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PPT – Approaching PNP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time PowerPoint presentation | free to view - id: dc2d2-NjJhZ

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Approaching PNP Can Soap Bubbles Solve The

Steiner Tree Problem In Polynomial Time?

- Long Ouyang
- Computer systems

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.

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.

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.

NP

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

Who cares?

- If PNP, hard problems are actually relatively

easy. - Implications Cryptography, Mapquest,

compression, scheduling, computation

How?

- Try to devise P algorithms to NP-Complete

problems. - Problem Turing arguments, Razborov-Rudich

barrier

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

Additional methods

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

Well take the soap, please

- 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

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

add additional points.

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.

Tools used

- OpenFOAM (computational fluid physics engine)
- Paraview (visualization engine)
- GeoSteiner '96 (exact STP solver)

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

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.

Ideal soap solution

Conclusions

- Agent-based modeling sucks for modeling fluids.
- Rigid-body physics sucks for modeling fluids.