T-tree algorithm is described in this paper
And T*-Tree is an improvement from T-tree for better use of query operations, including range queries and which contains all other good features of T-tree.
This algorithm is described in this paper "T*-tree: A Main Memory Database Index Structure for Real-Time Applications".
According to this research paper, T-Tree is faster than B-tree/B+tree when datasets fit in the memory.
I implemented T-Tree/T*Tree as they described in these papers and compared the performance with B-tree/B+tree, but B-tree/B+tree perform better than T-Tree/T*Tree in all test cases (insertion, deletion, searching).
I read that T-Tree is an efficient index structure for in-memory database, and it used by Oracle TimesTen. But my results did not show that.
If anyone may know the reason or have any comment about that, it will be great to hear from her (or him).
T-Trees are not a fundamental data structure in the same sense that AVL trees or B-trees are. They are just a hacked version of balanced binary trees and as such there may or may not be niche applications where they offer decent performance.
In this day and age they are bound to suffer horribly because of their poor locality, both in the sense of expected block/page transfer counts and in the sense of cache locality. The latter is evident since in all node accesses of a search except for the very last one, only the boundary values will be checked against the search key - all the rest is paged in or cached for nought.
Compare this to the excellent access locality of B-trees in general and B+trees in particular (not to mention cache-oblivious and cache-conscious versions that were designed explicitly with memory performance charactistics in mind).
Similar problems exist with the rebalancing. In the B-tree world many variations - starting with B+ and Blink - have been developed and perfected in order to achieve desired amortised performance characteristics, including aspects like concurrency (locking/latching) or the absence thereof. So most of the time you can simply go out and find a B-tree variation that fits your performance profile - or use the simple classic B+tree and be certain of decent results.
T-trees are more complicated than comparable B-trees and it seems that they have nothing to offer in the way of performance in general, given that the times of commodity hardware with a single-level memory 'hierarchy' have been gone for decades. Not only is the hard disk the new memory, the converse is also true and main memory is the new hard disk now. I.e. even without NUMA the cost of bringing data from main memory into the cache hierarchy is so high that it pays to minimise page transfers - which is precisely what B-trees and their variations do and the T-tree doesn't. Closer to the processor core it's the number of cache line accesses/transfers that matters but the picture remains the same.
In fact, if you take the idea of binary search - which is provably optimal - and think about ways of arranging the search keys in a manner that plays well with memory hierarchies (caches) then you invariably end up with something that looks uncannily like a B-tree...
If you program for performance then you'll find that winners are almost always located somewhere in the triangle between sorted arrays, B-trees and hashing. Even balanced binary trees are only competitive if their comparatively poor performance takes the back seat in the face of other considerations and key counts are fairly small, i.e. not more than a couple million.
Related
I was reading this answer
Efficient (and well explained) implementation of a Quadtree for 2D collision detection
and encountered this paragraph
All right, so actually quadtrees are not my favorite data structure for this purpose. I tend to prefer a grid hierarchy, like a coarse grid for the world, a finer grid for a region, and an even finer grid for a sub-region (3 fixed levels of dense grids, and no trees involved), with row-based optimizations so that a row that has no entities in it will be deallocated and turned into a null pointer, and likewise completely empty regions or sub-regions turned into nulls. While this simple implementation of the quadtree running in one thread can handle 100k agents on my i7 at 60+ FPS, I've implemented grids that can handle a couple million agents bouncing off each other every frame on older hardware (an i3). Also I always liked how grids made it very easy to predict how much memory they'll require, since they don't subdivide cells. But I'll try to cover how to implement a reasonably efficient quadtree.
This type of grid seems intuitive, it sort of sounds like a "N-order" grid, where instead of 4 child nodes, you have N child nodes per parent. N^3 can go much further than 4^3, which allows better precision with potentially (I guess) less branching (since there are many less nodes to branch).
I'm a little puzzled because I would intuitively use a single, or maybe 3 std::map with the proper < operator(), to reduce its memory footprint, but I'm not sure it would be so fast, since querying an AABB would mean stacking several accesses that are O(log n).
What exactly are those row-based optimizations he is talking about? Is this type of grid search common?
I have some understanding of a z order curve, and I'm not entirely satisfied with a quadtree.
It's my own quote. But that's based on a common pattern I encountered in my personal experience. Also, terms like "parent" and "child" are ones I'd largely discard when talking about grids. You just got a big 2-dimensional or N-dimensional table/matrix storing stuff. There's not really a hierarchy involved whatsoever -- these data structures are more comparable to arrays than trees.
"Coarse" and "Fine"
On "coarse" and "fine", what I meant there is that "coarse" search queries tend to be cheaper but give more false positives. A coarser grid would be one that is lower in grid resolution (fewer, larger cells). Coarse searches may involve traversing/searching fewer and larger grid cells. For example, say we want to see if an element intersects a point/dot in a gigantic cell (imagine just a 1x1 grid storing everything in the simulation). If the dot intersects the cell, we may get a whole lot of elements returned in that cell but maybe only one or none of them actually intersect the dot.
So a "coarse" query is broad and simple but not very precise at narrowing down the list of candidates (or "suspects"). It may return too many results and still leave a whole lot of processing required left to do to narrow down what actually intersects the search parameter*.
It's like in those detective shows when they search a database for a
possible killer, putting in "white male" might not require much
processing to list the results but might give way too many results to
properly narrow down the suspects. "Fine" would be the opposite and might require more processing of the database but narrow down the result to just one suspect.
This is a crude and flawed analogy but I hope it helps.
Often the key to broadly optimizing spatial indices before we get into things like memory optimizations whether we're talking spatial hashes or quadtrees is to find a nice balance between "coarse" and "fine". Too "fine" and we might spend too much time traversing the data structure (searching many small cells in a uniform grid, or spending too much time in tree traversal for adaptive grids like quadtrees). Too "coarse" and the spatial index might give back too many results to significantly reduce the amount of time required for further processing. For spatial hashes (a data structure I don't personally like very much but they're very popular in gamedev), there's often a lot of thought and experimentation and measuring involved in determining an optimal cell size to use.
With uniform NxM grids, how "coarse" or "fine" they are (big or small cells and high or low grid resolution) not only impacts search times for a query but can also impact insertion and removal times if the elements are larger than a point. If the grid is too fine, a single large or medium-sized element may have to be inserted into many tiny cells and removed from many tiny cells, using lots of extra memory and processing. If it's too coarse, the element may only have to be inserted and removed to/from one large cell but at costs to the data structure's ability to narrow down the number of candidates returned from a search query to a minimum. Without care, going too "fine/granular" can become very bottlenecky in practice and a developer might find his grid structure using gigabytes of RAM for a modest input size. With tree variants like quadtrees, a similar thing can happen if the maximum allowed tree depth is too high a value causing explosive memory use and processing when the leaf nodes of the quadtree store the tiniest cell sizes (we can even start running into floating-point precision bugs that wreck performance if the cells are allowed to be subdivided to too small a size in the tree).
The essence of accelerating performance with spatial indices is often this sort of balancing act. For example, we typically don't want to apply frustum culling to individual polygons being rendered in computer graphics because that's typically not only redundant with what the hardware already does at the clipping stage, but it's also too "fine/granular" and requires too much processing on its own compared to the time required to just request to render one or more polygons. But we might net huge performance improvements with something a bit "coarser", like applying frustum culling to an entire creature or space ship (an entire model/mesh), allowing us to avoid requesting to render many polygons at once with a single test. So I often use the terms, "coarse" and "fine/granular" frequently in these sorts of discussions (until I find better terminology that more people can easily understand).
Uniform vs. Adaptive Grid
You can think of a quadtree as an "adaptive" grid with adaptive grid cell sizes arranged hierarchically (working from "coarse' to "fine" as we drill down from root to leaf in a single smart and adaptive data structure) as opposed to a simple NxM "uniform" grid.
The adaptive nature of the tree-based structures is very smart and can handle a broad range of use cases (although typically requiring some fiddling of maximum tree depth and/or minimum cell size allowed and possibly how many maximum elements are stored in a cell/node before it subdivides). However, it can be more difficult to optimize tree data structures because the hierarchical nature doesn't lend itself so easily to the kind of contiguous memory layout that our hardware and memory hierarchy is so well-suited to traverse. So very often I find data structures that don't involve trees to be easier to optimize in the same sense that optimizing a hash table might be simpler than optimizing a red-black tree, especially when we can anticipate a lot about the type of data we're going to be storing in advance.
Another reason I tend to favor simpler, more contiguous data structures in lots of contexts is that the performance requirements of a realtime simulation often want not just fast frame rates, but consistent and predictable frame rates. The consistency is important because even if a video game has very high frame rates for most of the game but some part of the game causes the frame rates to drop substantially for even a brief period of time, the player may die and game over as a result of it. It was often very important in my case to avoid these types of scenarios and have data structures largely absent pathological worst-case performance scenarios. In general, I find it easier to predict the performance characteristics of lots of simpler data structures that don't involve an adaptive hierarchy and are kind of on the "dumber" side. Very often I find the consistency and predictability of frame rates to be roughly tied to how easily I can predict the data structure's overall memory usage and how stable that is. If the memory usage is wildly unpredictable and sporadic, I often (not always, but often) find the frame rates will likewise be sporadic.
So I often find better results using grids personally, but if it's tricky to determine a single optimal cell size to use for the grid in a particular simulation context, I just use multiple instances of them: one instance with larger cell sizes for "coarse" searches (say 10x10), one with smaller ones for "finer" searches (say 100x100), and maybe even one with even smaller cells for the "finest" searches (say 1000x1000). If no results are returned in the coarse search, then I don't proceed with the finer searches. I get some balance of the benefits of quadtrees and grids this way.
What I did when I used these types of representations in the past is not to store a single element in all three grid instances. That would triple the memory use of an element entry/node into these structures. Instead, what I did was insert the indices of the occupied cells of the finer grids into the coarser grids, as there are typically far fewer occupied cells than there are a total number of elements in the simulation. Only the finest, highest-resolution grid with the smallest cell sizes would store the element. The cells in the finest grid are analogous to the leaf nodes of a quadtree.
The "loose-tight double grid" as I'm calling it in one of the answers to that question is an expansion on this multi-grid idea. The difference is that the finer grid is actually loose and has cell sizes that expand and shrink based on the elements inserted to it, always guaranteeing that a single element, no matter how large or small, needs only be inserted to one cell in the grid. The coarser grid stores the occupied cells of the finer grid leading to two constant-time queries (one in the coarser tight grid, another into the finer loose grid) to return an element list of potential candidates matching the search parameter. It also has the most stable and predictable memory use (not necessarily the minimal memory use because the fine/loose cells require storing an axis-aligned bounding box that expands/shrinks which adds another 16 bytes or so to a cell) and corresponding stable frame rates because one element is always inserted to one cell and doesn't take any additional memory required to store it besides its own element data with the exception of when its insertion causes a loose cell to expand to the point where it has to be inserted to additional cells in the coarser grid (which should be a fairly rare-case scenario).
Multiple Grids For Other Purposes
I'm a little puzzled because I would intuitively use a single, or maybe 3 std::map with the proper operator(), to reduce its memory footprint, but I'm not sure it would be so fast, since querying an AABB would mean stacking several accesses that are O(log n).
I think that's an intuition many of us have and also probably a subconscious desire to just lean on one solution for everything because programming can get quite tedious and repetitive and it'd be ideal to just implement something once and reuse it for everything: a "one-size-fits-all" t-shirt. But a one-sized-fits-all shirt can be poorly tailored to fit our very broad and muscular programmer bodies*. So sometimes it helps to use the analogy of a small, medium, and large size.
This is a very possibly poor attempt at humor on my part to make my long-winded texts less boring to read.
For example, if you are using std::map for something like a spatial hash, then there can be a lot of thought and fiddling around trying to find an optimal cell size. In a video game, one might compromise with something like making the cell size relative to the size of your average human in the game (perhaps a bit smaller or bigger), since lots of the models/sprites in the game might be designed for human use. But it might get very fiddly and be very sub-optimal for teeny things and very sub-optimal for gigantic things. In that case, we might do well to resist our intuitions and desires to just use one solution and use multiple (it could still be the same code but just multiple instances of the same class instance for the data structure constructed with varying parameters).
As for the overhead of searching multiple data structures instead of a single one, that's something best measured and it's worth remembering that the input sizes of each container will be smaller as a result, reducing the cost of each search and very possibly improve locality of reference. It might exceed the benefits in a hierarchical structure that requires logarithmic search times like std::map (or not, best to just measure and compare), but I tend to use more data structures which do this in constant-time (grids or hash tables). In my cases, I find the benefits far exceeding the additional cost of requiring multiple searches to do a single query, especially when the element sizes vary radically or I want some basic thing resembling a hierarchy with 2 or more NxM grids that range from "coarse" to "fine".
Row-Based Optimizations
As for "row-based optimizations", that's very specific to uniform fixed-sized grids and not trees. It refers to using a separate variable-sized list/container per row instead of a single one for the entire grid. Aside from potentially reducing memory use for empty rows that just turn into nulls without requiring an allocated memory block, it can save on lots of processing and improve memory access patterns.
If we store a single list for the entire grid, then we have to constantly insert and remove from that one shared list as elements move around, particles are born, etc. That could lead to more heap allocations/deallocations growing and shrinking the container but also increases the average memory stride to get from one element in that list to the next which will tend to translate to more cache misses as a result of more irrelevant data being loaded into a cache line. Also these days we have so many cores so having a single shared container for the entire grid may reduce the ability to process the grid in parallel (ex: searching one row while simultaneously inserting to another). It can also lead to more net memory use for the structure since if we use a contiguous sequence like std::vector or ArrayList, those can often store the memory capacity of as many as twice the elements required to reduce the time of insertions to amortized constant time by minimizing the need to reallocate and copy the former elements in linear-time by keeping excess capacity.
By associating a separate medium-sized container per grid row or per column instead of gigantic one for the entire grid, we can mitigate these costs in some cases*.
This is the type of thing you definitely measure before and after though to make sure it actually improves overall frame rates, and probably attempt in response to a first attempt storing a single list for the entire grid revealing many non-compulsory cache misses in the profiler.
This might beg the question of why we don't use a separate teeny list container for every single cell in the grid. It's a balancing act. If we store that many containers (ex: a million instances of std::vector for a 1000x1000 grid possibly storing very few or no elements each), it would allow maximum parallelism and minimize the stride to get from one element in a cell to the next one in the cell, but that can be quite explosive in memory use and introduce a lot of extra processing and heap overhead.
Especially in my case, my finest grids might store a million cells or more, but I only require 4 bytes per cell. A variable-sized sequence per cell would typically explode to at least something like 24 bytes or more (typically far more) per cell on 64-bit architectures to store the container data (typically a pointer and a couple of extra integers, or three pointers on top of the heap-allocated memory block), but on top of that, every single element inserted to an empty cell may require a heap allocation and compulsory cache miss and page fault and very frequently due to the lack of temporal locality. So I find the balance and sweet spot to be one list container per row typically among my best-measured implementations.
I use what I call a "singly-linked array list" to store elements in a grid row and allow constant-time insertions and removals while still allowing some degree of spatial locality with lots of elements being contiguous. It can be described like this:
struct GridRow
{
struct Node
{
// Element data
...
// Stores the index into the 'nodes' array of the next element
// in the cell or the next removed element. -1 indicates end of
// list.
int next = -1;
};
// Stores all the nodes in the row.
std::vector<Node> nodes;
// Stores the index of the first removed element or -1
// if there are no removed elements.
int free_element = -1;
};
This combines some of the benefits of a linked list using a free list allocator but without the need to manage separate allocator and data structure implementations which I find to be too generic and unwieldy for my purposes. Furthermore, doing it this way allows us to halve the size of a pointer down to a 32-bit array index on 64-bit architectures which I find to be a big measured win in my use cases when the alignment requirements of the element data combined with the 32-bit index don't require an additional 32-bits of padding for the class/struct which is frequently the case for me since I often use 32-bit or smaller integers and 32-bit single-precision floating-point or 16-bit half-floats.
Unorthodox?
On this question:
Is this type of grid search common?
I am not sure! I tend to struggle a bit with terminology and I'll have to ask people's forgiveness and patience in communication. I started programming from early childhood in the 1980s before the internet was widespread, so I came to rely on inventing a lot of my own techniques and using my own crude terminology as a result. I got my degree in computer science about a decade and a half later when I reached my 20s and corrected some of my terminology and misconceptions but I've had many years just rolling my own solutions. So I am often not sure if other people have come across some of the same solutions or not, and if there are formal names and terms for them or not.
That makes communication with other programmers difficult and very frustrating for both of us at times and I have to ask for a lot of patience to explain what I have in mind. I've made it a habit in meetings to always start off showing something with very promising results which tends to make people more patient with my crude and long-winded explanations of how they work. They tend to give my ideas much more of a chance if I start off by showing results, but I'm often very frustrated with the orthodoxy and dogmatism that can be prevalent in this industry that can sometimes prioritize concepts far more than execution and actual results. I'm a pragmatist at heart so I don't think in terms of "what is the best data structure?" I think in terms of what we can effectively implement personally given our strengths and weaknesses and what is intuitive and counter-intuitive to us and I'm willing to endlessly compromise on the purity of concepts in favor of a simpler and less problematic execution. I just like good, reliable solutions that roll naturally off our fingertips no matter how orthodox or unorthodox they may be, but a lot of my methods may be unorthodox as a result (or not and I might just have yet to find people who have done the same things). I've found this site useful at rare times in finding peers who are like, "Oh, I've done that too! I found the best results if we do this [...]" or someone pointing out like, "What you are proposing is called [...]."
In performance-critical contexts, I kind of let the profiler come up with the data structure for me, crudely speaking. That is to say, I'll come up with some quick first draft (typically very orthodox) and measure it with the profiler and let the profiler's results give me ideas for a second draft until I converge to something both simple and performant and appropriately scalable for the requirements (which may become pretty unorthodox along the way). I'm very happy to abandon lots of ideas since I figure we have to weed through a lot of bad ideas in a process of elimination to come up with a good one, so I tend to cycle through lots of implementations and ideas and have come to become a really rapid prototyper (I have a psychological tendency to stubbornly fall in love with solutions I spent lots of time on, so to counter that I've learned to spend the absolute minimal time on a solution until it's very, very promising).
You can see my exact methodology at work in the very answers to that
question where I iteratively converged through lots of profiling and
measuring over the course of a few days and prototyping from a fairly orthodox quadtree to that
unorthodox "loose-tight double grid" solution that handled the largest
number of agents at the most stable frame rates and was, for me
anyway, much faster and simpler to implement than all the structures
before it. I had to go through lots of orthodox solutions and measure them though to generate the final idea for the unusual loose-tight variant. I always start off with and favor the orthodox solutions and start off inside the box because they're well-documented and understood and just very gently and timidly venture outside, but I do often find myself a bit outside the box when the requirements are steep enough. I'm no stranger to the steepest requirements since in my industry and as a fairly low-level type working on engines, the ability to handle more data at good frame rates often translates not only to greater interactivity for the user but also allows artists to create more detailed content of higher visual quality than ever before. We're always chasing higher and higher visual quality at good frame rates, and that often boils down to a combination of both performance and getting away with crude approximations whenever possible. This inevitably leads to some degree of unorthodoxy with lots of in-house solutions very unique to a particular engine, and each engine tends to have its own very unique strengths and weaknesses as you find comparing something like CryEngine to Unreal Engine to Frostbite to Unity.
For example, I have this data structure I've been using since childhood and I don't know the name of it. It's a straightforward concept and it's just a hierarchical bitset that allows set intersections of potentially millions of elements to be found in as little as a few iterations of simple work as well as traverse millions of elements occupying the set with just a few iterations (less than linear-time requirements to traverse everything in the set just through the data structure itself which returns ranges of occupied elements/set bits instead of individual elements/bit indices). But I have no idea what the name is since it's just something I rolled and I've never encountered anyone talking about it in computer science. I tend to refer to it as a "hierarchical bitset". Originally I called it a "sparse bitset tree" but that seems a tad verbose. It's not a particularly clever concept at all and I wouldn't be surprised or disappointed (actually quite happy) to find someone else discovering the same solution before me but just one I don't find people using or talking about ever. It just expands on the strengths of a regular, flat bitset in rapidly finding set intersections with bitwise OR and rapidly traverse all set bits using FFZ/FFS but reducing the linear-time requirements of both down to logarithmic (with the logarithm base being a number much larger than 2).
Anyway, I wouldn't be surprised if some of these solutions are very unorthodox, but also wouldn't be surprised if they are reasonably orthodox and I've just failed to find the proper name and terminology for these techniques. A lot of the appeal of sites like this for me is a lonely search for someone else who has used similar techniques and to try to find proper names and terms for them often to end in frustration. I'm also hoping to improve on my ability to explain them although I've always been so bad and long-winded here. I find using pictures helps me a lot because I find human language to be incredibly riddled with ambiguities. I'm also fond of deliberately imprecise figurative language which embraces and celebrates the ambiguities such as metaphor and analogy and humorous hyperbole, but I've not found it's the type of thing programmers tend to appreciate so much due to its lack of precision. But I've never found precision to be that important so long as we can convey the meaty stuff and what is "cool" about an idea while they can draw their own interpretations of the rest. Apologies for the whole explanation but hopefully that clears some things up about my crude terminology and the overall methodology I use to arrive at these techniques. English is also not my first language so that adds another layer of convolution where I have to sort of translate my thoughts into English words and struggle a lot with that.
I keep in mind that hash would be first thing I should resort to if I want to write an application which requests high lookup speed, and any other data structure wouldn't guarantee that.
But I got confused when saw some many post saying different, such as suffix tree, trie, to name a few.
So I wonder is hash always the best thing for high speed lookup? What if I want both high lookup speed and less space cost?
Is there any material (books or papers) lecturing about the data structures or algorithms **on high speed lookup and space efficiency? Any of this kind is highly appreciated.
So I wonder is hash always the best thing for high speed lookup?
No. As stated in comments:
There is never such a thing Best data structure for [some generic issue]. Everything is case dependent. Tries and radix trees might be great for strings, since you need to read the string anyway. arrays allows simplicity and great cache efficiency - and are usually the best for small scale static information
I once answered a related question of cases where a tree might be better then a hash table: Hash Table v/s Trees
What if I want both high lookup speed and less space cost?
The two might be self-contradicting. Even for the simple example of a hash table of size X vs a hash table of size 2*X. The bigger hash table is less likely to encounter collisions, and thus is expected to be faster then the smaller one.
Is there any material (books or papers) lecturing about the data
structures or algorithms on high speed lookup and space efficiency?
Introduction to Algorithms provide a good walk through on the main data structure used. Any algorithm developed is trying to provide a good space and time efficiency, but like said, there is a trade off, and some algorithms might be better for specific cases then others.
Choosing the right algorithm/data structure/design for the specific problem is what engineering is about, isn't it?
I assume you are talking about strings here, and the answer is "no", hashes are not the fastest or most space efficient way to look up strings, tries are. Of course, writing a hashing algorithm is much, much easier than writing a trie.
One thing you won't find in wikipedia or books about tries is that if you naively implement them with one node per letter, you end up with large numbers of inefficient, one-child nodes. To make a trie that really burns up the CPU you have to implement nodes so that they can have a variable number of characters. This, of course, is even harder than writing a plain trie.
I have written trie implementations that handle over a billion entries and I can tell you that if done properly it is insanely fast, nothing else compares.
One other issue with tries is that you have to write a custom heap, because if you just use some kind of generic memory management it will be slow. So in addition to implementing the trie, you have to implement the heap that the trie runs on. Pretty freakin complicated, but if you do it, you get batshit crazy speed.
Only a good implementation of hash will give you good performance. And you cannot compare hash with Trie for all situations. Situations where Trie is applicable, is fast, but it can be costly in terms of memory, (again dependent on implementation).
But have you measured performance? Or it is unnecessary optimization you are looking for. Did the map fail you?
That might also depend on the actual number of elements.
In complexity theory a hash is not bad, but complexity theory is only good if the actual number of elements is bigger than some threshold.
I.e. if you have only 2 elements, there is a faster method than a hash ;-)
Hash tables are a good general purpose structure but they can fail spectacularly if the hash function doesn't suit the input data. Worst case lookup is O(n). They also waste some space as you mentioned. Other general-purpose structures like balanced binary search trees have worse average case but better worst case performance than a hash table. This is important for real-time applications. A trie is a more special-purpose structure tailored to string lookup.
First, read this:
TPT paper
I was wondering what other options might exist for arranging nodes to boost performance. Anything from post-parent order in a byte array, like TPT's, to something more like a k-order b-tree; I'm wondering what good options are known at the moment?
A bit more on the problem:
I have an extremely fast way of finding elements within a sparse set, given some concept of adjacency to a given pointer. I was wondering how I could best take advantage of this in storing a patricia trie.
You can make assumptions about whether the trie will be random-access, read only, write-seldom, or add-only. Please note them if you do, but I've actually used a TPT and the gains were pretty significant so I'm willing to consider certain constraints.
Update
I guess in some senses this was a little unclear. What I'm looking for here is ways of arranging things in memory that optimize one performance metric or another. The TPTs, through some tricks, use node order to optimize disk reads and space-per-node. I'm curious about:
Total deletion, where the structure is removed from memory entirely.
Inserts, particularly in densely populated structures.
Deletes, again, particularly in densely populated structures.
A DAWG or a minimal DFA (see this question or the paper "How to squeeze a lexicon") may be even better than a TPT because the totel size is smaller.
I think that the zipper is a beautiful idea; it elegantly provides a way to walk a list or tree and make what appear to be local updates in a functional way.
Asymptotically, the costs appear to be reasonable. But traversing the data structure requires memory allocation at each iteration, where a normal list or tree traversal is just pointer chasing. This seems expensive (please correct me if I am wrong).
Are the costs prohibitive? And what under what circumstances would it be reasonable to use a zipper?
I can provide one solid data point: John Dias and I had a paper in the 2005 ML Workshop where we compared the cost of using a zipper to represent control-flow graphs with the cost of using mutable record fields in Objective Caml. We were very pleasantly surprised to find that the performance of our compiler with the zipper-based control-flow graphs was actually slightly better than the performance using a traditional data structure with mutable records linked by pointers. We couldn't find serious analysis tools to tell us exactly why the zipper was faster, but I suspect the reason was that there were fewer invariants to maintain and so relatively fewer pointer assignments. It's also possible that the optimizer was smart enough to amortize some of the allocation costs incurred by the zipper. In any case, the zipper can be used in an optimizing compiler, and there is actually a slight performance benefit.
I recently came across the data structure known as a skip list. It seems to have very similar behavior to a binary search tree.
Why would you ever want to use a skip list over a binary search tree?
Skip lists are more amenable to concurrent access/modification. Herb Sutter wrote an article about data structure in concurrent environments. It has more indepth information.
The most frequently used implementation of a binary search tree is a red-black tree. The concurrent problems come in when the tree is modified it often needs to rebalance. The rebalance operation can affect large portions of the tree, which would require a mutex lock on many of the tree nodes. Inserting a node into a skip list is far more localized, only nodes directly linked to the affected node need to be locked.
Update from Jon Harrops comments
I read Fraser and Harris's latest paper Concurrent programming without locks. Really good stuff if you're interested in lock-free data structures. The paper focuses on Transactional Memory and a theoretical operation multiword-compare-and-swap MCAS. Both of these are simulated in software as no hardware supports them yet. I'm fairly impressed that they were able to build MCAS in software at all.
I didn't find the transactional memory stuff particularly compelling as it requires a garbage collector. Also software transactional memory is plagued with performance issues. However, I'd be very excited if hardware transactional memory ever becomes common. In the end it's still research and won't be of use for production code for another decade or so.
In section 8.2 they compare the performance of several concurrent tree implementations. I'll summarize their findings. It's worth it to download the pdf as it has some very informative graphs on pages 50, 53, and 54.
Locking skip lists is insanely fast. They scale incredibly well with the number of concurrent accesses. This is what makes skip lists special, other lock based data structures tend to croak under pressure.
Lock-free skip lists are consistently faster than locking skip lists but only barely.
transactional skip lists are consistently 2-3 times slower than the locking and non-locking versions.
locking red-black trees croak under concurrent access. Their performance degrades linearly with each new concurrent user. Of the two known locking red-black tree implementations, one essentially has a global lock during tree rebalancing. The other uses fancy (and complicated) lock escalation but still doesn't significantly outperform the global lock version.
lock-free red-black trees don't exist (no longer true, see Update).
transactional red-black trees are comparable with transactional skip-lists. That was very surprising and very promising. Transactional memory, though slower if far easier to write. It can be as easy as quick search and replace on the non-concurrent version.
Update
Here is paper about lock-free trees: Lock-Free Red-Black Trees Using CAS.
I haven't looked into it deeply, but on the surface it seems solid.
First, you cannot fairly compare a randomized data structure with one that gives you worst-case guarantees.
A skip list is equivalent to a randomly balanced binary search tree (RBST) in the way that is explained in more detail in Dean and Jones' "Exploring the Duality Between Skip Lists and Binary Search Trees".
The other way around, you can also have deterministic skip lists which guarantee worst case performance, cf. Munro et al.
Contra to what some claim above, you can have implementations of binary search trees (BST) that work well in concurrent programming. A potential problem with the concurrency-focused BSTs is that you can't easily get the same had guarantees about balancing as you would from a red-black (RB) tree. (But "standard", i.e. randomzided, skip lists don't give you these guarantees either.) There's a trade-off between maintaining balancing at all times and good (and easy to program) concurrent access, so relaxed RB trees are usually used when good concurrency is desired. The relaxation consists in not re-balancing the tree right away. For a somewhat dated (1998) survey see Hanke's ''The Performance of Concurrent Red-Black Tree Algorithms'' [ps.gz].
One of the more recent improvements on these is the so-called chromatic tree (basically you have some weight such that black would be 1 and red would be zero, but you also allow values in between). And how does a chromatic tree fare against skip list? Let's see what Brown et al. "A General Technique for Non-blocking Trees" (2014) have to say:
with 128 threads, our algorithm outperforms Java’s non-blocking skiplist
by 13% to 156%, the lock-based AVL tree of Bronson et al. by 63% to 224%, and a RBT that uses software transactional memory (STM) by 13 to 134 times
EDIT to add: Pugh's lock-based skip list, which was benchmarked in Fraser and Harris (2007) "Concurrent Programming Without Lock" as coming close to their own lock-free version (a point amply insisted upon in the top answer here), is also tweaked for good concurrent operation, cf. Pugh's "Concurrent Maintenance of Skip Lists", although in a rather mild way. Nevertheless one newer/2009 paper "A Simple Optimistic skip-list Algorithm" by Herlihy et al., which proposes a supposedly simpler (than Pugh's) lock-based implementation of concurrent skip lists, criticized Pugh for not providing a proof of correctness convincing enough for them. Leaving aside this (maybe too pedantic) qualm, Herlihy et al. show that their simpler lock-based implementation of a skip list actually fails to scale as well as the JDK's lock-free implementation thereof, but only for high contention (50% inserts, 50% deletes and 0% lookups)... which Fraser and Harris didn't test at all; Fraser and Harris only tested 75% lookups, 12.5% inserts and 12.5% deletes (on skip list with ~500K elements). The simpler implementation of Herlihy et al. also comes close to the lock-free solution from the JDK in the case of low contention that they tested (70% lookups, 20% inserts, 10% deletes); they actually beat the lock-free solution for this scenario when they made their skip list big enough, i.e. going from 200K to 2M elements, so that the probability of contention on any lock became negligible. It would have been nice if Herlihy et al. had gotten over their hangup over Pugh's proof and tested his implementation too, but alas they didn't do that.
EDIT2: I found a (2015 published) motherlode of all benchmarks: Gramoli's "More Than You Ever Wanted to Know about Synchronization. Synchrobench, Measuring the Impact of the Synchronization on Concurrent Algorithms": Here's a an excerpted image relevant to this question.
"Algo.4" is a precursor (older, 2011 version) of Brown et al.'s mentioned above. (I don't know how much better or worse the 2014 version is). "Algo.26" is Herlihy's mentioned above; as you can see it gets trashed on updates, and much worse on the Intel CPUs used here than on the Sun CPUs from the original paper. "Algo.28" is ConcurrentSkipListMap from the JDK; it doesn't do as well as one might have hoped compared to other CAS-based skip list implementations. The winners under high-contention are "Algo.2" a lock-based algorithm (!!) described by Crain et al. in "A Contention-Friendly Binary Search Tree" and "Algo.30" is the "rotating skiplist" from "Logarithmic data structures for
multicores". "Algo.29" is the "No hot spot non-blocking skip
list". Be advised that Gramoli is a co-author to all three of these winner-algorithm papers. "Algo.27" is the C++ implementation of Fraser's skip list.
Gramoli's conclusion is that's much easier to screw-up a CAS-based concurrent tree implementation than it is to screw up a similar skip list. And based on the figures, it's hard to disagree. His explanation for this fact is:
The difficulty in designing a tree that is lock-free stems from
the difficulty of modifying multiple references atomically. Skip lists
consist of towers linked to each other through successor pointers and
in which each node points to the node immediately below it. They are
often considered similar to trees because each node has a successor
in the successor tower and below it, however, a major distinction is
that the downward pointer is generally immutable hence simplifying
the atomic modification of a node. This distinction is probably
the reason why skip lists outperform trees under heavy contention
as observed in Figure [above].
Overriding this difficulty was a key concern in Brown et al.'s recent work.
They have a whole separate (2013) paper "Pragmatic Primitives for Non-blocking Data Structures"
on building multi-record LL/SC compound "primitives", which they call LLX/SCX, themselves implemented using (machine-level) CAS. Brown et al. used this LLX/SCX building block in their 2014 (but not in their 2011) concurrent tree implementation.
I think it's perhaps also worth summarizing here the fundamental ideas
of the "no hot spot"/contention-friendly (CF) skip list. It addapts an essential idea from the relaxed RB trees (and similar concrrency friedly data structures): the towers are no longer built up immediately upon insertion, but delayed until there's less contention. Conversely, the deletion of a tall tower can create many contentions;
this was observed as far back as Pugh's 1990 concurrent skip-list paper, which is why Pugh introduced pointer reversal on deletion (a tidbit that Wikipedia's page on skip lists still doesn't mention to this day, alas). The CF skip list takes this a step further and delays deleting the upper levels of a tall tower. Both kinds of delayed operations in CF skip lists are carried out by a (CAS based) separate garbage-collector-like thread, which its authors call the "adapting thread".
The Synchrobench code (including all algorithms tested) is available at: https://github.com/gramoli/synchrobench.
The latest Brown et al. implementation (not included in the above) is available at http://www.cs.toronto.edu/~tabrown/chromatic/ConcurrentChromaticTreeMap.java Does anyone have a 32+ core machine available? J/K My point is that you can run these yourselves.
Also, in addition to the answers given (ease of implementation combined with comparable performance to a balanced tree). I find that implementing in-order traversal (forwards and backwards) is far simpler because a skip-list effectively has a linked list inside its implementation.
In practice I've found that B-tree performance on my projects has worked out to be better than skip-lists. Skip lists do seem easier to understand but implementing a B-tree is not that hard.
The one advantage that I know of is that some clever people have worked out how to implement a lock-free concurrent skip list that only uses atomic operations. For example, Java 6 contains the ConcurrentSkipListMap class, and you can read the source code to it if you are crazy.
But it's not too hard to write a concurrent B-tree variant either - I've seen it done by someone else - if you preemptively split and merge nodes "just in case" as you walk down the tree then you won't have to worry about deadlocks and only ever need to hold a lock on two levels of the tree at a time. The synchronization overhead will be a bit higher but the B-tree is probably faster.
From the Wikipedia article you quoted:
Θ(n) operations, which force us to visit every node in ascending order (such as printing the entire list) provide the opportunity to perform a behind-the-scenes derandomization of the level structure of the skip-list in an optimal way, bringing the skip list to O(log n) search time. [...]
A skip list, upon which we have not
recently performed [any such] Θ(n) operations, does not
provide the same absolute worst-case
performance guarantees as more
traditional balanced tree data
structures, because it is always
possible (though with very low
probability) that the coin-flips used
to build the skip list will produce a
badly balanced structure
EDIT: so it's a trade-off: Skip Lists use less memory at the risk that they might degenerate into an unbalanced tree.
Skip lists are implemented using lists.
Lock free solutions exist for singly and doubly linked lists - but there are no lock free solutions which directly using only CAS for any O(logn) data structure.
You can however use CAS based lists to create skip lists.
(Note that MCAS, which is created using CAS, permits arbitrary data structures and a proof of concept red-black tree had been created using MCAS).
So, odd as they are, they turn out to be very useful :-)
Skip Lists do have the advantage of lock stripping. But, the runt time depends on how the level of a new node is decided. Usually this is done using Random(). On a dictionary of 56000 words, skip list took more time than a splay tree and the tree took more time than a hash table. The first two could not match hash table's runtime. Also, the array of the hash table can be lock stripped in a concurrent way too.
Skip List and similar ordered lists are used when locality of reference is needed. For ex: finding flights next and before a date in an application.
An inmemory binary search splay tree is great and more frequently used.
Skip List Vs Splay Tree Vs Hash Table Runtime on dictionary find op