A cover tree is a tree data structure used for the partitiong of metric spaces to speed up operations like nearest neighbor, -nearest neighbor, or range searches. In this blog post, I introduce cover trees, their uses, their properties, and I measure the effect of the dimension of the metric space on the run-time in an experiment with synthetic data.

# Tag Archives: programming

# Another Advantage of Garbage Collection

Most literature about garbage collection contains a list of advantages of garbage collection. The lists of advantages known to me omit one advantage: certain concurrent algorithms require garbage collection.

I will demonstrate this point with an example in the programming language C, specifically the revision C11. Consider a singly-linked list where the pointer to the head of the list is concurrently read and written by multiple threads:

struct node { size_t value; struct node* p_next; }; typedef struct node node; _Atomic(node*) p_head = ATOMIC_VAR_INIT(NULL);

The singly-linked list is considered empty if `p_head`

is a null pointer. The thread T1 is only reading the list:

void* T1(void* args) { node* p = atomic_load(&p_head); // more computations // stop referencing the head of the list return NULL; }

The thread T2 removes the first element of the singly-linked list by trying to overwrite the pointer stored in `p_head`

:

void* T2(void* args) { node* p = NULL; node* p_next = NULL; node* p_expected = NULL; do { p = atomic_load(&p_head); if( !p ) break; p_next = p->p_next; p_expected = p; } while(!atomic_compare_exchange_strong(&p_head, &p_expected, p_next)); // ensure other threads stopped referencing p free(p); return NULL; }

T2 relies on compare-and-swap in line 16 to detect interference of other threads.

After successfully updating `p_head`

, the memory referenced by `p`

needs to be freed after all threads stopped referencing this memory and in general, this requires a garbage collector. Waiting does not help because the threads holding references might have been stopped by the operating system. Scanning the stack, the heap, and the other threads' CPU registers is not possible in many languages or not covered by the programming language standard and besides, such a scan is an integral part of any tracing garbage collector.

In the introduction I wrote that certain concurrent algorithms **require** garbage collection and more accurately, it should say: In the absence of special guarantees, certain concurrent algorithms require garbage collection. For example, if we can guarantee that threads hold their references to the singly-linked list only for a certain amount of time, then there is no need for garbage collection and this fact is used in the Linux kernel when using the read-copy-update mechanisms.

# Master's Thesis: Projection Methods for Generalized Eigenvalue Problems

My master's thesis deals with dense and sparse solvers for generalized eigenvalue problems (GEPs) with Hermitian positive semidefinite matrices. Key results are

- structure-preserving backward error bounds computable in linear time,
- the runtime of GSVD-based dense GEP solvers is within factor 5 of the fastest GEP solver with Netlib LAPACK in my tests,
- computing the GSVD directly is up to 20 times slower than the computation by means of QR factorizations and the CS decomposition with Netlib LAPACK in my tests,
- given a pair of matrices with 2x2 block structure, I show how to minimize eigenvalue perturbation by off-diagonal blocks with the aid of graph algorithms, and
- I propose a new multilevel eigensolver for sparse GEPs that is able to compute up to 1000 eigenpairs on a cluster node with two dual-core CPUs and 16 GB virtual memory limit for problems with up to 150,000 degrees of freedom in less than eleven hours.

The revised edition of the thesis with fixed typos is here (PDF), the source code is available here, and the abstract is below. In February, I already gave a talk on the preliminary thesis results; more details can be found in the corresponding blog post.

## Abstract

This thesis treats the numerical solution of generalized eigenvalue problems (GEPs) , where , are Hermitian positive semidefinite (HPSD). We discuss problem and solution properties, accuracy assessment of solutions, aspect of computations in finite precision, the connection to the finite element method (FEM), dense solvers, and projection methods for these GEPs. All results are directly applicable to real-world problems.

We present properties and origins of GEPs with HPSD matrices and briefly mention the FEM as a source of such problems.

With respect to accuracy assessment of solutions, we address quickly computable and structure-preserving backward error bounds and their corresponding condition numbers for GEPs with HPSD matrices. There is an abundance of literature on backward error measures possessing one of these features; the backward error in this thesis provides both.

In Chapter 3, we elaborate on dense solvers for GEPs with HPSD matrices. The standard solver reduces the GEP to a standard eigenvalue problem; it is fast but requires positive definite mass matrices and is only conditionally backward stable. The QZ algorithm for general GEPs is backward stable but it is also much slower and does not preserve any problem properties. We present two new backward stable and structure preserving solvers, one using deflation of infinite eigenvalues, the other one using the generalized singular value decomposition (GSVD). We analyze backward stability and computational complexity. In comparison to the QZ algorithm, both solvers are competitive with the standard solver in our tests. Finally, we propose a new solver combining the speed of deflation with the ability of GSVD-based solvers to handle singular matrix pencils.

Finally, we consider black-box solvers based on projection methods to compute the eigenpairs with the smallest eigenvalues of large, sparse GEPs with Hermitian positive definite matrices (HPD). After reviewing common methods for spectral approximation, we briefly mention ways to improve numerical stability. We discuss the automated multilevel substructuring method (AMLS) before analyzing the impact of off-diagonal blocks in block matrices on eigenvalues. We use the results of this thesis and insights in recent papers to propose a new divide-and-conquer eigensolver and to suggest a change that makes AMLS more robust. We test the divide-and-conquer eigensolver on sparse structural engineering matrices with 10,000 to 150,000 degrees of freedom.

2010 Mathematics Subject Classification. 65F15, 65F50, 65Y04, 65Y20.

**Edit**: Revised master's thesis from April 2016 (PDF)

# Advice: Implementing a Solver using LAPACK

For my master's thesis I implemented multiple solvers for structured generalized eigenvalue problems using LAPACK. In this post, I will briefly discuss a method to simplify the memory management and ways to catch programming errors as early as possible when implementing a solver for linear algebra problems that uses LAPACK. The advice in this post only supplements good programming practices like using version control systems and automated tests.

# Type Coercions and Floating-Point Types

Consider a language where we only allow automatic, implicit type conversions (coercion) among numeric types if every value of the source type can be represented as a value of the target type, i.e., there is no truncation and no round-off. Let us call this kind of coercion *value-preserving coercion*. With such a strict coercion rule, an unsigned integer with four bytes can be coerced into a signed integer with eight bytes but the compiler (interpreter) will not coerce signed integers to unsigned integers. In this post, we highlight a reason why coercions from single-precision to double-precision floating-point types may be undesirable although they are value preserving.

# Matlab: Existing Files and mkdir

In Matlab, a directory called `foo`

can be created by calling `mkdir('foo')`

. If `mkdir`

is successful, then the function returns logical 1; otherwise it returns logical 0. Now consider the case where Matlab is run on Linux and there is already a file called `foo`

in the current directory. Linux does not permit the existence of a file and a directory of the same name in the same folder so the call to `mkdir`

should fail. Instead, `mkdir`

will leave the file intact, return logical 1, and print the misleading warning "Directory already exists". That is, there is no *directory* called `foo`

after this supposedly successful function call.

The peculiarity of this behavior has been reported to Mathworks in September 2015. The example above was tested with Matlab R2015b and R2014a on Linux.

# Implicit Type Conversions Gone Wrong, C++ Edition

Consider the following setup:

#include <cstddef> using namespace std; // ... size_t n = 100; size_t* array = new size_t[n];

The goal is to set every element of the array to the fixed value `x`

.

In C++, an idiomatic way to solve this problem are the functions `std::fill`

(link to documentation) and `std::fill_n`

(link to documentation) from the STL header `algorithm`

. In addition to the fixed value `x`

, `std::fill`

requires a pair of output iterators specifying a half-open range as its arguments whereas the additional argument to `std::fill_n`

is an output iterator pointing to the first element and the *length* of the container. That is, we could assign to the elements of the array above as follows:

#include <cstddef> #include <algorithm> using namespace std; // ... size_t n = 100; size_t* array = new size_t[n]; fill(array, array+n, x); fill_n(array, n, x);

What made me write this blog post is the following line of code:

fill_n(array, array+n, x);

Here, I accidentally used `std::fill_n`

in place of `std::fill`

but the code still compiles because of the automatic, implicit type conversion (type coercion) from `std::size_t*`

to `std::size_t`

. With warnings enabled (`-Wextra -Wall -std=c++11 -pedantic`

) neither g++ 4.8.5 nor clang++ 3.5 warn about this line and yet this piece of code causes a segmentation fault on my computer whenever it is executed.

# Matlab: Unawareness of Partial Path Names may cause Bugs

In a filesystem, there are two kinds of paths: relative paths and absolute paths. Absolute paths always point to the same target while the target of a relative path depends on the current working directory. In Matlab, paths are called "path names" and there is a third kind of path name: the partial path name. It is defined as follows:

A partial path name is the last portion of a full path name for a location on the MATLAB search path.

(The Matlab search path is similar to the environment variable PATH and is used to locate Matlab files.) In consequence of this definition, a path name may be treated by Matlab as a partial path name even if the user is unaware of them.

In this post, I will demonstrate Matlab behavior that is unexpected if you are not aware of the existence of partial path names. I will also explain how this causes a bug in xUnit 3 and xUnit 4.

Continue reading Matlab: Unawareness of Partial Path Names may cause Bugs