1. When Existence is Enough Dan Kalman American University

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

3. Existence Proofs Some collection of problems to solve Example: ax 2 + bx + c = 0 Some collection of problems to solve Example: ax 2 + bx + c = 0 Not all instances are solvable Not all instances are solvable Conditions which assure existence of a solution Example: b 2 – 4ac ≥ 0 Conditions which assure existence of a solution Example: b 2 – 4ac ≥ 0 Not a prescription for finding a solution Not a prescription for finding a solution WHAT GOOD IS THAT? WHAT GOOD IS THAT?

4. 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 communication system Subject area: designing a satellite communication system General Problem: Can a given system design handle a projected data load? General Problem: Can a given system design handle a projected data load? Resource Allocation problem: who gets to talk to which satellite when? 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 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 Result was used in a very practical way

5. 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

6. Geometric Framework x-y-z coordinate system x-y-z coordinate system Earth = sphere centered at (0,0,0) Earth = sphere centered at (0,0,0) Equator in xy plane Equator in xy plane Earth rotates around z axis Earth rotates around z axis Ground Stations travel in horizontal circles around z axis Ground Stations travel in horizontal circles around z 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, we can calculate its position at any time 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 Given latitude and longitude of ground station, we can calculate its position at any time

7. Instantaneous Visibility Geometric Models for visibility Geometric Models for visibility At instant, positions of satellites and stations given by motion models At instant, positions of satellites and stations given by motion models Constraints described in terms of lines, angles, cones Constraints described in terms of lines, angles, cones Line of sight from station to satellite is computed as a vector Line of sight from station to satellite is computed as a vector Vector methods used to compute angles Vector methods used to compute angles Communication possible when satellite can see the station Communication possible when satellite can see the station

11. Discrete Time Step Model Compute positions of all satellites and stations at one fixed time Compute positions of all satellites and stations at one fixed time Determine which satellites can see which stations Determine which satellites can see which stations Advance time by one minute, repeat all calculations Advance time by one minute, repeat all calculations Repeat many many times Repeat many many times For a 24 hour simulation, repeat 1440 times For a 24 hour simulation, repeat 1440 times

12. Design Problem Fixed ground stations Fixed ground stations Predefined connection time requirements Predefined connection time requirements LARGE range of choices for satellite orbits LARGE range of choices for satellite orbits For a given set of orbits, can all connect time requirements be met? For a given set of orbits, can all connect time requirements be met?

13. 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 for each time step A separate vertex for each satellite for each time step Edges indicate visibility Edges indicate visibility Visibility graph Visibility graph

15. Problem Formulation Edge Count = degree Edge Count = degree At station vertex degree = amount of connect time At station vertex degree = amount of connect time Assignment Subgraph: Degree 1 at each satellite/time vertex Assignment Subgraph: Degree 1 at each satellite/time vertex Given: Visibility graph and required degree at each station vertex Given: Visibility graph and required degree at each station vertex To Find: Assignment subgraph that meets all requirements To Find: Assignment subgraph that meets all requirements Existence Question: does a solution exist? Existence Question: does a solution exist?

17. History Gordan's Problem: Is there a finite set of invariants that can be used to generate all the rest? 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 David Hilbert gives existence proof of a solution, 1888 Gordan: ``This is not mathematics. This is theology.'' Gordan: ``This is not mathematics. This is theology.'' Felix Klein: ``wholly simple and, therefore, logically compelling.'' Felix Klein: ``wholly simple and, therefore, logically compelling.'' History sided with Hilbert and Klein 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) 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)

18. Hilbert Gordon Klein

19. Satellites: Necessary Conditions ConReq: connection requirement for a station ConReq: connection requirement for a station ConReq for a station must be  visibility graph degree for the station ConReq for a station must be  visibility graph degree for the station Total of all ConReqs must be  number of satellite-time nodes Total of all ConReqs must be  number of satellite-time nodes These are two extreme cases of a more general constraint 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 For any subset of stations, the sum of ConReqs must be  the number of satellite-time nodes connected to the subset

21. Necessary and Sufficient Chris Reed approach Chris Reed approach With n stations, 2 n - 1 necessary conditions With n stations, 2 n - 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 exist! If all conditions are met.... A solution must exist! ``Bed rest, plenty of fluids, and a good hard proof.'' ``Bed rest, plenty of fluids, and a good hard proof.'' The marriage problem in graph theory The marriage problem in graph theory

22. Existence Result Finding the assignment subgraph is computationally prohibitive Finding the assignment subgraph is computationally prohibitive Checking all the conditions is computationally feasible Checking all the conditions is computationally feasible We can easily compute whether the assignment problem is solvable We can easily compute whether the assignment problem is solvable We cannot find the solution We cannot find the solution This is exactly what is meant by an existence result This is exactly what is meant by an existence result

23. VISREV Legacy Code Legacy Code Compute visibility graph AND right graph statistics to check all the necessary conditions 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 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, 2 10 - 1  1000 conditions to check Computationally intensive: 1440 time steps, 20 satellites, 10 stations, 2 10 - 1  1000 conditions to check Several hours on Cyber 7600 number cruncher Several hours on Cyber 7600 number cruncher Probably a few minutes on a PC today Probably a few minutes on a PC today

24. Minimal Transmission Rate Chris Reed Idea Chris Reed Idea Connection requirements inversly proportional to transmission rate Connection requirements inversly proportional to transmission rate Assignment Problem Unsolvable: try over with increased transmission rate Assignment Problem Unsolvable: try over with increased transmission rate Assignment Problem Solvable: try over with decreased transmission rate Assignment Problem Solvable: try over with decreased transmission rate Find smallest transmission rate that permits assignment problem to be solved Find smallest transmission rate that permits 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 assignment problem. Existence is enough. NOTE: never need to actually find the solution to assignment problem. Existence is enough.

25. Brute Force Computation 109 system designs 109 system designs Computer time charges assessed by the second Computer time charges assessed by the second Submit 15 jobs in the morning and tie up computers all day Submit 15 jobs in the morning and tie up computers all day In one week: $80,000 of computer charges In one week: $80,000 of computer charges Chris Reed: don’t worry about computer charges – it’s “funny money” Chris Reed: don’t worry about computer charges – it’s “funny money”

28. Epilog: Funny Money Near the end of the brute force attack Near the end of the brute force attack Crossing El Segundo Blvd on my way to the LAAFS gym Crossing El Segundo Blvd on my way 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''