# Introduction

An Octree is a recursive, axis-aligned, spatial partitioning data structure commonly used in computer graphics to optimize collision detection, nearest neighbor search, frustum culling and more. Conceptually, an Octree is a simple data structure. However, when digging deeper into the literature one will find many interesting, not very well-known techniques for optimizing and extending Octrees. This is why I decided to write a series of blog posts about Octree techniques not widely covered on the Internet. This series will consist of five posts covering the following topics:

1. Preliminaries, insertion strategies and maximum tree depth
2. Different node representations for memory footprint reduction
3. Non-static Octrees to support moving objects and inserting objects not falling into the root cell
4. Loose Octrees for optimizing insertion and culling hotspots
5. Accessing a node’s neighbors

I’ll publish the posts of this series one by one during the next few weeks, starting today with the first one on preliminaries, insertion strategies and an upper bound for the maximum Octree depth. Thanks for reading!

# Preliminaries

An Octree hierarchically subdivides a finite 3D volume into eight disjoint octants. In the following, octants are also called nodes in the context of tree data structures and cells in the context of space. An Octree’s root cell encloses the entire world. Sometimes, Octrees are introduced as subdividing space into cubes instead of arbitrarily sized boxes. Generally, arbitrarily sized boxes work equally well and there’s no need to adjust the root cell’s size to have the shape of a cube. However, cube-shaped cells slightly speed-up cell subdivision computations and the cell size can be stored as just one float per node instead of three. The general structure of an Octree is illustrated in the figure below.

An Octree is constructed by recursively subdividing space into eight cells until the remaining number of objects in each cell is below a pre-defined threshold, or a maximum tree depth is reached. Every cell is subdivided by three axis-aligned planes, which are usually placed in the middle of the parent node. Thus, each node can have up to eight children. The possibility not to allocate certain child nodes allows, in contrast to regular grids, to store sparsely populated worlds in Octrees.

# Insertion strategies

Points are dimensionless and thereby have no spatial extent. Thus, if points are stored in an Octree, they can be unambiguously assigned to exactly one node. However, if objects with spatial extent like polygons are stored in an Octree, their midpoint can fall into a cell while the object it-self straddles the cell boundary. In that case there are basically three options.

1. The object in question is split along the boundaries of the straddling cells and each part is inserted into its corresponding cell. This approach has two disadvantages. First, the splitting operation might imply costly computations. Second, the data structures are more complicated, because the split-off objects need to be stored somewhere.

2. The object in question is added to each cell it straddles. This option is disadvantageous when the Octree needs to be updated, because it can contain duplicate references to the same object. Furthermore, the culling performance gets worse, because the same object might be found in more than one visible cell. Additionally, special care must be taken when subdividing cells. If after subdividing a cell all objects straddle the very same newly created sub-cell(s), all objects will be inserted again into the same sub-cell(s) causing yet another subdivision step. This results in an infinite loop, only stopped when the maximum tree depth is reached.

3. The object in question is stored in the smallest Octree cell it’s completely enclosed by. This option results in many unnecessary tests, because objects are stored in inner nodes instead of leaf nodes in case they straddle any child cell further down the tree.

In some of the following posts of this series I’ll come back to the different insertion strategies and show in which situation which of the strategies is advantageous. Especially, Loose Octrees are a particularly nice way of overcoming most of the downsides discussed above.

# Maximum tree depth

Let’s assume an Octree contains $M$ points. As described in the previous section, each of the points can only fall exactly into one node. Is it possible to establish a relation between the number of points $M$ and the maximum Octree depth $D_\text{max}$? It turns out, that the number of Octree nodes (=> the Octree depth) is not limited by the number of points. The reason is, that if the points are distributed closely enough in multiple widespread clusters, the number of Octree nodes can grow arbitrarily large. Look at the following figure for an illustration.

In order to split any of the two point clusters, the Octree must be subdivided a few times first. As the points inside the clusters can be arbitrarily close and the clusters can be arbitrarily far away from each other, the number of subdivision steps, and therefore the number of Octree nodes is not limited by the size of the point set. That shows that generally, Octrees cannot be balanced as we are used to from traditional tree data structures.

Nevertheless, we can come up with another upper bound for the maximum tree depth. Let’s assume a cubic Octree for simplicity. Given the minimum distance $d_\text{min}$ between any two points in the point set and the side length of the root cell $s$, it can be shown that the maximum Octree depth is limited by $\log\frac{s}{d_\text{min}}+\log\sqrt{3}\geq D_\text{max}$. The following proof for this upper bound is rather simple.
The maximum distance between any two points in a cell at depth $k$ is given by $\sqrt{3(s/2^k)^2}=\sqrt{3}\frac{s}{2^k}$. Any inner node encloses at least two points, otherwise it would be a leaf node. Hence, the maximum distance between any two points in this cell is guaranteed to be bigger than the minimum distance $d_\text{min}$ between any two points of the point set. Therefore, it holds that $d_\text{min}\leq\sqrt{3}\frac{s}{2^k}\Leftrightarrow k\leq\log\sqrt{3}+\log\frac{s}{d_\text{min}}$.

That’s it for today. Stay tuned for the next article!

# Slides for “Beyond the Limits – The Usage of C++ in the Demoscene”

At the C++ User Group Berlin meeting this May, Eivind Liland and I gave a talk about the usage of C++ in the demoscene. Below is the abstract of the talk. The slides can be found in the downloads section.

Eivind and David are demosceners since decades. In this talk they’re going to show you two of their award-winning, real-time graphics demos, both highly optimized for different limitations and platforms, and both written in C++.

Turtles all the Way Down by Brain Control (2013) is a 64k-intro for the PC. It’s an almost 5 minutes long audio-visual journey using cutting edge algorithms in the areas of computer graphics, generative art and music synthesis. Being a 64k-intro, all textures, 3D objects and music fit into a single executable of merely 65.536 bytes.
Matt Current by Shitfaced Clowns (2007) is a demo for the Gameboy Advance. It features at that time never-seen-before graphics effects and a software-rendered 3d engine that pushes the device’s hardware to their limits. One prevailing opinion is that only by coding in 100% assembly one can push such platforms beyond their limits. Eivind will explain how they used C++ to carefully squeeze the maximum out of every cycle of the GBA’s 16 MHz CPU.

Though seemingly esoteric, all the techniques employed to realize these demos have their application in professional software development nowadays. In times of GHz multi-core processors, GPUs and terabyte hard-drives, performance critical code and compact code for embedded and mobile platforms still plays an important role. Eivind and David are going to guide you through the process of creating these graphics demos. They talk about the used algorithms and tools keeping the focus of how C++ was used to do the job.

# Introduction

Direct memory access (DMA) is conceptually easy, but without experience in hardware design or driver development it can be cumbersome to understand. In this blog post I will explain what DMA is and how it evolved during the last decades. My goal is to make it comprehensible especially for people without experience in hardware design or driver development.

DMA allows computer devices of certain hardware sub-systems to directly access system memory and other device’s memory independently of the CPU. This enables the CPU to keep on working concurrently on other task while long lasting memory operations take place; considerably boosting overall system performance. DMA is used by different hardware like graphics cards, sound cards, network cards and disk drive controllers. DMA is rather a concept than a specific technology. There is no specification which describes in detail how DMA transfers work. Even on the contrary, the concept of directly accessing memory without CPU interaction is employed in many different hardware sub-systems in today’s computers. The most typical application is communicating with peripheral devices plugged into a bus system like ATA, SATA, PCI or PCI Express. Beyond that, DMA transfers are used for intra-core communication in micro processors and even to copy data from the memory of one computer into the memory of another computer over the network via remote DMA (don’t mix up this technology with NVIDIA’s new GPUDirect RDMA feature).

To give a concrete example, imagine you’re playing an open world computer game which loads new game assets on demand from your hard disk. Large amounts of game data must be copied over from hard disk into system RAM. Without DMA the CPU would be actively involved in each and every memory transfer operation. Consequently, less computing time would be left for other game play related tasks like AI or physics. In times of multi-core processors this seems less like a problem. However, as data volumes and work load sizes are ever growing, off-loading large memory transfer operations from the CPU is also today absolutely essential in order to achieve high system performance.

# How DMA evolved over time

In my experience many software people think that DMA nowadays still works as it did in the old days. I guess this is because it’s the more intuitive way to think about DMA. Back then, extension devices did not actively take part in DMA transfers, but there was a DMA controller (e.g. the Intel 8237, first used in the IBM PC in 1981) which enabled DMA transfers between system memory and device I/O over the good old Industrial Standard Architecture (ISA) bus. The DMA controller could be programmed by the CPU to perform a number of memory transfers on behalf of the CPU. This way of accomplishing DMA transfers is also known as third party DMA. At that time the system bus was identical to the ISA expansion bus. To account for reduced bus performance in situations where CPU and DMA controller needed to access the bus simultaneously, different DMA modes (cycle stealing, transparent and burst) could be used. When the first IBM AT clones came out, the expansion bus got physically separated from the system bus using an ISA bridge. This was necessary because the AT clones had CPUs running at higher frequencies than the expansion bus. In the figure below the single bus and the separated bus architectures are depicted.

With the introduction of the conventional Peripheral Component Interface (PCI) bus architecture in 1992, the DMA controller became obsolete because of a technique called bus mastering, or first party DMA. PCI DMA transfers were implemented by allowing only one device at a time to access the bus. This device is called the bus master. While the bus master holds the bus it can perform memory transfers without CPU interaction. The fundamental difference between bus mastering and the use of a DMA controller is that DMA compatible devices must contain a DMA engine driving the memory transfers. As multiple PCI devices can master the bus, an arbitration scheme is required to avoid that more than one device drives the bus simultaneously. The advantage of bus mastering is a significant latency reduction because communication with the third party DMA controller is avoided. Additionally, each device’s DMA engine can be specifically optimized for the sort of DMA transfers it performs.

Today’s computers don’t contain DMA controllers anymore. If they do so, it’s only to support legacy buses like e.g. ISA, often by simulating an ISA interface using a Low Pin Count (LPC) bus bridge. In 2004 the PCI successor and latest peripheral computer bus system PCI Express (PCIe) was introduced. PCIe turned the conventional PCI bus from a true bus architecture, with several devices physically sharing the same bus, into a serial, packet-switched, point-to-point architecture; very similar to how packet-switched networks function. PCIe connects each device with a dedicated, bi-directional link to a PCIe switch. As a result, PCIe supports full duplex DMA transfers of multiple devices at the same time. All arbitration logic is replaced by the packet routing logic implemented in the PCIe switches. While PCIe is entirely different to PCI on the hardware level, PCIe preserves backwards compatibility with PCI on the driver level. Newer PCIe devices can be detected and used by PCI drivers without explicit support for the PCIe standard. Though, the new PCIe features cannot be used of course.

# DMA from a driver developer’s perspective

Now you know what DMA is and how it fits into a computer’s hardware architecture. So let’s see how DMA can be used in practice to speed up data heavy tasks. Since the dawn of DMA the driver (software) must prepare any peripheral DMA transfers, because only the operating system (OS) has full control over the memory system (we see later why this is important), the file system and the user-space processes. In the first step, the driver determines the source and destination memory addresses for the transfer. Next, the driver programs the hardware to perform the DMA transfer. The major difference between PCI/PCIe DMA and legacy ISA DMA is the way a DMA transfer is initiated. For PCI/PCIe no uniform, device independent way to initiate DMA transfers exists anymore, because each device contains its own, proprietary DMA engine. In contrast, the legacy DMA controller is always the same.

First, the peripheral device’s DMA engine is programmed with the source and destination addresses of the memory ranges to copy. Second, the device is signaled to begin the DMA transfer. Fair enough, but how can the driver know when the DMA transfer has finished? Usually, the device raises interrupts to inform the CPU about transfers that have finished. For each interrupt an interrupt handler, previously installed by the driver, is called and the finished transfer can be acknowledged accordingly by the OS (e.g. signaling the block I/O layer that a block has been read from disk and control can be handed back to the user-space process which requested this block). Back in the times of high latency spinning disks an slow network interfaces this was sufficient. Today, however, we’ve got solid state disks (SSD) and gigabit, low-latency network interfaces. To avoid completely maxing out the system by a vast number of interrupts, a common technique is to hold back and queue up multiple interrupts on the device until e.g. a timeout triggers, a certain number of interrupts are pending or any other condition suiting the application is met. This technique is known as interrupt coalescing. Obviously, the condition is always a trade-off between low latency and high throughput. The more frequently new interrupts are raised, the quicker the OS and its waiting processes are informed about finished memory transfers. However, if the OS is interrupted less often it can spend more time on other jobs.

DMA seems to be a nice feature in theory, but how does transferring large continuous memory regions play together with virtual memory? Virtual memory is usually organized in chunks of 4 KiB, called pages. Virtual memory is continuous as seen from a process’ point-of-view thanks to page tables and the memory management unit (MMU). However, it’s non-continuous as seen from the device point-of-view, because there is no MMU between the PCIe bus and the memory controller (well, some CPUs have an IO-MMU but let’s keep things simple). Hence, in a single DMA transfer only one page could be copied at a time. To overcome this limitation OS usually provide a scatter/gather API. Such an API chains together multiple page-sized memory transfers by creating a list of addresses of pages to be transferred.

# Take home message

DMA is an indispensable technique for memory-heavy, high-performance computing. Over the last decades, the entire bus system and DMA controller concept was superseded by moving the DMA controller into the devices and using a point-to-point bus architecture. This reduced latency, made concurrent DMA transfers possible and allowed for device specific DMA engine optimizations. For the drivers less has changed. They are still responsible for initiating the DMA transfers. Though, today, instead of programming a DMA controller in a device independent way, drivers must program device specific DMA engines. Therefore, programming DMA transfers and processing DMA status information can look very different depending on the device.

# Principles

Given is an arbitrary integer variable. How to find the index of the least significant bit (LSB) of the first 1-bit sequence of length >= n? Assuming n=4, let’s consider the following example of a random 32-bit integer value. The index we’re looking for is 10 in this case (indicated by the V character).

    31      24 | 23      16 | 15    V 8 | 7      0
MSB  01000111  |  11111101  |  10111100 | 01101001  LSB

Using a series of bit-wise and and shift-right operations the index of the LSB of the first 1111 sequence in the integer x can be found with the following trick.

x &= x>>1;
x &= x>>2;
index = __builtin_ffs(x)-1; // use _BitScanForward in Visual C++

After the first statement every 1 in x indicates the start of a 11 sequence. After the second statement every 1 in x indicates the start of a 1111 sequence. In the last statement the GCC intrinsic __builtin_ffs() (use _BitScanForward() if you’re on Visual C++) returns the bit position of the first set bit, starting from the LSB.
Note, that it doesn’t work to shift by four bits at once because it’s necessary to combine neighboring 1-bits to make sure that there are no 0-bits in-between. The following example illustrates how shifting by 3 bits wrongly yields two isolated 1-bits. In contrast, shifting by 2 bits correctly yields a sequence of 2 bits which can be further reduced into a single 1-bit indicating the start of the 1111 sequence.

 shift by 2    shift by 3
01111010      01111010
& 00011110    & 00001111
= 00011010    = 00001010
ok           wrong

# Arbitrary sequence lengths

By cleverly choosing the number of bits to shift, it’s even possible to extend this construction to find bit sequences which length is not a power of two. As the order of the and-shift-right operations has no relevance, the following algorithm can be used to compute the number of bits to shift in order to find the n-bit sequence index. The sum of shifted bits must be equal to n-1 and the number of bits to shift is halved in each iteration. Therefore, the total number of executed iterations is ceil(log2(n)).

int FindBitSeqIndexLsb(int x, int n)
{
assert(n >= 0 && n <= 32);

while (n > 1)
{
const int shiftBits = n>>1;
x &= (unsigned)x>>shiftBits; // shift in zeros from left
n -= shiftBits;
}

return __builtin_ffs(x)-1; // use _BitScanForward in Visual C++
}

# Exact sequence length

The described method finds bit sequences of length >= n. In case you’re looking for a bit sequence of exactly n bits, the following statement can be inserted right before the LSB scan is performed. This statement masks out any 1-bit which has a 1 on its left or right side. All the remaining 1-bits are isolated and indicate the start of a sequence of exactly n bits.

mask = (~(x<<1))&(~((unsigned)x>>1)); // shift in zeros from left and right
x &= mask;

# Sequence alignment

To account for aligned bit sequences, unaligned 1-bits can be simply masked out from x before the LSB scan is performed. For example, to regard only bit sequences starting at nibble boundaries x can be modified with the operation x &= 0x11111111 (it is 0x...11 = 0b...00010001) to clear all bits not starting at an index which is a multiple of four.

# Introduction

Oriented bounding boxes are an important tool for visibility testing and collision detection in computer graphics. In this post I want to talk about how to compute the oriented minimum bounding box (OMBB) of an arbitrary polygon in two dimensions. As a polygon just enforces an ordering on a set of points (vertices), everything described in the following equally applies to simple point sets. Minimum in this context refers to the area of the bounding box. A minimum oriented bounding box is also known as smallest-area enclosing rectangle. However, I will stick to the former term throughout this article as it is more frequently used in the computer graphics world.

The easiest way of computing a bounding box for a polygon is to determine the minimum and maximum $x$- and $y$- coordinates of its vertices. Such an axis aligned bounding box (AABB) can be computed trivially but it’s in most cases significantly bigger than the polygon’s OMBB. Finding the OMBB requires some more work as the bounding box’ area must be minimized, constrained by the location of the polygon’s vertices. Look at the following figure for an illustration (AABB in blue, OMBB in red).

The technique for computing OMBBs presented in the following consists of two detached steps. In the first step the convex hull of the input polygon is computed. If the polygon is convex this step can be omitted because a convex polygon is equal to its convex hull. In the second step the Rotating Calipers method is employed on the convex hull to compute the resulting OMBB. I will focus on the Rotating Calipers method because it’s not very widely known in comparison to the numerous ways of computing convex hulls.

# Convex hulls

In less mathematical but more illustrative terms the convex hull of a set of $n$ points can be described as the closed polygonal chain of all outer points of the set, which entirely encloses all set elements. You can picture it as the shape of a rubber band stretched around all set elements. The convex hull of a set of two-dimensional points can be efficiently computed in $O(n\log n)$. In the figure below the convex hull of the vertices of a concave polygon is depicted.

There are numerous algorithms for computing convex hulls: Quick Hull, Gift Wrapping (also known as Jarvis March), Graham’s Algorithm and some more. I’ve chosen the Gift Wrapping algorithm for my implementation because it’s easy to implement and provides good performance in case $n$ is small or the polygon’s convex hull contains only a few vertices. The runtime complexity is $O(nh)$, where $h$ is the number of vertices in the convex hull. In the general case Gift Wrapping is outperformed by other algorithms. Especially, when all points are part of the convex hull. In that case the complexity degrades to $O(n^2)$.

As there are many good articles on the Gift Wrapping algorithm available online, I won’t describe it another time here. Instead I want to focus on the lesser-known Rotating Calipers method for computing OMBBs. However, take care that your convex hull algorithm correctly handles collinear points. If multiple points lie on a convex hull edge, only the spanning points should end up in the convex hull.

# Rotating Calipers

Rotating Calipers is a versatile method for solving a number of problems from the field of computational geometry. It resembles the idea of rotating a dynamically adjustable caliper around the outside of a polygon’s convex hull. Originally, this method was invented to compute the diameter of convex polygons. Beyond that, it can be used to compute OMBBs, the minimum and maximum distance between two convex polygons, the intersection of convex polygons and many things more.

The idea of using the Rotating Calipers method for computing OMBBs is based on the following theorem, establishing a connection between the input polygon’s convex hull and the orientation of the resulting OMBB. The theorem was proven in 1975 by Freeman and Shapira1:

The smallest-area enclosing rectangle of a polygon has a side collinear with one of the edges of its convex hull.

Thanks to this theorem the number of OMBB candidates is dramatically reduced to the number of convex hull edges. Thus, the complexity of the Rotating Calipers method is linear if the convex hull is already available. If it isn’t available the overall complexity is bound by the cost of computing the convex hull. An example of a set of OMBB candidates (red) for a convex hull (green) is depicted in the figure below. Note, that there are as many OMBB candidates as convex hull edges and each OMBB candidate has one side flush with one edge of the convex hull.

To determine the OMBB of a polygon, first, two orthogonally aligned pairs of parallel supporting lines through the convex hull’s extreme points are created. The intersection of the four lines forms a rectangle. Next, the lines are simultaneously rotated about their supporting points until one line coincides with an edge of the convex hull. Each time an edge coincides, the four lines form another rectangle / OMBB candidate. This process is repeated until each convex hull edge once coincided with one of the four caliper lines. The resulting OMBB is the OMBB candidate with the smallest area. The entire algorithm is outlined step by step below.

1. Compute the convex hull of the input polygon.
2. Find the the extreme points $p_\text{min}=(x_\text{min},y_\text{min})^T$ and $p_\text{max}=(x_\text{max},y_\text{max})^T$ of the convex hull.
3. Construct two vertical supporting lines at $x_\text{min}$ and $x_\text{max}$ and two horizontal ones at $y_\text{min}$ and $y_\text{max}$.
4. Initialize the current minimum rectangle area $A_\text{min}=\infty$.
5. Rotate the supporting lines until one coincides with an edge of the convex hull.
1. Compute the area $A$ of the current rectangle.
2. Update the minimum area and store the current rectangle if $A.
6. Repeat step 5 until all edges of the convex hull coincided once with one of the supporting lines.
7. Output the minimum area rectangle stored in step 5.2.

In practice, in every iteration the smallest angle $\phi_\text{min}$ between each caliper line and its associated, following convex hull edge is determined. Then, all caliper lines are rotated at once by $\phi_\text{min}$ and the associated convex hull edge of the caliper line enclosing the smallest angle is advanced to the next convex hull edge.

# Wrap up

Rotating Calipers is a very elegant method for computing OMBBs in two dimensions. O’Rourke generalized it to three dimensions, yielding an algorithm of cubic runtime complexity. However, in practice approximation algorithms are used for three dimensional data because they’re usually faster. Beyond that, it's worth knowing the Rotating Calipers technique as it can be employed with minor changes to numerous other geometric problems. Depending on the programming language the implementation of the entire algorithm, including the convex hull computation, requires merely 150-200 lines of code. My sample implementation in Javascript can be found in my github repository.

1. H. Freeman, R. Shapira. Determining the minimum-area encasing rectangle for an arbitrary closed curve. Communications of the ACM, 18 Issue 7, July 1975, Pages 409-413

# Mutex lock guards in C++11

The new concurrency library of C++11 comes with two different classes for managing mutex locks: namely std::lock_guard and std::unique_lock. How do they compare with each other and in which situation which of the two classes should be preferably used?

The std::lock_guard class keeps its associated mutex locked during the entire life time by acquiring the lock on construction and releasing the lock on destruction. This makes it impossible to forget unlocking a critical section and it guarantees exception safety because any critical section is automatically unlocked when the stack is unwound after an exception was thrown. The std::lock_guard class should be used when a limited scope, like a class method, is to be locked.

void Foo::Bar()
{
std::lock_guard<std::mutex> guard(this->Mutex);
// mutex is locked now
}   // mutex is unlocked when lock guard goes out of scope

In contrast, the std::unique_lock class is a lot more flexible when dealing with mutex locks. It has the same interface as std::lock_guard but provides additional methods for explicitly locking and unlocking mutexes and deferring locking on construction. By passing std::defer_lock instead of std::adopt_lock the mutex remains unlocked when a std::unique_lock instance is constructed. The lock can then be obtained later by calling lock() on the std::unique_lock instance or alternatively, by passing it to the std::lock() function. To check if a std::unique_lock currently owns its associated mutex the owns_lock() method can be used. Hence, the mutex associated with a std::unique_lock doesn’t have to be locked (sometimes also referred to as owned) during the lock guard’s entire life time. As a consequence, the ownership of a std::unqiue_lock can be transferred between instances. This is why std::unique_lock is movable whereas std::lock_guard is not. Thus, more flexible locking schemes can be implemented by passing around locks between scopes.

For example a std::unique_lock can be returned from a function, or instances of a class containing a std::unique_lock attribute can be stored in containers. Consider the following example in which a mutex is locked in the function Foo(), returned to the function Bar() and only then unlocked on destruction.

std::mutex Mutex;

std::unique_lock<std::mutex> Foo()
{
std::unique_lock<std::mutex> lock(Mutex);
return lock;
// mutex isn't unlocked here!
}

void Bar()
{
auto lock = Foo();
}   // mutex is unlocked when lock goes out of scope

Keeping std::unique_lock’s additional lock status up-to-date induces some additional, minimal space and speed overhead in comparison to std::lock_guard. Hence, as a general rule, std::lock_guard should be preferably used when the additional features of std::unique_lock are not needed.

# Introduction

Binary search finds the index of a specified value (also called search key) in a pre-sorted array. Binary search is a very efficient and elegant algorithm which is used in a large number of data structures and algorithms. It has a runtime complexity of O(log N), where N is the number of elements to be searched through, because in each step the remaining number of array elements is halved.

Even though, the binary search algorithm is generally extremely fast I recently designed a data structure where the binary search was the major bottleneck. The said data structure performs successively a vast number of binary searches on a very large, static array of integers. The array can have a size of up to 100 billion unsigned 32-bit integers which corresponds to around 372.5 GB of raw data. To search for the first occurrence of an arbitrary key in an array of 100 billion elements a binary search touches floor(log2(N)+1)) = 30 elements, with N = 100000000000.

This is not an article about how to implement the binary search algorithm. More information on that can be found in plenty of resources online. The implementation of the optimization technique described in the rest of this post can be found in the accompanying github repository.

# Memory hierarchy

Considering the size of the data set the first few array accesses are guaranteed to cause cache misses; or even worse page misses depending on the amount of available main memory. On machines with little main memory a page miss requires the operating system to fetch the corresponding block containing the required array element from disk (obviously this would require some additional code logic). Accessing main memory is roughly 200x slower than accessing the first level cache. The random access time of a modern SSD is typically under 0.1 ms, which is roughly 1000x slower than accessing main memory. The random access times of spinning disks vary considerably and are usually in the order of a few milliseconds. Assuming a random access time of 5 ms, accessing a spinning disk is roughly 50000x slower than accessing main memory. Take these numbers with a grain of salt. They are by no means accurate and will be considerably different measured on concrete systems. However, the overall orders of magnitude are correct and give an intuitive feeling of how costly array accesses can be.

# Look-up tables

The previous paragraph should have cleared up why array accesses can be costly, especially when data is stored "far away" from the CPU. As a consequence it seems reasonable to try reducing the runtime costs of a binary search by reducing the number of array accesses needed for a search. This is a widely used optimization technique: an increased memory consumption is traded for increased performance.

The number of array accesses can be reduced by limiting the initial search range from left = 0, right = N-1 to anything smaller which is guaranteed to still contain the search key. The simple binary structure of unsigned integers allows us to use some of the high bits as an index into a look-up table (LUT) containing the reduced search range for the corresponding key. The more bits are used the bigger is the resulting speed-up, but the more memory is used. Using 16 bits requires only a 512 KB LUT. Using 24 bits requires already a 128 MB LUT. Let's consider exemplary a 16-bit LUT where the highest 16-bit of the search key are mapped. Each LUT entry maps to a sub-range in the value array that can contain up to 65536 different values.

0 [0x00000..0x0ffff] -> left = index of first element >= 0, right = index of last element < 65536
1 [0x10000..0x1ffff] -> left = index of first element >= 65536, right = index of last element < 131072
2 [0x20000..0x2ffff] -> left = index of first element >= 131072, right = index of last element < 196608
...

Let's consider a search for the key 114910 which is in hex 0x0001c0de. In order to obtain the LUT index for the key a right-shift by 16 bits is performed: lutIdx = key>>16, which results in lutIdx = 0x0001 = 1. The sub-range stored in the LUT at element 1 is guaranteed to contain all values in the range 65536 <= key < 131072. Following, the binary search, taking the search key, as well as the interval start and end as arguments, can be invoked by calling:

BinarySearch(key, Lut[lutIdx].left, Lut[lutIdx].right);

This is very little, simple and efficient code but how to populate the LUT? It turns out that populating the LUT can be done very efficiently in O(N) with a single, sequential pass over the input array. Even though, all elements must be touched once to populate the LUT, the sequential nature of the algorithm allows prefetching required data in advance and exploits the largely increased performance characteristics of spinning disks for sequential reads. The algorithm to populate the LUT is depicted in C++ below.

std::vector<std::pair<size_t, size_t>> Lut;

void InitLut(const std::vector<uint32_t> &vals, size_t lutBits)
{
// pre-allocate memory for LUT
const size_t lutSize = 1<<lutBits;
const size_t shiftBits = 32-lutBits; // for 32-bit integers!
Lut.resize(lutSize);

// populate look-up-table
size_t thresh = 0, last = 0;

for (ssize_t i=0; i<(ssize_t)vals.size()-1; i++)
{
const size_t nextThresh = vals[i+1]>>shiftBits;
Lut[thresh] = {last, i};

if (nextThresh > thresh) // more than one LUT entry to fill?
{
last = i+1;
for (size_t j=thresh+1; j<=nextThresh; j++)
Lut[j] = {last, i+1};
}

thresh = nextThresh;
}

// set remaining thresholds that couldn't be found
for (size_t i=thresh; i<Lut.size(); i++)
Lut[i] = {last, vals.size()-1};
}

The LUT is implemented as an array of pairs. Each pair represents the interval the corresponding LUT entry maps to. To populate the LUT, the outer for-loop iterates over the pre-sorted array and calculates the number of LUT entries to fill between two successive array elements. The inner for-loop fills the LUT accordingly. As the start of each LUT entry can be derived from the end of the previous one, the last variable keeps track of the end of the last interval. The intervals are non-overlapping and it holds that the end of an interval is the beginning of the next interval minus one: Lut[i].second = Lut[i+1].first-1. As a consequence, the memory size of the LUT can be halved by storing only the interval starts and computing the interval ends as described. For simplicity reasons the depicted code is without this improvement, in contrast to the code in the accompanying github repository.

# Other data types

The presented LUT-based optimization only works for data types that are compatible with a simple bit-wise comparison. This means that more significant bits must have a bigger influence on the value's magnitude than less significant bits. So do signed integers and floats work as well? As long as they're positive there are no problems, because for any two positive signed integers or floats x and y it holds that if x <= y it follows that UIR(x) <= UIR(y), where UIR() denotes the binary unsigned integer representation. However, negative values cause problems. Hence, when creating the LUT and when doing a range look-up the array elements and the search key must be mapped to an ordered range using a bijective mapping as described below.

## Signed integers

Signed integers are stored in two's complement, in which the most significant bit (MSB) represents the sign: if the MSB is set the value is negative, otherwise it's positive. Therefore, signed integers compare greater using bit-wise comparison than unsigned ones. It turns out, that by simply flipping the sign bit the ordering can be fixed. It's important to note that flipping the sign bit is only enough if two's complement is used. If a simple signed magnitude representation is used negative values are not automatically reversed. In that case not only the sign bit but also all the remaining bits have to flipped when the sign bit is set. For clarity look at the following example of two's complement integers where FSB() denotes the function that flips the sign bit: FSB(-6 = 11111010) = 01111010 = 122 < FSB(-5 = 11111011) = 01111011 = 123 < FSB(2 = 00000010) = 10000010 = 130. In contrast, if a signed magnitude representation is used flipping the sign bit is not sufficient anymore as the following example illustrates: -6 < -5, however FSB(-6 = 10000110) = 00000110 = 6 > FSB(-5 = 10000101) = 00000101 = 5.

## Floating point

Floating point values (I'm talking about IEEE 754) are a bit more nasty than signed integers. Their signed magnitude representation has exactly the same problem as signed magnitude represented integers have, outlined in the previous section. To overcome this problem all the remaining bits just have to be flipped as well if the sign bit was set. For further information read the article of Michael Herf about Radix Sort for floats on Stereopsis. One possible implementation in C++ is:

// taken from Michael Herf's article on Stereopsis
uint32_t MapFloat(float val)
{
// 1) flip sign bit
// 2) if sign bit was set flip all other bits as well
const uint32_t cv = (uint32_t &)val;
}
I analyzed the performance gain of the LUT optimization by searching 10 million times through an array of 1 billion 32-bit integers. The integers were uniformly distributed in the interval [0, 0xffffffff]. In each search a randomly determined key from the data set was used. The performance was measured on a laptop with a Core(TM) i7-3612QM CPU (Ivy Bridge) at 2.1 GHz with 6 MB L3 cache and 8 GB RAM. The results of the standard binary search and the LUT optimized binary search for different LUT sizes are depicted in the table below. The speed-up is related to C++ standard binary search function std::lower_bound(), not my own implementation used for the range reduced binary search.
std::lower_bound() - 9.373 secs 1.07 m -
The 16-bit LUT binary search is more than 2.5x, the 24-bit LUT binary search is even almost 5x faster than C++'s standard binary search implementation. The amount of required additional code is neglectable and the LUT memory size can be configured granularly by adjust the number of bits used to index the LUT. Thanks to the O(N) complexity of the LUT population algorithm the discussed optimization is even feasible in situations where the underlying value array changes from time to time. Especially, in situations where parts of the data set are only resident on hard disk the reduced number of array accesses saves a considerable amount of work. By using simple mappings the LUT optimization can be applied as well to signed integer and even floating point values. For a sample implementation checkout the accompanying github repository.