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

10
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

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
add additional points.
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.
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