When Existence is When Existence is EnoughEnough

Dan KalmanDan Kalman

American UniversityAmerican University

A chemist, a physicist, an engineer A chemist, a physicist, an engineer and a mathematician …and a mathematician …

Existence Proofs Existence Proofs Some collection of problems to solveSome collection of problems to solve

Example: Example: axax2 2 + + bxbx + + cc = 0 = 0 Not all instances are solvable Not all instances are solvable Conditions which assure existence of a Conditions which assure existence of a

solution solution Example: Example: bb22 – 4 – 4acac ≥ ≥ 00

Not a prescription for finding a solution Not a prescription for finding a solution WHAT GOOD IS THAT? WHAT GOOD IS THAT?

A Real Practical Example A Real Practical Example Example is both real and practical Example is both real and practical Worked on in the aerospace industry in LA Worked on in the aerospace industry in LA Subject area: designing a satellite Subject area: designing a satellite

communication system communication system General Problem: Can a given system General Problem: Can a given system

design handle a projected data load? design handle a projected data load? Resource Allocation problem: who gets to Resource Allocation problem: who gets to

talk to which satellite when? talk to which satellite when? Existence result tells if the load can be Existence result tells if the load can be

handled, but does not tell how to allocate handled, but does not tell how to allocate the resources the resources

Result was used in a very practical way Result was used in a very practical way

Computer Model Overview Computer Model Overview Geometric framework Geometric framework Earth and satellite motion Earth and satellite motion Instantaneous visibility Instantaneous visibility Discrete time step model Discrete time step model

Geometric Framework Geometric Framework xx--yy--zz coordinate system coordinate system Earth = sphere centered at (0,0,0) Earth = sphere centered at (0,0,0) Equator in Equator in xyxy plane plane Earth rotates around Earth rotates around zz axis axis Ground Stations travel in horizontal circles Ground Stations travel in horizontal circles

around around zz axis axis Satellites travel in ellipses, one focus at (0,0,0) Satellites travel in ellipses, one focus at (0,0,0) Given initial position and velocity of a satellite, Given initial position and velocity of a satellite,

we can calculate its position at any time we can calculate its position at any time Given latitude and longitude of ground station, Given latitude and longitude of ground station,

we can calculate its position at any time we can calculate its position at any time

Instantaneous Visibility Instantaneous Visibility Geometric Models for visibility Geometric Models for visibility At instant, positions of satellites and At instant, positions of satellites and

stations given by motion models stations given by motion models Constraints described in terms of lines, Constraints described in terms of lines,

angles, cones angles, cones Line of sight from station to satellite is Line of sight from station to satellite is

computed as a vector computed as a vector Vector methods used to compute angles Vector methods used to compute angles Communication possible when satellite Communication possible when satellite

can can seesee the station the station

Discrete Time Step Model Discrete Time Step Model

Compute positions of all satellites Compute positions of all satellites and stations at one fixed time and stations at one fixed time

Determine which satellites can see Determine which satellites can see which stations which stations

Advance time by one minute, repeat Advance time by one minute, repeat all calculations all calculations

Repeat many many times Repeat many many times For a 24 hour simulation, repeat For a 24 hour simulation, repeat

1440 times 1440 times

Design Problem Design Problem

Fixed ground stations Fixed ground stations Predefined connection time Predefined connection time

requirements requirements LARGE range of choices for satellite LARGE range of choices for satellite

orbits orbits For a given set of orbits, can all For a given set of orbits, can all

connect time requirements be met? connect time requirements be met?

Graph Theory Formulation Graph Theory Formulation Bi-Partite Graph: Two sets of vertices Bi-Partite Graph: Two sets of vertices One vertex for each ground station One vertex for each ground station A separate vertex for each satellite A separate vertex for each satellite

for each time step for each time step Edges indicate Edges indicate

visibility visibility Visibility graph Visibility graph

Problem Formulation Problem Formulation Edge Count = Edge Count = degreedegree At station vertexAt station vertex

degree = amountdegree = amountof connect time of connect time

Assignment Subgraph: Assignment Subgraph: Degree 1 at each Degree 1 at each satellite/time vertexsatellite/time vertex

Given: Visibility graph and required degree Given: Visibility graph and required degree at each station vertex at each station vertex

To Find: Assignment subgraph that meets To Find: Assignment subgraph that meets all requirements all requirements

Existence Question: does a solution exist? Existence Question: does a solution exist?

History History Gordan's Problem: Is there a finite set of Gordan's Problem: Is there a finite set of

invariants that can be used to generate all the invariants that can be used to generate all the rest? rest?

David Hilbert gives existence proof of a solution, David Hilbert gives existence proof of a solution, 1888 1888

Gordan: ``This is not mathematics. This is Gordan: ``This is not mathematics. This is theology.'' theology.''

Felix Klein: ``wholly simple and, therefore, Felix Klein: ``wholly simple and, therefore, logically compelling.'' logically compelling.''

History sided with Hilbert and Klein History sided with Hilbert and Klein Gordan best remembered for being wrong about Gordan best remembered for being wrong about

existence proofs. His statement ``has echoed in existence proofs. His statement ``has echoed in mathematics long after his own mathematical mathematics long after his own mathematical work has fallen silent.'' (Constance Reid) work has fallen silent.'' (Constance Reid)

Hilbert

Gordon Klein

Satellites: Necessary Conditions Satellites: Necessary Conditions ConReq: connection requirement for a ConReq: connection requirement for a

station station ConReq for a station must be ConReq for a station must be visibility visibility

graph degree for the station graph degree for the station Total of all ConReqs must be Total of all ConReqs must be number of number of

satellite-time nodes satellite-time nodes These are two extreme cases of a more These are two extreme cases of a more

general constraint general constraint For any subset of stations, the sum of For any subset of stations, the sum of

ConReqs must be ConReqs must be the number of the number of satellite-time nodes connected to the satellite-time nodes connected to the subset subset

Necessary Necessary and and Sufficient Sufficient

Chris Reed approachChris Reed approach With With nn stations, 2 stations, 2nn - 1 necessary conditions - 1 necessary conditions Check them all Check them all If one fails, no solution If one fails, no solution If all conditions are met .... A solution must If all conditions are met .... A solution must

exist! exist! ``Bed rest, plenty of fluids, and a good ``Bed rest, plenty of fluids, and a good

hard proof.'' hard proof.'' The marriage problem in graph theory The marriage problem in graph theory

Existence Result Existence Result

Finding the assignment subgraph is Finding the assignment subgraph is computationally prohibitive computationally prohibitive

Checking all the conditions is Checking all the conditions is computationally feasible computationally feasible

We can easily compute whether the We can easily compute whether the assignment problem is solvable assignment problem is solvable

We cannot find the solution We cannot find the solution This is exactly what is meant by an This is exactly what is meant by an

existence resultexistence result

VISREV VISREV Legacy Code Legacy Code Compute visibility graph AND right graph Compute visibility graph AND right graph

statistics to check all the necessary statistics to check all the necessary conditions conditions

I added logic to do all the tests, and report I added logic to do all the tests, and report on the solvability of the assignment on the solvability of the assignment problem problem

Computationally intensive: 1440 time Computationally intensive: 1440 time steps, 20 satellites, 10 stations, steps, 20 satellites, 10 stations, 221010 - 1 - 1 1000 conditions to check 1000 conditions to check

Several hours on Cyber 7600 number Several hours on Cyber 7600 number cruncher cruncher

Probably a few minutes on a PC today Probably a few minutes on a PC today

Minimal Transmission Rate Minimal Transmission Rate Chris Reed Idea Chris Reed Idea Connection requirements inversly proportional to Connection requirements inversly proportional to

transmission rate transmission rate Assignment Problem Unsolvable: try over with Assignment Problem Unsolvable: try over with

increased transmission rate increased transmission rate Assignment Problem Solvable: try over with Assignment Problem Solvable: try over with

decreased transmission rate decreased transmission rate Find smallest transmission rate that permits Find smallest transmission rate that permits

assignment problem to be solved assignment problem to be solved This provides a comparison between system designs This provides a comparison between system designs NOTE: never need to actually find the solution to NOTE: never need to actually find the solution to

assignment problem. assignment problem. Existence is enough.Existence is enough.

Brute Force Computation Brute Force Computation 109 system designs 109 system designs Computer time charges assessed by Computer time charges assessed by

the second the second Submit 15 jobs in the morning and tie Submit 15 jobs in the morning and tie

up computers all day up computers all day In one week: $80,000 of computer In one week: $80,000 of computer

charges charges Chris Reed: don’t worry about Chris Reed: don’t worry about

computer charges – it’s “funny money” computer charges – it’s “funny money”

Epilog: Funny Money Epilog: Funny Money

Near the end of the brute force Near the end of the brute force attack attack

Crossing El Segundo Blvd on my way Crossing El Segundo Blvd on my way to the LAAFS gym to the LAAFS gym

Met up with Chris Reed Met up with Chris Reed STOP the analysis!!!STOP the analysis!!! ``It's not funny'' ``It's not funny''