OUTLINE Regression: why and how Spectra: Linear system solvers Graphs: tree embeddings
PROBLEM Given: matrix A, vector b Size of A : n-by-n m non-zeros
SPECIAL STRUCTURE OF A A = Deg – Adj Deg : diag(degree) Adj : adjacency matrix [Gremban-Miller `96]: extensions to SDD matrices ` A ij =deg(i) if i=j w(ij) otherwise
UNSTRUCTURED GRAPHS Social network Intermediate systems of other algorithms are almost adversarial
NEARLY LINEAR TIME SOLVERS [SPIELMAN-TENG ‘04] Input : n by n graph Laplacian A with m non-zeros, vector b Where : b = A x for some x Output : Approximate solution x’ s.t. |x-x’| A <ε|x| A Runtime : Nearly Linear. O(m log c n log(1/ε)) expected runtime is cost per bit of accuracy. Error in the A -norm: |y| A =√y T A y.
HOW MANY LOGS Runtime : O(mlog c n log(1/ ε)) Value of c: I don’t know [Spielman]: c≤70 [Koutis]: c≤15 [Miller]: c≤32 [Teng]: c≤12 [Orecchia]: c≤6 When n = 10 6, log 6 n > 10 6
PRACTICAL NEARLY LINEAR TIME SOLVERS [KOUTIS-MILLER-P `10] Input : n by n graph Laplacian A with m non-zeros, vector b Where : b = A x for some x Output : Approximate solution x’ s.t. |x-x’| A <ε|x| A Runtime : O(mlog 2 n log(1/ ε)) runtime is cost per bit of accuracy. Error in the A -norm: |y| A =√y T A y.
PRACTICAL NEARLY LINEAR TIME SOLVERS [KOUTIS-MILLER-P `11] Input : n by n graph Laplacian A with m non-zeros, vector b Where : b = A x for some x Output : Approximate solution x’ s.t. |x-x’| A <ε|x| A Runtime : O(mlogn log(1/ ε)) runtime is cost per bit of accuracy. Error in the A -norm: |y| A =√y T A y.
STAGES OF THE SOLVER Iterative Methods Spectral Sparsifiers Low Stretch Spanning Trees
ITERATIVE METHODS Numerical analysis: Can solve systems in A by iteratively solving spectrally similar, but easier, B
WHAT IS SPECTRALLY SIMILAR? A ≺ B ≺ k A for some small k Ideas from scalars hold! A ≺ B : for any vector x, |x| A 2 < |x| B 2 [Vaidya `91] : Since A is a graph, B should be too! [Vaidya `91] : Since G is a graph, H should be too!
`EASIER’ H Goal: H with fewer edges that’s similar to G Ways of easier: Fewer vertices Fewer edges Can reduce vertex count if edge count is small
GRAPH SPARSIFIERS Sparse equivalents of graphs that preserve something Spanners: distance, diameter. Cut sparsifier: all cuts. What we need: spectrum
WHAT WE NEED: ULTRASPARSIFIERS [Spielman-Teng `04] : ultrasparsifiers with n- 1+O(mlog p n/k) edges imply solvers with O(mlog p n) running time. Given: G with n vertices, m edges parameter k Output: H with n vertices, n-1+O(mlog p n/k) edges Goal: G ≺ H ≺ kG ``
SPECTRAL SPARSIFICATION BY EFFECTIVE RESISTANCE [Spielman-Srivastava `08] : Setting P(e) to W(e)R(u,v)O(logn) gives G ≺ H ≺ 2G* *Ignoring probabilistic issues [Foster `49] : ∑ e W(e)R(e) = n-1 Spectral sparsifier with O(nlogn) edges Ultrasparsifier? Solver???
THE CHICKEN AND EGG PROBLEM How to find effective resistance? [Spielman-Srivastava `08] : use solver [Spielman-Teng `04] : need sparsifier
OUR WORK AROUND Use upper bounds of effective resistance, R’(u,v) Modify the problem
RAYLEIGH’S MONOTONICITY LAW Rayleigh’s Monotonicity Law: R(u, v) only increase when edges are removed ` Calculate effective resistance w.r.t. a tree T
SAMPLING PROBABILITIES ACCORDING TO TREE Sample Probability: edge weight times effective resistance of tree path ` Goal: small total stretch stretch
GOOD TREES EXIST Every graph has a spanning tree with total stretch O(mlogn) O(mlog 2 n) edges, too many! ∑ e W(e)R’(e) = O(mlogn) Hiding loglogn
‘GOOD’ TREE??? Unit weight case: stretch ≥ 1 for all edges ` Stretch = 1+1 = 2
WHAT ARE WE MISSING? Need: G ≺ H ≺ k G n-1+O(mlog p n/k) edges Generated: G ≺ H ≺ 2 G n-1+O(mlog 2 n) edges `` Haven’t used k!
USE K, SOMEHOW Tree is good! Increase weights of tree edges by factor of k ` G ≺ G’ ≺ k G
RESULT Tree heavier by factor of k Tree effective resistance decrease by factor of k ` Stretch = 1/k+1/k = 2/k
NOW SAMPLE? Expected in H: Tree edges: n-1 Off tree edges: O(mlog 2 n/k) ` Total: n- 1+O(mlog 2 n/k)
BUT WE CHANGED G! G ≺ G’ ≺ k G G’ ≺ H ≺ 2 G’ ` G ≺ H ≺ 2k G
WHAT WE NEED: ULTRASPARSIFIERS [Spielman-Teng `04] : ultrasparsifiers with n-1+O(mlog p n/k) edges imply solvers with O(mlog p n) running time. Given: G with n vertices, m edges parameter k Output: H with n vertices, n-1+O(mlog p n/k) edges Goal: G ≺ H ≺ kG `` G ≺ H ≺ 2k G n-1+O(mlog 2 n/k) edges
Input: Graph Laplacian G Compute low stretch tree T of G T ( log 2 n) T H G + T H Sample T (H) Solve G by iterating on H and solving recursively, but reuse T PSEUDOCODE OF O(MLOGN) SOLVER
EXTENSIONS / GENERALIZATIONS [Koutis-Levin-P `12] : sparsify mildly dense graphs in O(m) time [Miller-P `12] : general matrices: find ‘simpler’ matrix that’s similar in O(m+n 2.38+a ) time. ``
SUMMARY OF SOLVERS Spectral graph theory allows one to find similar, easier to solve graphs Backbone: good trees ``
SOLVERS USING GRAPH THEORY Fast solvers for graph Laplacians use combinatorial graph theory
OUTLINE Regression: why and how Spectra: linear system solvers Graphs: tree embeddings
LOW STRETCH SPANNING TREE Sampling probability: edge weight times effective resistance of tree path Unit weight case: length of tree path Low stretch spanning tree: small total stretch
DIFFERENT THAN USUAL TREES n 1/2 -by-n 1/2 unit weighted mesh stretch(e)= O(1)total stretch = Ω(n 3/2 )stretch(e)=Ω(n 1/2 ) ‘haircomb’ is both shortest path and max weight spanning tree
LOW STRETCH SPANNING TREES [Elkin-Emek-Spielman-Teng `05], [Abraham-Bartal-Neiman `08]: Any graph has a spanning tree with total stretch O(mlogn) Hiding loglogn
ISSUE: RUNNING TIME Algorithms given by [Elkin-Emek-Spielman-Teng `05], [Abraham-Bartal-Neiman `08] take O(nlog 2 n+mlogn) time Reason: O(logn) shortest paths
SPEED UP [Koutis-Miller-P `11] : Round edge weights to powers of 2 k=logn, total work = O(mlogn) [Orlin-Madduri-Subramani-Williamson `10]: Shortest path on graphs with k distinct weights can run in O(mlog m/n k) time Hiding loglogn, we actually improve these
[Blelloch-Gupta-Koutis-Miller-P- Tangwongsan. `11] : current framework parallelizes to O(m 1/3+a ) depth Combine with Laplacian paradigm fast parallel graph algorithms `` PARALLEL ALGORITHM?
Before this work: parallel time > state of the art sequential time Our result: parallel work close to sequential, and O(m 2/3 ) time PARALLEL GRAPH ALGORITHMS?
FUNDAMENTAL PROBLEM Long standing open problem: theoretical speedups for BFS / shortest path in directed graphs Sequential algorithms are too fast!
First step of framework by [Elkin-Emek-Spielman-Teng `05] : `` PARALLEL ALGORITHM? shortest path
Workaround: use earlier algorithm by [Alon-Karp-Peleg-West `95] Idea: repeated clustering Based on ideas from [Cohen `93, `00] for approximating shortest path PARALLEL TREE EMBEDDING