The Spectral Relation between the Cube-Connected Cycles and the Shuffle-Exchange Network
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1 The Spectral Relation between the Cube-Connected Cycles and the Shuffle-Exchange Network Christian Riess 1 Rolf Wanka 2 Volker Strehl 3 February 29, Pattern Recognition Lab (CS 5) 2 Hardware-Software Co-Design (CS 12), 3 Artificial Intelligence (CS 8)
2 Spectra of Networks Spectral set ( ) contains information on Network throughput Fault-tolerance Known spectra: Linear array Cycle Hypercube Butterfly De Bruijn PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 2
3 Spectra of Networks Spectral set ( Network throughput Fault-tolerance Known spectra: Linear array Cycle Hypercube Butterfly De Bruijn ) contains information on PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 3
4 Parallel Computations on the Hypercube -dimensional Hypercube: a popular architecture for parallel computations Versatile connection structure for many algorithms Finite differences Spanning trees Connected components Sorting networks Nearest neighbor search Chinese remaindering Drawback: degree in every node PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 4
5 Hypercubic: Cube-Connected Cycles Network -dimensional Cube-Connected Cycles : nodes Starting from a hypercube, every node is replaced by a cycle Runs hypercube algorithms with constant slowdown Constant degree PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 5
6 Hypercubic: Shuffle-Exchange Network -dimensional Shuffle-Exchange Network : nodes Connections according to bit pattern: Cyclic left- or right shifts Flip (exchange) of the last bit Runs hypercube algorithms with constant slowdown Constant degree PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 6
7 Hypercubic: Shuffle-Exchange Network -dimensional Shuffle-Exchange Network : nodes Connections according to bit pattern: Cyclic left- or right shifts Flip (exchange) of the last bit Runs hypercube algorithms with constant slowdown Constant degree PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 7
8 Hypercubic: Shuffle-Exchange Network -dimensional Shuffle-Exchange Network : nodes Connections according to bit pattern: Cyclic left- or right shifts Flip (exchange) of the last bit Runs hypercube algorithms with constant slowdown Constant degree PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 8
9 Hypercubic: Shuffle-Exchange Network -dimensional Shuffle-Exchange Network : nodes Connections according to bit pattern: Cyclic left- or right shifts Flip (exchange) of the last bit Runs hypercube algorithms with constant slowdown Constant degree PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 9
10 Mathematical Tools for Network Characterization Network = Graph (represented by its adjacency matrix) Spectral graph theory: Examine relationship between the network and its Eigenvalues/Eigenvectors A small example: Adjacency: Eigenvalues: Average degree: PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 10
11 Bounds from Graph Spectra Isoperimetric number Expansion property Routing number Chromatic number Independence number Bisection width PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 11
12 Bounds from Graph Spectra Isoperimetric number Expansion property Routing number Chromatic number Independence number Bisection width PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 12
13 Bounds from Graph Spectra Isoperimetric number Expansion property Routing number Message destinations in Chromatic number Independence number Bisection width PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 13
14 Bounds from Graph Spectra Isoperimetric number Expansion property Routing number Message destinations in Chromatic number Independence number Bisection width PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 14
15 Bounds from Graph Spectra Isoperimetric number Expansion property Routing number Message destinations in Number nodes Chromatic number Independence number Bisection width 2 nd largest Eigenvalue PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 15
16 Spectral Relation of CCC and SE Decomposing CCC Decomposing SE Graph transform Spectrum is identical Similarity transform Spectrum is identical Relation of CCC and SE Matching of subgraphs If odd: If even: PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 16
17 Decomposing CCC Spectrum-preserving graph editing PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 17
18 Decomposing CCC Spectrum-preserving graph editing PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 18
19 Decomposing CCC Spectrum-preserving graph editing PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 19
20 Decomposing CCC Spectrum-preserving graph editing PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 20
21 Decomposing CCC Spectrum-preserving graph editing PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 21
22 Decomposing CCC Spectrum-preserving graph editing PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 22
23 Decomposing CCC Spectrum-preserving graph editing disconnected subgraphs PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 23
24 Decomposing CCC Spectrum-preserving graph editing cospectral disconnected subgraphs PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 24
25 Decomposing SE Removal of exchange edges PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 25
26 Decomposing SE Removal of exchange edges PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 26
27 Decomposing SE Removal of exchange edges Similarity transform with PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 27
28 Decomposing SE Removal of exchange edges Similarity transform with PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 28
29 Decomposing SE Removal of exchange edges Similarity transform with PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 29
30 Decomposing SE Removal of exchange edges Similarity transform with PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 30
31 Decomposing SE Removal of exchange edges cospectral Similarity transform with PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 31
32 Relating the Circles We characterize the mappings i.e. Cospectral graph to Cospectral graph to PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 32
33 Relating the Circles We characterize the mappings i.e. Cospectral graph to Cospectral graph to PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 33
34 Relating the Circles We characterize the mappings i.e. Cospectral graph to Cospectral graph to PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 34
35 Relating the Circles We characterize the mappings i.e. Cospectral graph to Cospectral graph to PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 35
36 Result If odd: If even: Note that is not the only difference. We illustrate that in a second. PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 36
37 Result If odd: There is always a complete eigenvalue mapping If even: Note that is not the only difference. We illustrate that in a second. PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 37
38 Result If odd: There is always a complete eigenvalue mapping If even: There is never a complete eigenvalue mapping Note that is not the only difference. We illustrate that in a second. PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 38
39 Relating the Circles Let Denote e.g. as (by the diagonal entries) Aperiodic case: a cycle appears in both graphs Sometimes, this does not work! compresses periodic cycles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 39
40 Relating the Circles Let Denote e.g. as (by the diagonal entries) Aperiodic case: a cycle appears in both graphs Circle in Sometimes, this does not work! compresses periodic cycles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 40
41 Relating the Circles Let Denote e.g. as (by the diagonal entries) Aperiodic case: a cycle appears in both graphs Circle in Circle in Sometimes, this does not work! compresses periodic cycles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 41
42 Relating the Circles Let Denote e.g. as (by the diagonal entries) Aperiodic case: a cycle appears in both graphs Circle in Eigenvalues (numerically) Circle in Sometimes, this does not work! compresses periodic cycles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 42
43 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 43
44 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 44
45 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 45
46 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 46
47 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 47
48 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 48
49 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 49
50 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 50
51 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 51
52 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 52
53 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 53
54 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 54
55 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 55
56 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 56
57 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 57
58 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 58
59 Non-trivial correspondences between EVs and Circles PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 59
60 Conclusion Graph spectra characterize parallel networks d-dimensional Cube-Connected Cycles (CCC) and Shuffle Exchange (SE) Network offer similar properties PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 60
61 Conclusion Graph spectra characterize parallel networks d-dimensional Cube-Connected Cycles (CCC) and Shuffle Exchange (SE) Network offer similar properties Spectral relation: Spectral sets are equal when odd Spectral sets differ when even several eigenvalues are accidentally equal PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 61
62 Conclusion Graph spectra characterize parallel networks d-dimensional Cube-Connected Cycles (CCC) and Shuffle Exchange (SE) Network offer similar properties Spectral relation: Spectral sets are equal when odd Spectral sets differ when even several eigenvalues are accidentally equal PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 62
63 Conclusion Graph spectra characterize parallel networks d-dimensional Cube-Connected Cycles (CCC) and Shuffle Exchange (SE) Network offer similar properties Spectral relation: Spectral sets are equal when odd Spectral sets differ when even several eigenvalues are accidentally equal Future work: pin down the accidental PASA 2012 Riess et al. CS 5, 8, 12 Spectral Relation Cube Connected Cycles and Shuffle-Exchange Network 63
64 Thank you for your attention.
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