When a given square sparse matrix can be permuted into a lower triangular form?

For some, the question in the title of this post could look trivial. After all, they have been doing LU decomposition in Matlab and using Matlab’s mldivide (or “backslash”) to solve systems with the factor matrices. These are usually done without worrying if the pivoting took place and permuted those factor matrices (Matlab used to call them psychologically triangular matrices. For “why?” and “who said that?”, see here). Because, the help page of mldivide says that it tests if a matrix “is square, is not diagonal, looks triangular, is actually triangular, there are zeros in the diagonal, is a permuted triangular”. So this must be taken care of…

Davis’s “Direct Methods for Sparse Linear Systems” book (a real gem) asks in its Exercise 3.7:

…Consider a matrix $\mathbf{A}$ that may be a permuted upper or lower triangular matrix with both rows and columns permuted by unknown permutations $\mathbf{P}$ and $\mathbf{Q}$. Write an algorithm that determines if the matrix is in this form and, if so, solves $\mathbf{A}x = b$….

Then the book hints an algorithm, which builds the permutation matrices essentially by picking a row with a single nonzero, ordering that row and the associated column, and then by deleting both from the matrix and so on. But this is ok, only if we can recover a full diagonal! What if, at some point, we have rows or columns with no entries at all? The “algorithm” does not handle this case.

It is perhaps ok in the context of LU decomposition, in which case one is warned of about the numerical rank deficiency during LU decomposition (before one attempts to solve the linear systems). But in general it is not ok; at Mathworks, folks are aware of this (see here). If the matrix is not coming from LU decomposition, then we may well be in trouble. We are back to the question: When a given square sparse matrix can be permuted into a lower triangular form?

Some other cases are also easy. If there is a perfect matching in the given matrix, then all standard matching algorithms find it quickly, and after permuting the matching entries to the diagonal, we can check if the matrix is reducible into $1\times 1$ blocks (or the directed graph of $\mathbf{B}=\mathbf{AQ}$ is acyclic, where $\mathbf{Q}$ permutes the columns of $\mathbf{A}$ to have a zero-free diagonal in $\mathbf{B}$). Every step of this is linear time—even computing the matching with the standard tricks). So this case is handled. The question is complicated when we do not have a perfect matching. It is in fact NP-complete to decide if a given square sparse matrix can be permuted into a lower/upper triangular form [1] with unsymmetric permutations.

So writing it once more to emphasize it:

It is easily decidable to test if $\mathbf{A}$ can be symmetrically permuted to a lower/upper triangular form. The directed graph of $\mathbf{A}$ should be acyclic (or $\mathbf{PAP}^T$ is lower/upper triangular).

It is NP-complete to decide if a given square sparse matrix can be permuted to a lower/upper triangular form. Otherwise said, one can find two permutation matrices $\mathbf{P}$ and $\mathbf{Q}$ such that $\mathbf{PAQ}$ is lower/upper triangular only with non-polynomial time algorithms, assuming P$\neq$ NP.

The topic was discussed elsewhere before.

References

1. Guillaume Fertin, Irena Rusu, and Stéphane Vialette:
Obtaining a triangular matrix by independent row-column permutations. ISAAC 2015: 165–175 (doi)

On the degree-1 and degree-2 reduction rules

Karp–Sipser (KS) [5] heuristic is a well-known method to obtain high quality matchings in undirected graphs, bipartite graphs, and hypergraphs.  This heuristic is based on two reduction rules. The first rule says that if there is a vertex with degree one, then we can match that vertex with its unique neighbor and remove both from the graph without any loss.  The second rule says that if there is a vertex $v$ with degree two, then its neighbors can be coalesced, and the vertex $v$ can be discarded. A maximum matching on the reduced graph can be extended to a maximum matching on the original graph. If there is no degree-1 or degree-2 vertices, then a randomly selected edge is used to match two vertices, upon which both of them are removed.

The (full) heuristic with the degree-2 reduction rule turns out to be difficult to analyse. That is why most studies are conserved with the variant with the degree-1 reduction rule. This variant has excellent practical performance [6,7], good theoretical guarantees for certain classes of random graphs [2,5], and can be made to have an approximation ratio around 0.866 [3] for bipartite graphs and a close-by ratio on general graphs [4]. There are graphs for which KS with only the degree-1 reduction can obtain worse results. The following is an example and a table from [3].

The bipartite graph instances giving hard time to KS with the degree-1 rule correspond to the following matrices (rows form one part, and the columns form the other part) . The rows and columns are split into two sets in the middle, yielding a $2\times 2$ structure. The $(1,1)$-block is full, the $(2,2)$-block is empty, and there is an identity matrix at the other two blocks. In order to hide the perfect matching composed of those two identity matrices, additional nonzeros are added to the last $t$ rows in the first row block, and the last $t$ columns in the first column block so that those rows and columns become dense. For $n=3200$, and with different thickness $t\in \{2,4,8,16, 32\}$ in the full rows, KS with only the degree-1 rule obtains the performance shown in the table.

$t$ Performance
2 0.782
4 0.704
8 0.707
16 0.685
32 0.670

We want to now turn attention to the full KS (that is with the degree-1 and degree-2 reduction rules). Anastos and Frieze [1] state in the abstract of a recent arXiv paper:

In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser (FOCS 1981) proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results in the orignal paper suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least for random cubic graphs. We show that w.h.p. the first algorithm will find a matching of size $n/2-O(\log n)$ on a random cubic graph (indeed on a random graph with degrees in $\{3, 4\}$). We also show that the algorithm can be adapted to find a perfect matching w.h.p. in $O(n)$ time, as opposed to $O(n^{3/2})$ time for the worst-case.

This wonderful result shows improvement with the degree-2 reductions (over using only degree-1 reductions): instead of leaving out $O(n^{1/5})$ vertices as KS with the degree-1 rule only, it leaves out only $\log n$ vertices. We can also create instances for which KS with both reduction rules obtains perfect matchings, whereas that with the degree-1 reduction obtains inferior results. These instances are upper (or lower) Hessenberg matrices, or more sparser versions of it (for example if we discard all but the first and the last subdiagonal nonzeros). A figure is shown below with only two subdiagonal nonzeros. On these instances, an immediate application of the degree-2 reduction triggers a chain of degree-1 reductions, until we have a $2\times 2$ matrix, whose perfect matching is obtained by a degree-2 and then a degree-1 reduction. On the other hand, KS with only the degree-1 rule obtains the performance shown in the table for different $n$ (the performance reported in the table is average of 10 different random runs on the same instance).

$n$ Performance
500 0.828
1000 0.790
5000 0.754

There is a catch though. While there is a linear time, straightforward implementation of the degree-1 rule, the straightforward implementation of the degree-2 rule can be quadratic in the worst case; consider repetitive merges with the same vertex which will happen with an Hessenberg matrix. We can perhaps assume that the worst cases are rare in practice, but nonetheless a worst-case quadratic run time complexity is a lot!

References

1. Michael Anastos and Alan Frieze, Finding perfect matchings in random cubic graphs in linear time, arXiv preprint arXiv:1808.00825, Nov., 2018. (url)
2. Jonathan Aronson, Alan Frieze, and Boris G. Pittel, Maximum matchings in sparse random graphs: Karp-Sipser revisited, Random Structures & Algorithms, 12 (1998), pp. 111-177.
3. Fanny Dufossé, Kamer Kaya, and Bora Uçar, Two approximation algorithms for bipartite matching on multicore architectures, Journal of Parallel and Distributed Computing, 85 (2015), pp. 62–78. (doi)
4. Fanny Dufossé, Kamer Kaya, Ioannis Panagiotas, and Bora Uçar, Approximation algorithms for maximum matchings in undirected graphs, Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing, Bergen, Norway, 2018, pp. 56–65. (doi)
5. Richard M. Karp and Michael Sipser, Maximum matching in sparse random graphs, 22nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1981, pp. 364–375. (doi)
6. Johannes Langguth, Fredrik Manne, and Peter Sanders, Heuristic initialization for bipartite matching problems, Journal of Experimental Algorithmics, 15 (2010), pp. 1.1–1.22. (doi)
7. Jakob Magun, Greedy matching algorithms, an experimental study, Journal of Experimental Algorithmics, 3 (1998). (doi)

9 November 2018

Today was a great day at the school. There were two remarkable events. The first one is the 30th anniversary of the lab LIP at ENS Lyon (here is the program). The second was that the ENS Lyon awarded Marc Snir with an honorary doctorate degree (Docteur Honoris Causa). The page announcing this event is in French and google translate does a good job in translating it to English. Yves Robert had slides for introducing Marc during the event.

Many of us know Marc (ever heard of the MPI standard and the book “MPI: The complete reference“). His work span complexity theory, to MPI standard, to parallel computing systems. And oh, he speaks French.

I was lucky to see his talk during the 30th anniversary of the LIP (but unlucky to miss the ceremony of Docteur Honoris Causa). He gave an overview of his involvement with building parallel machines: BlueGene, Blue Waters, SP/Vulcan, and others. His talk has many whimsical observations. Here are some:

• A supercomputer research prototype is an oxymoron.
• A supercomputer research design is either boring or unpractical.
• The main contribution [of all the supercomputer design projects]: The projects educated a generation of researchers.
• Theory informs practice, but should not be taken literally.

After stating

Often theory follows practice, rather than practice following theory

he discussed how his paper with Upfal and Felperin was motivated by Vulcan’s practically well behaving design of $\log N + O(1)$ stages. Back then, the theory demonstrated $2\log N$ stages to avoid worst cases. The cited paper shows $\log N + \log\log N$, where the extra term is $O(1)$ for practical purposes.

After the talk, I wondered which computer he helped to build was his favorite. He said, more or less,

I created them, so all are my favorite !

References

Eli Upfal, Sergio Felperin, and Marc Snir, Randomized routing with shorter paths, IEEE Transactions on Parallel and Distributed Systems, 7(4), pp. 356–362, 1996. (doi)

After CSC18

It has been long time we last blogged, due to Bora’s other professional engagements. There has been many things in between, including the 8th SIAM Workshop on CSC (CSC18), June 6–8, 2018, Bergen, Norway. The best paper of CSC18 by Kevin Deweese and John R. Gilbert, entitled Evolving Difficult Graphs for Laplacian Solvers is our subject.

Kevin is currently a PhD student at the University of California,  Santa Barbara, working on provably fast Laplacian solvers. See his web page for a few of his papers with experimental evaluation (most of the similar solvers are hard to implement).

Here is the abstract of the subject paper (Link to paper) by Kevin and John:

We seek to discover Laplacian linear systems that stress the ability of existing Laplacian solver packages to solve them efficiently. We employ a genetic algorithm to explore the problem space of graphs of fixed size and edge density. The goal is to measure the gap between theoretical and existing Laplacian solvers, by trying to find worst case example graphs for existing solvers. These problems may have little use inside any real world application, but they give great insight into solver behavior. We report performance results of our genetic algorithm, and explore the properties of the evolved graphs.

Kevin and John focus on the combinatorial solver by Kelner, Orecchia, Sidford, and Zhu (arXiv link) known as KOSZ on the provably fast solver side, and PCG with a Jacobi preconditioner on the traditional side. The genetic algorithms by Kevin and John create populations of graphs by starting with an initial population. Then, a Laplacian solver is run on all graphs and those which required the most work to solve are selected as parents. A random vertex is swapped between the selected parents to yield new individuals. Random edge mutations of the form edge removal and replacement are performed. The techniques are versatile: they are used for creating hard instances for KOSZ and PCG; they are also used to create instances in which the performance of KOSZ with respect to that of PCG varies. On a reported instance, KOSZ outperforms PCG by a factor of 2, and on another one PCG outperforms KOSZ by a factor of 140! In all experiments, the performance is measured in terms of the number of arithmetic operations. Future work includes combining different instances to yield larger problems which stress both solvers to understand our abilities (beware solvers!).

Recent news on the minimum fill-in problem

We have just seen a a paper in the Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2017. The proceedings is available online for the curious. A paper from the proceedings is the subject of this post.

Minimum fill-in: Inapproximability and almost tight lower bounds,
by, Yixin Cao and R. B. Sandeep (url).

The minimum fill-in problem is one of the core problems in sparse direct methods. This problem is NP-complete [6]. The NP-completeness of the problem was first conjectured to be true in 1976 by Rose, Tarjan, and Lueker [5] in terms of the elimination process on undirected graphs. Then Rose and Tarjan [4] proved in 1978 that finding an elimination ordering on a directed graph that gives minimum fill-in is NP-complete (there was apparently a glitch in the proof which was rectified by Gilbert [1] two years later). Finally, Yannakakis [6] proved the NP-completeness of the minimum fill-in problem on undirected graphs in 1981.

The sparse matrix community needs a heuristic to tackle the minimum fill-in problem, as direct methods are used in many applications. There are quite efficient and effective heuristics for this purpose, which are variants of the approximate minimum degree and incomplete nested dissection algorithms. These heuristics are efficiently implemented in programs that are wide spread. The mentioned heuristics do not have any performance guarantees (except for classes of graphs such as graphs with good separators) but they are very-well established. So there is an NP-complete problem, practitioners have some heuristics and are happy with what they have.

On the other hand, the minimum fill-in problem poses many challenges to the theoreticians. Natanzon, Shamir, and Sharan [2] obtained an algorithm that approximates the minimum fill-in ($k$) by bounding the fill by $8k^2$ in $O(k n^3)$ time, for a graph with $n$ vertices. The paper by Cao and Sandeep shows that unless P=NP, there is no polynomial time approximation scheme (PTAS) for the minimum fill-in problem. A PTAS is an algorithm which takes an instance of an optimization problem and a parameter $\epsilon > 0$ and, in polynomial time, produces a solution that is within a factor $1 + \epsilon$ of the optimum. This is a bad news in a way: no simple heuristic with a performance guarantee for theoretically oriented practitioners is in view.

The paper by Cao and Sandeep contains two other theorems, ruling out algorithms with approximation $1+\epsilon$ and with a running time $2^{O(n^{1-\delta})}$, and exact algorithms with running time in $2^{O(k^{1/2-\delta})}\cdot n^{O(1)}$, assuming exponential time hypothesis (ETH). Here $n$ is again the number of vertices, $k$ is an integer parameterizing the fill-in, and $\delta$ is a small positive constant. ETH posits that the satisfiability problem with at most three variables per clause cannot be solved in $2^{o(p+q)}$ time, where $p$ and $q$ denote the number of clauses and variables, respectively, in the problem.

If you are interested in these problems, then see the tree-width and minimum fill-in challenges. The minimum tree-width is related to the shortest elimination tree over all elimination orderings of an undirected graph, which one of us had proved to be NP-complete [3].

References

1. John R. Gilbert, A note on the NP-completeness of vertex elimination on directed graphs, SIAM Journal on Algebraic and Discrete Methods, 1 (1980), pp. 292–294 (link).
2. Assaf Natanzon, Ron Shamir, and Roded Sharan, A polynomial approximation for the minimum fill-in problem, SIAM Journal on Computing, 30, (2000), pp. 1067–1079 (link).
3. Alex Pothen, The complexity of optimal elimination trees,
Tech. Report, Department of Computer Science, Penn State University, 1988 (link).
4. Donald J. Rose and Robert E. Tarjan, Algorithmic aspects of vertex elimination in directed graphs, SIAM Journal on Applied Mathematics, 34 (1978), pp. 176–197 (link).
5. Donald J. Rose, Robert E. Tarjan, and George S. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM Journal on Computing, 5 (1976), pp. 266–283 (link).
6. Mihalis Yannakakis, Computing the minimum fill-in is NP-complete, SIAM Journal on Algebraic and Discrete Methods, 2 (1981), pp. 77–79 (link).

After CSC’16 (cont’): The best paper

Now that the Proceedings of the CSC Workshop 2016 has appeared (link), we can look at the papers. In this post, we are going to look at the best paper by Manne, Naim, Lerring, and Halappanavar, “On stable marriages and greedy matchings” (doi).

As noted in the citation of the award, this paper brings together  recent half-approximation algorithms  for  weighted matchings  and the classical  Gale-Shapley and McVitie-Wilson algorithms for the Stable Marriage problem. This is helpful in two ways.

First, the authors show that a  recent half-approximation algorithm for computing greedy matchings, the Suitor algorithm and its variants,  is equivalent to  the classical algorithms for the Stable Marriage problem. This correspondence enables the authors to propose multi-core and many-core parallel algorithms for the Stable Marriage problem, based on the greedy matching algorithms. This way, if I am not mistaken, they develop the first GPU algorithm for the Stable Marriage problem.

Second, the extensive theoretical work on the Stable Marriage algorithms  explains the  behavior of the Suitor matching algorithm. The worst-case number of proposals in the Suitor algorithm can be $O(n^2)$,  where n is the number of vertices  in the bipartite graph. However, if the weights are assigned randomly, the expected number of proposals is $O(n \log n)$, which follows from a classical analysis of Donald Knuth [1, 2].

This work represents a nice marriage between theory and practice: the practical algorithms for the greedy matching help in designing parallel algorithms for the Stable Marriage problem, and the theoretical understanding of the Stable Marriage problem sheds light into the behavior of the greedy weighted matching algorithms.

Vocabulary

Stable marriage problem: This is described on a full bipartite graph $G=(L\cup R, L\times R)$.  Each vertex on the left has a total ranking of the vertices on the right; similarly each vertex on the right has a total ranking of all vertices on the left. The aim is to find a matching $M$ such that no $l\in L$ and $r\in R$  would obtain a higher ranked partner if they were to abandon their current partners in $M$ and rematch with each other.

Greedy matching algorithm: Here we compute approximations to matchings of maximum  weight in weighted graphs. The Greedy algorithm  considers edges in a non-increasing order of weights and the heaviest remaining edge $(u,v)$ is added to the matching, whereupon all edges incident on $u$ and $v$ are removed. The Greedy matching is a half-approximate matching.

Suitor algorithm: This is a proposal based algorithm [3]. Vertices can propose in any order, however, each vertex proposes to a neighbor with the  heaviest weight  that already does not have a better weight offer to match with it.  A vertex could annul the proposal received by a neighbor if it has a better weight edge to offer the neighbor.  When two vertices propose to each other, they are matched. The Suitor algorithm computes the same matching as the one obtained by the Greedy algorithm and the Locally Dominant edge algorithm described by Robert Preis [4]!

References

1. Vicki Knoblauch, Marriage matching: A conjecture of Donald Knuth, Economics Working Papers. Paper 200715, http://digitalcommons.uconn.edu/econ_wpapers/200715, 2007.
2. Donald E. Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, Vol. 10 American Mathematical Society, (1997).
3. Fredrik Manne and Mahantesh Halappanavar, New effective multithreaded matching algorithms, in Proc. IPDPS 2014, IEEE 28th International Parallel and Distributed Processing Symposium, Phoenix, AZ, USA, pp. 519–528, 2014.
4. Robert Preis, Linear time $1/2$-approximation algorithm for maximum weighted matching in general graphs, in Proc. STACS 99, 16th Annual Symposium on Theoretical Aspects of Computer Science Trier, Germany, pp. 259–269, 1999.

After CSC16 (cont’)

Umit V. Çatalyürek presented our work on the directed acyclic graph (DAG) partitioning problem. In this problem, we are given a DAG $G=(V,E)$ and an integer $k\geq 2$. The aim is to partition the vertex set $V$ into $k$ parts $V_1,\ldots, V_k$ in such a way that the parts have (almost) equal weight and the sum of the costs of all those arcs having their endpoint in different parts minimized. Vertices can have weights, and the edges can have costs. Up to now, all is standard. What is not standard is that the quotient graph of the parts should be acyclic. In other words, the directed graph $G'=(V', E')$, where $V'=\{V_1,\ldots,V_k\}$ and $V_i\rightarrow V_j\in E'$ iff $v_i\rightarrow v_j\in E$ for some $v_i\in V_i$ and $v_j\in V_j$, should be acyclic.

John R. Gilbert wanted to understand the complexity of the problem, with respect to the undirected version. He is an expert on the subject matter (see, e.g., [2]). He asked what happens if we orient the edges of the $n\times n$ model problem. If you are not familiar with this jargon, it is the $n\times n$ mesh with each node being connected to its immediate neighbor in the four main directions, if those neighbors exist. See the small image for an $8\times 8$ example.

Partitioning these kind of meshes is a very hard problem. Gary Miller had mentioned their optimal partitioning in his invited talk (about something else). Rob Bisseling [Ch. 4, 1] has a great text about partitioning these meshes and their cousins in 3D. I had surveyed known methods in a paper with Anaël Grandjean [3]. In particular, Anaël found about discrete isoperimetric problems, showed that the shape of an optimal partition at a corner, or inside the mesh was discussed in the related literature. He also showed that the Cartesian partitioning is optimal for edge cut.  Anaël also proposed efficient heuristics which produced connected components. See the cited papers for nice images. Our were based on our earlier work with Umit [4].

Anyhow,  let’s return back to acyclic partitioning of DAGs, and John’s question. He suggested that we should look at the electrical spiral heater to get an orientation. This orientation results in a total order of the vertices. The figures below show the ordering of the $8\times 8$ and the $16\times 16$ meshes. Only some of the edges are shown; all edges of the mesh, including those that are not shown are from the lower numbered vertex to the higher numbered one.

As seen in the figures above, the spiral ordering is a total order and there is only one way to cut the meshes into two parts with the same number of vertices; blue and red show the two parts.

Theorem: Consider the $n\times n$ mesh whose edges are oriented following the electrical spiral heater ordering. The unique acyclic cut with $n^2/2$ vertices in each side has $n^2-4n+3$ edges in cut, for $n\geq 8$.

The theorem can be proved by observing that the blue vertices in the border (excluding the corners) has one arc going to a red vertex; those in the interior, except the one labeled $n^2/2$ has 2 such arcs;  the vertex labeled $n^2/2$ has three such arcs. The condition $n\geq 8$ comes from the fact that we assumed that there are blue vertices in the interior of the mesh. This is a lot of edges to cut!

Later, John said that he visited the Maxwell Museum of Anthropology at UNM after the CSC16 workshop, and saw that similar designs by the original native New Mexicans.

References

1. Rob H. Bisseling, Parallel Scientific Computation: A Structured Approach using BSP and MPI, 1st ed, Oxford University Press, 2004.
2. John R. Gilbert, Some nested dissection order is nearly optimal. Inf. Process. Lett. 26(6): 325–328 (1988).
3. Anaël Grandjean and Bora Uçar, On partitioning two dimensional finite difference meshes for distributed memory parallel computers. PDP 2014: 9–16.
4. Bora Uçar and Umit V. Çatalyürek, On the scalability of hypergraph models for sparse matrix partitioning. PDP 2014: 593–600.
5. Da-Lun Wang and Ping Wang, Discrete isoperimetric problems, SIAM Journal on Applied Mathematics, 32(4):860–870 (1977).

After CSC16

CSC16 was held last week. Kudos to Assefaw and Erik as the chair of the workshop.

There are so much to talk about. We will have a series of posts about the workshop and related things. Here are some bits and pieces.

The workshop had three invited talks, 19 contributed talks, and eight posters, and attended by 60+ people. There will be a proceedings with 11 papers. The proceedings will be published by SIAM and will be hosted at its publication platform.

We had also celebrated the 60th birthdays of Alex Pothen and Rob Bisseling.

There was a best paper award. It went to Fredrik Manne, Md. Naim, Håkon Lerring, and Mahantesh Halappanavar for their paper titled On Stable Marriages and Greedy Matching. Congratulations. The citation by the best paper award committee (Uwe Naumann, Alex Pothen, and Sivan Toledo) reads as:

for the way the paper brings together several decades of work on stable marriages with the more recent work on approximation algorithms for weighted matchings, and the consequences for the average case complexity of the latter algorithms.

A heads up: the CSC18 meeting will likely be in Bergen, Norway. Erik cracked a joke about this in saying that the best paper awardees should take on organizing the next meeting.

CSC Mini-symposium at SIAM PP16

SIAM Conference on Parallel Processing for Scientific Computing (PP16) took place in Paris, April 12–15, 2016. This was the first SIAM PP outside the US.

As announced before, we had a mini-symposium on CSC in three sessions. SIAM keeps records of the program, the speakers and the abstracts (here). As the organizers, we thought that we should take this one step ahead and make the pdf’s of the talks available as well, wherever possible.

Here is the list of talks, in the order of speaker line-up.

1. Computational surgery: Visualization with augmented matrices (Alex Pothen)

Authors: Alex Pothen (Purdue University, USA) and Mu Wang (Purdue University, USA)

Abstract. Not yet

Talk: No files yet.

Comments: Alex had a video showcasing the use of the solver, in real-time, updating the mesh and showing the result of the surgery. The video was prepared with the help of professionals. The audience was all captive and silent during the video!

2. Parallel combinatorial algorithms in sparse matrix computation? (Esmond Ng)

Authors: Mathias Jacquelin (Lawrence Berkeley National Laboratory, USA), Esmond Ng (Lawrence Berkeley National Laboratory, USA), Barry Peyton (Dalton State College, USA), Yili Zheng (Lawrence Berkeley National Laboratory, USA), and Kathy Yelick, (Lawrence Berkeley National Laboratory, USA).

Abstract. Combinatorial techniques are used in several phases of sparse matrix computation. For large-scale problems, while numerical phases are often executed in parallel, most of these combinatorial techniques are serial and can become bottlenecks. We are investigating the extent to which some of the combinatorial techniques can be performed in parallel.

Talk: No files yet.

Comments: RCM is discussed as the showcase. I think Aydin and Ariful were also involved (second slide of the talk had this information). Given the group’s experience in distributed memory BFS, it is of surprise that the RCM is implemented based on this. The target was not small-world graphs, neither social network graphs; graphs with large diameters were at the focus. So the parallelization problem is somehow tough. Sorting (by the vertex degrees) is required for a formal RCM (I could not catch the details of which sorting algorithm was used). This step incurred cost and was detrimental to performance. Maybe, in an application one can skip sorting and obtain a variant of RCM (after all, RCM is a heuristic). Also, Esmond pointed that the motivation for this work is that the matrix/graph could be already distributed in another context. Instead of collecting the global data to a central processor, solving it there, and distributing the result back to everyone, one could possibly solve the problem in parallel. In RCM,  pseudo-peripheral nodes are used traditionally. They are again found by BFS. There are recent work for finding the diameters in graphs with a few rounds of BFS. Maybe review this.

3. Parallel graph matching algorithms using matrix algebra (Ariful Azad)

Authors: Ariful Azad (Lawrence Berkeley National Laboratory, USA) and Aydin Buluç (Lawrence Berkeley National Laboratory, USA).

Abstract. We present distributed-memory parallel algorithms for computing matchings in bipartite graphs. We consider both exact and approximate algorithms for cardinality and weighted matching problems. We substitute the asynchronous data access patterns of traditional matching algorithms by a small subset of more structured, bulk-synchronous functions based on matrix algebra. Relying on communication-avoiding algorithms for the underlying matrix-algebric modules, different matching algorithms achieve good speedups on tens of thousands of cores on current supercomputers.

4. On the Birkhoff–von Neumann decomposition (Bora Uçar)

Authors: Michele Benzi (Emory University, USA), Fanny Dufossé (Inria Lille-Nord Europe, France),  Kamer Kaya (Sabanci University, Turkey), Alex Pothen (Purdue University, USA), and Bora Uçar (CNRS and ENS Lyon, France).

Abstract. Not yet

Talk: (boraUcar-pp16) file.

5. A Partitioning problem for load balancing and reducing communication from the field of quantum chemistry (Edmond Chow)

Authors: Edmond Chow (Georgia Institute of Technology, USA)

Abstract. We present a combinatorial problem and potential solutions arising in parallel computational chemistry. The Hartree-Fock (HF) method has a very complex data access pattern. Much research has been devoted over 20 years for parallelizing this important method, based primarily on intuition and experience. A formal approach for parallelizing HF while reducing communication may come from graph and hypergraph partitioning. Besides providing a potential solution, this approach may also shed light on the optimality of existing approaches.

Talk: (edmondChow-pp16) file.

6. Community detection on GPU (Fredrik Manne)

Authors: Md Naim (University of Bergen, Norway) and Fredrik Manne (University of Bergen, Norway,)

Abstract. There has been considerable interest in community detection for finding the modularity structure in real world data. Such data sets can arise from social networks as well as various scientific domains. The Louvain method is one popular method for this problem as it is simple and fast. It can also be used to detect hierarchical structures in the data. However, its inherently sequential nature and cache unfriendly workloads makes it difficult to parallelize. This is particularly true for co-processor architectures. In this work we show how these obstacles can be overcome and present results from implementing the algorithm on a GPU.

Talk: (fredrikManne-pp16) file.

Comments: Md Naim could not attend to the conference (dommage), and Fredrik replaced him.

7. Scalable parallel algorithms for de novo assembly of complex genomes (Evangelos Georganas)

Authors: Evangelos Georganas (University of California, Berkeley, USA)

Abstract. A critical problem for computational genomics is the problem of de novo genome assembly: the development of robust scalable methods for transforming short randomly sampled sequences into the contiguous and accurate reconstruction of complex genomes. While advanced methods exist for assembling the small and haploid genomes of prokaryotes, the genomes of eukaryotes are more complex. We address this challenge head on by developing HipMer, an end-to-end high performance de novo assembler designed to scale to massive concurrencies. HipMer employs an efficient Unified Parallel C (UPC) implementation and computes the assembly of the human genome in only 8.4 minutes using 15,360 cores of a Cray XC30 system.

Talk: (evangelosGeorganas-pp16) file.

8. Faster and more scalable sparse matrix-matrix multiplication (Aydin Buluç)

Authors: Aydin Buluç (Lawrence Berkeley National Laboratory, USA)

Abstract. We present a faster and more scalable implementation of the sparse matrix-matrix multiplication (SpGEMM) kernel. The implementation exploits multiple levels of parallelism, using a scalable three-dimensional algorithm for inter-node parallelism and multithreaded subroutines for intra-node parallelism. The three-dimensional formalism has characteristics that are special for the sparse case, which we thoroughly explain. We then provide results on applications in Markov graph clustering and Algebraic Multigrid based graph coarsening.

Talk: (aydinBuluc-pp16) file.

9. Directed graph partitioning (Umit V. Catalyurek)

Authors: Julien Herrmann (The Ohio State University, USA), Umit V. Catalyurek (The Ohio State University, USA), Kamer Kaya (Sabanci University, Turkey), and Bora Uçar (CNRS and ENS Lyon, France).

Abstract. In scientific computing directed graphs are commonly used for modeling dependencies among entities. However, while modeling some of the problems as graph partitioning problems, directionality is generally ignored. Accurate modeling of some of the problems necessitates to take the directionally into account, which adds additional constraints that cannot be easily addressed in the current state-of-the-art partitioning methods and tools. In this talk, we will discuss some example problems, models and potential solution approaches for them.

Talk: (pdf) file.

10. Parallel approximation algorithms for b-Edge Covers and data privacy (Arif Khan)

Authors: Arif Khan (Purdue University, USA), and Alex Pothen (Purdue University, USA).

Abstract. We propose a new 3/2-approximation algorithm, called LSE for computing $b$-Edge Cover and its application to a data privacy problem called adaptive $latex k$-Anonymity. $b$-Edge Cover is a special case of the well-known  Set Multicover problem and also a generalization of Edge Cover problem in graphs. The objective is to choose a subset of $latex C$ edges in the graph with weights on the edges, such that at least a specified number $latex b(v)$ of edges in $latex C$ are incident on each vertex $latex v$ and the sum of edge weights is minimized. We implement the algorithm on serial and shared-memory parallel processors and compare its performance against a collection of inherently sequential approximation algorithms that have been proposed for the Set Multicover problem. With LSE, i) we propose the first shared-memory parallel algorithm for the adaptive $latex k$-Anonymity problem and ii) give new theoretical results regarding privacy guarantees which are significantly stronger than the best known previous results.

Talk: (arifKhan-pp16) file.

11. Clustering sparse matrices with information from both numerical values and pattern (Daniel Ruiz)

Authors: Iain S. Duff (Science & Technology Facilities Council, United Kingdom and CERFACS, Toulouse, France), Philip Knight (University of Strathclyde, United Kingdom), Sandrine Mouysset (Université de Toulouse, France), Daniel Ruiz (ENSEEIHT, France), and Bora Uçar (CNRS and ENS Lyon, France).

Abstract. Considering any square fully indecomposable matrix $A$, we can apply a two-sided diagonal scaling to $latex |A|$ to render it into doubly stochastic form. The Perron-Frobenius theorem is a key tool to exploit and we aim to use spectral properties of doubly stochastic matrices to reveal hidden block structure in matrices. We also combine this with classical graph analysis techniques to design partitioning algorithms for large sparse matrices based on both numerical values and pattern information.

Talk: (danielRuiz-pp16) file.

12. Parallel graph coloring on manycore architectures (Mehmet Deveci)

Authors: Mehmet Deveci (Sandia National Laboratories, USA), Erik Boman (Sandia National Laboratories, USA), and Siva Rajamanickam (Sandia National Laboratories, USA).

Abstract. In scientific computing, the problem of finding sets of independent tasks is usually addressed with graph coloring. We study performance portable graph coloring algorithms for many-core architectures. We propose a novel edge-based algorithm and enhancements of the speculative Gebremedhin-Manne algorithm that exploit architectures. We show superior quality and execution time of the proposed algorithms on GPUs and Xeon Phi compared to previous work. We present effects of coloring on applications such as Gauss-Seidel preconditioned solvers.

Talk: (mehmetDeveci-pp16) file.

On HPC Days in Lyon

Last week (6–8 April, 2016) we had an incredible meeting called HPC Days in Lyon. This three day event featured only invited long talks and invited mini-symposia talks. The meeting was organized with generous support of Labex MILYON.

Rob Bisseling and Alex Pothen contributed to a mini-symposium on combinatorial scientific computing.

Rob talked about hypergraph partitioning and how to use it with an iterative solver. We usually get this question: how many mat-vecs (or iterations in a solver) one needs to perform to offset the use of hypergraph partitioning. Rob’s main point in this talk was that one can estimate the number of iterations and spend some more time partitioning the hypergraph, if the number of iterations allow it. He has an ongoing project of optimally bisecting sparse matrices (see the link); his talk included updates (theoretical and practical) to this project. He says he adds a matrix a day to this page. As of now, there are 263 matrices. Chapeau! as the French say.

Also, he said (well maybe slipped out after a few glasses of Côtes du Rhône) that the new edition of his book (Parallel Scientific Computation: A Structured Approach using BSP and MPI) will be coming out.  There are new materials; in particular a few sections on sorting algorithms and a complete chapter on graph algorithms (mainly matching). Stay tuned! Rob will be at SIAM PP next week. I will try to get more  information about his book.

[I have just realized that I did not put Alex’s photo anywhere yet. So let’s have his face too.]

Alex discussed approximation algorithms to matching and $b$-matching problems. He took up the challenge of designing parallel algorithms for matching problems, where the concurrency is usually limited. He discussed approximation algorithms with provable guarantees and great parallel performance for the $b$-matching and a related edge cover problem. He also discussed an application of these algorithms in a data privacy problem he has been working on.

Alex arrived earlier to Lyon and we did some work. With Alex, we always end up discussing matching problems. This was not exception. We looked at the foundations of bottleneck matching algorithms. Alex and I will be attending SIAM PP16 next week. If you know/like these algorithms, please attend CSC mini-symposia so that we can talk.

The talk was for 90 minutes, without break! His talk was very engaging and illuminating. I enjoyed very much and appreciated how he communicates deep math to innocent (or ignorant;)) computer scientists. His two books Iterative methods for sparse linear systems (2nd edition) and Numerical Methods for Large Eigenvalue Problems (2nd Edition) are available at his web page and attest this.
Here is a crash course on Krylov subspace methods from his talk.

Let $x_0$ be an initial guess and $r_0=b-Ax_0$ be the initial residual.
Define $K_m=\textrm{span}\{r_0, Ar_0,\ldots,A^{m-1}r_0\}$ and $L_m$ another subspace of dimension $m$.
Basic Krylov step is then:
$x_m=x_0 + \delta$ where $\delta\in K_m$ and $b-Ax_m \perp L_m$.

At this point, the reader/listener gets the principle and starts wondering what are the choices of $L_m$ that make sense? How do I keep all $m$ vectors? How do I get something orthogonal to them? Yousef had another slide:

1. $L_m=K_m$; class of Galerkin or orthogonal projection methods (e.g., CG), where $\|x^*-\tilde{x}\|=\min_{z\in K_m}\|x^*-z\|_{A}.$
2. $L_m=AK_m$; class of minimal residual methods (e.g., ORTHOMIN, GMRES) where $\|b-A\tilde{x}\|_2=\min_{z\in K_m}\|b-Az\|_2$.

So we learned the alternatives for $L_m$, and we probably guessed correctly that we do not need to keep all $m$ vectors in all cases (e.g., CG), sometimes need all (e.g., GMRES without restart), and even if we need we can short-cut and restart. Getting orthogonal vectors could be tougher, especially if we do not store all the $m$ vectors. Now that we have a guide, a feeling, and a few questions we can turn to resources to study.