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Twice-ramanujan sparsifiers
TLDR
It is proved that every graph has a spectral sparsifier with a number of edges linear in its number of vertices, and an elementary deterministic polynomial time algorithm is given for constructing H, which approximates G spectrally at least as well as a Ramanujan expander with dn/2 edges approximates the complete graph.
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Graph sparsification by effective resistances
TLDR
A key ingredient in the algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure from which the authors can query the approximate effective resistance between any two vertices in a graph in O(log n) time.
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Spectral Sparsification of Graphs
TLDR
It is proved that every graph has a spectral sparsifier of nearly linear size, and an algorithm is presented that produces spectralSparsifiers in time $O(m\log^{c}m)$, where $m$ is the number of edges in the original graph and $c$ is some absolute constant.
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Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems
We present algorithms for solving symmetric, diagonally-dominant linear systems to accuracy ε in time linear in their number of non-zeros and log (κf (A) ε), where κf (A) is the condition number of…
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Efficient erasure correcting codes
TLDR
A simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs is introduced and a simple criterion involving the fractions of nodes of different degrees on both sides of the graph is obtained which is necessary and sufficient for the decoding process to finish successfully with high probability.
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Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time
TLDR
The smoothed analysis of algorithms is introduced, which is a hybrid of the worst-case and average-case analysis of algorithm performance and shows that the shadow-vertex simplex algorithm has polynomial smoothed complexity.
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Practical loss-resilient codes
We present randomized constructions of linear-time encodable and decodable codes that can transmit over lossy channels at rates extremely close to capacity. The encod-ing and decoding algorithms for…
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Expander codes
  • M. Sipser, D. Spielman
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations…
  • 20 November 1994
We present a new class of asymptotically good, linear error-correcting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel…
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Exponential algorithmic speedup by a quantum walk
TLDR
A black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer is constructed and it is proved that no classical algorithm can solve the problem in subexponential time.
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Interlacing families II: Mixed characteristic polynomials and the Kadison{Singer problem
We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the Kadison{Singer problem. The rst is Weaver’s conjecture…
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