When Existence is
Enough
Dan Kalman
American University
A chemist, a physicist, an engineer
and a mathematician …
Existence Proofs
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Some collection of problems to solve
Example: ax2 + bx + c = 0
Not all instances are solvable
Conditions which assure existence of a
solution
Example: b2 – 4ac ≥ 0
Not a prescription for finding a solution
WHAT GOOD IS THAT?
A Real Practical Example
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Example is both real and practical
Worked on in the aerospace industry in LA
Subject area: designing a satellite
communication system
General Problem: Can a given system
design handle a projected data load?
Resource Allocation problem: who gets to
talk to which satellite when?
Existence result tells if the load can be
handled, but does not tell how to allocate
the resources
Result was used in a very practical way
Computer Model Overview
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Geometric framework
Earth and satellite motion
Instantaneous visibility
Discrete time step model
Geometric Framework
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x-y-z coordinate system
Earth = sphere centered at (0,0,0)
Equator in xy plane
Earth rotates around z axis
Ground Stations travel in horizontal circles
around z axis
Satellites travel in ellipses, one focus at (0,0,0)
Given initial position and velocity of a satellite,
we can calculate its position at any time
Given latitude and longitude of ground station,
we can calculate its position at any time
Instantaneous Visibility
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Geometric Models for visibility
At instant, positions of satellites and
stations given by motion models
Constraints described in terms of lines,
angles, cones
Line of sight from station to satellite is
computed as a vector
Vector methods used to compute angles
Communication possible when satellite can
see the station
Discrete Time Step Model
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Compute positions of all satellites
and stations at one fixed time
Determine which satellites can see
which stations
Advance time by one minute, repeat
all calculations
Repeat many many times
For a 24 hour simulation, repeat
1440 times
Design Problem
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Fixed ground stations
Predefined connection time
requirements
LARGE range of choices for satellite
orbits
For a given set of orbits, can all
connect time requirements be met?
Graph Theory Formulation
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Bi-Partite Graph: Two sets of vertices
One vertex for each ground station
A separate vertex for each satellite
for each time step
Edges indicate
visibility
Visibility graph
Problem Formulation
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Edge Count = degree
At station vertex
degree = amount
of connect time
Assignment Subgraph:
Degree 1 at each
satellite/time vertex
Given: Visibility graph and required
degree at each station vertex
To Find: Assignment subgraph that meets
all requirements
Existence Question: does a solution exist?
History
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Gordan's Problem: Is there a finite set of
invariants that can be used to generate all the
rest?
David Hilbert gives existence proof of a solution,
1888
Gordan: ``This is not mathematics. This is
theology.''
Felix Klein: ``wholly simple and, therefore,
logically compelling.''
History sided with Hilbert and Klein
Gordan best remembered for being wrong about
existence proofs. His statement ``has echoed in
mathematics long after his own mathematical
work has fallen silent.'' (Constance Reid)
Gordon
Klein
Hilbert
Satellites: Necessary Conditions
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ConReq: connection requirement for a
station
ConReq for a station must be  visibility
graph degree for the station
Total of all ConReqs must be  number of
satellite-time nodes
These are two extreme cases of a more
general constraint
For any subset of stations, the sum of
ConReqs must be  the number of
satellite-time nodes connected to the
subset
Necessary and Sufficient
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Chris Reed approach
With n stations, 2n - 1 necessary
conditions
Check them all
If one fails, no solution
If all conditions are met .... A solution
must exist!
``Bed rest, plenty of fluids, and a good
hard proof.''
The marriage problem in graph theory
Existence Result
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Finding the assignment subgraph is
computationally prohibitive
Checking all the conditions is
computationally feasible
We can easily compute whether the
assignment problem is solvable
We cannot find the solution
This is exactly what is meant by an
existence result
VISREV
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Legacy Code
Compute visibility graph AND right graph
statistics to check all the necessary
conditions
I added logic to do all the tests, and
report on the solvability of the assignment
problem
Computationally intensive: 1440 time
steps, 20 satellites, 10 stations,
210 - 1  1000 conditions to check
Several hours on Cyber 7600 number
cruncher
Probably a few minutes on a PC today
Minimal Transmission Rate
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Chris Reed Idea
Connection requirements inversly proportional to
transmission rate
Assignment Problem Unsolvable: try over with
increased transmission rate
Assignment Problem Solvable: try over with
decreased transmission rate
Find smallest transmission rate that permits
assignment problem to be solved
This provides a comparison between system designs
NOTE: never need to actually find the solution to
assignment problem. Existence is enough.
Brute Force Computation
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109 system designs
Computer time charges assessed by
the second
Submit 15 jobs in the morning and tie
up computers all day
In one week: $80,000 of computer
charges
Chris Reed: don’t worry about
computer charges – it’s “funny money”
Epilog: Funny Money
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Near the end of the brute force
attack
Crossing El Segundo Blvd on my way
to the LAAFS gym
Met up with Chris Reed
STOP the analysis!!!
``It's not funny''