ETH Zurich – Distributed Computing – www.disco.ethz.ch Klaus-Tycho Förster, Rijad Nuridini, Jara Uitto, Roger Wattenhofer Lower Bounds for the Capture.

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Presentation transcript:

ETH Zurich – Distributed Computing – Klaus-Tycho Förster, Rijad Nuridini, Jara Uitto, Roger Wattenhofer Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond

How to catch a robber on a graph? The game of Cops and Robbers

The rules of the game

The Cop is placed first C

The Robber may then choose a placement CR

Next, they alternate in moves CR

CR

CR

CR

The Cop won! C

C

How many moves does the cop need?

Catch multiple? RRRC

Upper bound to catch ℓ robbers

Lower bound to catch ℓ robbers … …

C … …

CR R R … …

CR R R … …

CR R R … …

R CR R … …

R CR R … …

CR R R … …

CR R R … …

R CR R … …

R C R … …

R C … …

R C … …

CR R … …

Summary so far

What about multiple cops and one robber?

Let’s start with two cops and one robber …

R C C … one cop not enough

Let’s start with two cops and one robber C R C …

Beyond two cops? … ??

Multiple cops and one robber …

Summary so far

What about multiple cops and multiple robbers?

Are we done? Multiple cops and multiple robbers … … … …

… … … …

… … … … C Prevents robbers from crossing 

C ……

C ……

C

C

C

Construction of the ring … … …… …

Robber placement Robbers choose side with less cops R R R

Robber strategy R R R

R R R

R R R C Can catch half of the robbers!

Robber strategy R R R C

R R R C

R R R C

R R R C

Summary

ETH Zurich – Distributed Computing – Klaus-Tycho Förster, Rijad Nuridini, Jara Uitto, Roger Wattenhofer Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond