# Approaching PNP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time - PowerPoint PPT Presentation

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## Approaching PNP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time

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### How does soap do this? Soap, in water, acts as a surfactant, ... OpenFOAM computation of soap action on vertices. Comparison of exact solution with soap solution ... – PowerPoint PPT presentation

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Title: Approaching PNP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time

1
Approaching PNP Can Soap Bubbles Solve The
Steiner Tree Problem In Polynomial Time?
• Long Ouyang
• Computer systems

2
Introduction
• Decision problems Ask yes/no questions.
• Two classes of problems, P and NP
• P Problems that can be solved in time polynomial
to the size of the input by a deterministic
Turing machine.
• NP Problems that can be solved in time
polynomial to the size of the input by a
nondeterministic Turing machine.

3
Turing machines (not important)
Deterministic -At most one entry for each
combination of symbol and state.
Non-deterministic -More than one entry for each
combination of symbol and state.
4
What does this mean?
• With regards to modern computers
• Problems in P can be solved in polynomial time.
• Solutions to problems in NP can be verified in
polynomial time.
• Problems in P take relatively less time to solve,
problems in NP take relatively more.

5
NP
• Problems in NP
• Traveling salesman problem
• Hamiltonian path problem
• Partition problem
• Multiprocessor scheduling
• Bin packing
• Sudoku
• Tetris

6
Who cares?
• If PNP, hard problems are actually relatively
easy.
• Implications Cryptography, Mapquest,
compression, scheduling, computation

7
How?
• Try to devise P algorithms to NP-Complete
problems.
• Problem Turing arguments, Razborov-Rudich
barrier

8
So what do we do?
• Physical systems often in nature, physical
systems reduce a situation to its lowest energy
state (optimizing energy).
• Soap films
• Spin glasses
• Folding proteins
• Bubbles

9
• Quantum computing
• Using DNA as non-deterministic Turing machines.
• Time travel
• Quantum computing
• Anthropic principles

10
• Pros
• Its inexpensive, compared to time travel.
• Reduces PNP to a problem in digital physics.
• Cons
• Makes formal proof at the least, very difficult
• Optimistically, at best, provides experimental
run-time data

11
The Steiner Problem
• Soap is rumored to solve the Steiner Tree Problem
(STP).

Steiner Tree Problem Description Given a
weighted graph G, G(V,E,w), where V is the set of
vertices, E is the set of edges, and w is the set
of weights, and S, a subset of V, find the subset
of G that contains S and has the minimum
weight. Simply put Find the minimum spanning
tree for a bunch of vertices, given that you can
12
How does soap do this?
• Soap, in water, acts as a surfactant, which
decreases the surface tension of the water.
• This acts to minimize the surface energy of the
liquid.
• This should minimize surface area (graph weight),
and solve the problem.

13
Tools used
• OpenFOAM (computational fluid physics engine)
• Paraview (visualization engine)
• GeoSteiner '96 (exact STP solver)

14
Design
• Generation of random vertices, appropriate mesh
for OpenFOAM
• Solution of STP (where nodes are the random
vertices) by GeoSteiner '96
• OpenFOAM computation of soap action on vertices
• Comparison of exact solution with soap solution

15
Soap model
• Thin box filled with soap water.
• Pegs connect the same parallel faces of the box
(nodes)
• There's a small drain at the bottom of the box.

16
Ideal soap solution
17
Conclusions
• Agent-based modeling sucks for modeling fluids.
• Rigid-body physics sucks for modeling fluids.