑖=1 𝑖
2
𝑛
𝑑
𝑖=1 𝑖
.
Theorem: The graph with 𝑛 edges that has
maximum randomized search ratio is the one with
equal length edges, i.e. 𝜌 ≤ (𝑛 + 1)/2.
Further directions
• Computational complexity of finding
randomized search ratio?
• Continuous version…