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I recently heard about ternary search in which we divide an array into 3 parts and compare. Here there will be two comparisons but it reduces the array to n/3. Why don't people use this much?
Actually, people do use k-ary trees for arbitrary k.
This is, however, a tradeoff.
To find an element in a k-ary tree, you need around k*ln(N)/ln(k) operations (remember the change-of-base formula). The larger your k is, the more overall operations you need.
The logical extension of what you are saying is "why don't people use an N-ary tree for N data elements?". Which, of course, would be an array.
A ternary search will still give you the same asymptotic complexity O(log N) search time, and adds complexity to the implementation.
The same argument can be said for why you would not want a quad search or any other higher order.
Searching 1 billion (a US billion - 1,000,000,000) sorted items would take an average of about 15 compares with binary search and about 9 compares with a ternary search - not a huge advantage. And note that each 'ternary compare' might involve 2 actual comparisons.
Wow. The top voted answers miss the boat on this one, I think.
Your CPU doesn't support ternary logic as a single operation; it breaks ternary logic into several steps of binary logic. The most optimal code for the CPU is binary logic. If chips were common that supported ternary logic as a single operation, you'd be right.
B-Trees can have multiple branches at each node; a order-3 B-tree is ternary logic. Each step down the tree will take two comparisons instead of one, and this will probably cause it to be slower in CPU time.
B-Trees, however, are pretty common. If you assume that every node in the tree will be stored somewhere separately on disk, you're going to spend most of your time reading from disk... and the CPU won't be a bottleneck, but the disk will be. So you take a B-tree with 100,000 children per node, or whatever else will barely fit into one block of memory. B-trees with that kind of branching factor would rarely be more than three nodes high, and you'd only have three disk reads - three stops at a bottleneck - to search an enormous, enormous dataset.
Reviewing:
Ternary trees aren't supported by hardware, so they run less quickly.
B-tress with orders much, much, much higher than 3 are common for disk-optimization of large datasets; once you've gone past 2, go higher than 3.
The only way a ternary search can be faster than a binary search is if a 3-way partition determination can be done for less than about 1.55 times the cost of a 2-way comparison. If the items are stored in a sorted array, the 3-way determination will on average be 1.66 times as expensive as a 2-way determination. If information is stored in a tree, however, the cost to fetch information is high relative to the cost of actually comparing, and cache locality means the cost of randomly fetching a pair of related data is not much worse than the cost of fetching a single datum, a ternary or n-way tree may improve efficiency greatly.
What makes you think Ternary search should be faster?
Average number of comparisons:
in ternary search = ((1/3)*1 + (2/3)*2) * ln(n)/ln(3) ~ 1.517*ln(n)
in binary search = 1 * ln(n)/ln(2) ~ 1.443*ln(n).
Worst number of comparisons:
in ternary search = 2 * ln(n)/ln(3) ~ 1.820*ln(n)
in binary search = 1 * ln(n)/ln(2) ~ 1.443*ln(n).
So it looks like ternary search is worse.
Also, note that this sequence generalizes to linear search if we go on
Binary search
Ternary search
...
...
n-ary search ≡ linear search
So, in an n-ary search, we will have "one only COMPARE" which might take upto n actual comparisons.
"Terinary" (ternary?) search is more efficient in the best case, which would involve searching for the first element (or perhaps the last, depending on which comparison you do first). For elements farther from the end you're checking first, while two comparisons would narrow the array by 2/3 each time, the same two comparisons with binary search would narrow the search space by 3/4.
Add to that, binary search is simpler. You just compare and get one half or the other, rather than compare, if less than get the first third, else compare, if less than get the second third, else get the last third.
Ternary search can be effectively used on parallel architectures - FPGAs and ASICs. For example if internal FPGA memory required for search is less than half of the FPGA resource, you can make a duplicate memory block. This would allow to simultaneously access two different memory addresses and do all comparisons in a single clock cycle. This is one of the reasons why 100MHz FPGA can sometimes outperform the 4GHz CPU :)
Here's some random experimental evidence that I haven't vetted at all showing that it's slower than binary search.
Almost all textbooks and websites on binary search trees do not really talk about binary trees! They show you ternary search trees! True binary trees store data in their leaves not internal nodes (except for keys to navigate). Some call these leaf trees and make the distinction between node trees shown in textbooks:
J. Nievergelt, C.-K. Wong: Upper Bounds for the Total Path Length of Binary Trees,
Journal ACM 20 (1973) 1–6.
The following about this is from Peter Brass's book on data structures.
2.1 Two Models of Search Trees
In the outline just given, we supressed an important point that at first seems
trivial, but indeed it leads to two different models of search trees, either of
which can be combined with much of the following material, but one of which
is strongly preferable.
If we compare in each node the query key with the key contained in the
node and follow the left branch if the query key is smaller and the right branch
if the query key is larger, then what happens if they are equal? The two models
of search trees are as follows:
Take left branch if query key is smaller than node key; otherwise take the
right branch, until you reach a leaf of the tree. The keys in the interior node
of the tree are only for comparison; all the objects are in the leaves.
Take left branch if query key is smaller than node key; take the right branch
if the query key is larger than the node key; and take the object contained
in the node if they are equal.
This minor point has a number of consequences:
{ In model 1, the underlying tree is a binary tree, whereas in model 2, each
tree node is really a ternary node with a special middle neighbor.
{ In model 1, each interior node has a left and a right subtree (each possibly a
leaf node of the tree), whereas in model 2, we have to allow incomplete
nodes, where left or right subtree might be missing, and only the
comparison object and key are guaranteed to exist.
So the structure of a search tree of model 1 is more regular than that of a tree
of model 2; this is, at least for the implementation, a clear advantage.
{ In model 1, traversing an interior node requires only one comparison,
whereas in model 2, we need two comparisons to check the three
possibilities.
Indeed, trees of the same height in models 1 and 2 contain at most approximately
the same number of objects, but one needs twice as many comparisons in model
2 to reach the deepest objects of the tree. Of course, in model 2, there are also
some objects that are reached much earlier; the object in the root is found
with only two comparisons, but almost all objects are on or near the deepest
level.
Theorem. A tree of height h and model 1 contains at most 2^h objects.
A tree of height h and model 2 contains at most 2^h+1 − 1 objects.
This is easily seen because the tree of height h has as left and right subtrees a
tree of height at most h − 1 each, and in model 2 one additional object between
them.
{ In model 1, keys in interior nodes serve only for comparisons and may
reappear in the leaves for the identification of the objects. In model 2, each
key appears only once, together with its object.
It is even possible in model 1 that there are keys used for comparison that
do not belong to any object, for example, if the object has been deleted. By
conceptually separating these functions of comparison and identification, this
is not surprising, and in later structures we might even need to define artificial
tests not corresponding to any object, just to get a good division of the search
space. All keys used for comparison are necessarily distinct because in a model
1 tree, each interior node has nonempty left and right subtrees. So each key
occurs at most twice, once as comparison key and once as identification key in
the leaf.
Model 2 became the preferred textbook version because in most textbooks
the distinction between object and its key is not made: the key is the object.
Then it becomes unnatural to duplicate the key in the tree structure. But in
all real applications, the distinction between key and object is quite important.
One almost never wishes to keep track of just a set of numbers; the numbers
are normally associated with some further information, which is often much
larger than the key itself.
You may have heard ternary search being used in those riddles that involve weighing things on scales. Those scales can return 3 answers: left is lighter, both are the same, or left is heavier. So in a ternary search, it only takes 1 comparison.
However, computers use boolean logic, which only has 2 answers. To do the ternary search, you'd actually have to do 2 comparisons instead of 1.
I guess there are some cases where this is still faster as earlier posters mentioned, but you can see that ternary search isn't always better, and it's more confusing and less natural to implement on a computer.
Theoretically the minimum of k/ln(k) is achieved at e and since 3 is closer to e than 2 it requires less comparisons. You can check that 3/ln(3) = 2.73.. and 2/ln(2) = 2.88.. The reason why binary search could be faster is that the code for it will have less branches and will run faster on modern CPUs.
I have just posted a blog about the ternary search and I have shown some results. I have also provided some initial level implementations on my git repo I totally agree with every one about the theory part of the ternary search but why not give it a try? As per the implementation that part is easy enough if you have three years of coding experience.
I found that if you have huge data set and you need to search it many times ternary search has an advantage.
If you think you can do better with a ternary search go for it.
Although you get the same big-O complexity (ln n) in both search trees, the difference is in the constants. You have to do more comparisons for a ternary search tree at each level. So the difference boils down to k/ln(k) for a k-ary search tree. This has a minimum value at e=2.7 and k=2 provides the optimal result.
I was wondering whether the AVL-tree or Red-black-tree balancing method (which happens at each node insertion by simple rotation) could be extended to any given binary tree at any stage of the binary tree construction.
I found that following AVL or Red-black rotating rules for any given unbalanced binary tree lead to tie situations in which you can choose at least two way to balance the tree. If you follow AVL rule you'll get same level difference which means there are at least two nodes which you can choose in order to balance the tree.
Apparently, any decision you make for balancing leads to a different balanced tree (perhaps one has a better situation in terms of level difference to the other). I tested this procedure on a simple example, I'm not sure if it's applicable in every case so I want to know:
if it could be generalized
if there is an example that proves that it could not be generalized?
Why even think about this situation?
It's fun to know!
suppose a binary tree data structure (and you don't want to use AVL or red-black). So you don't want to balance all the time but you may want to balance sometimes in the middle of construction of the tree.
A simple example:
Suppose our elements consist of {1, 3, 5, 7}
RR: rotate right
RL: rotate left
The given binary tree would look like this:
If you balance it with RR (pivot 5) then RL (pivot 3) it'll be as following (same tree if AVL approach used from beginning):
If you balance it with LL (pivot 5):
The Wikipedia page for Rendezvous hashing (Highest Random Weight "HRW") makes the following claim:
While it might first appear that the HRW algorithm runs in O(n) time, this is not the case. The sites can be organized hierarchically, and HRW applied at each level as one descends the hierarchy, leading to O(log n) running time, as in.[7]
I got a copy of the referenced paper, "Hash-Based Virtual Hierarchies for Scalable Location Service in Mobile Ad-hoc Networks." However the hierarchy referenced in their paper seems to be very specific to their application domain. As far as I can discern, there is no clear indication of how to generalize the method. The Wikipedia remark makes it seem like log is the general case.
I looked at a few general HRW implementations, and none of them seemed to support anything better than linear time. I gave it some thought, but I don't see any way to organize sites hierarchically without causing parent nodes to cause inefficient remapping when they drop out, significantly defeating the main advantage of HRW.
Does anybody know how to do this? Alternatively, is Wikipedia incorrect about there being a general way to implement this in log time?
Edit: Investigating mcdowella's approach:
OK, I think I see how this could work. But you need a little more than you've specified.
If you just do what you've described, you get in a situation where each leaf probably just has either zero or one nodes in it, and there's significant variance in how many nodes are in the leaf-most subtrees. If you swap using HRW at each level with just making the whole thing a regular search tree, you get exactly the same effect. Essentially, you've got an implementation of consistent hashing, along with its flaw of having unequal loading between buckets. Computing the combined weights, the defining implementation of HRW, adds nothing; you're better off just doing a search at each level, since it saves doing the hashes, and can be implemented without looping over each radix value
It's fixable though: you just need to be using HRW to choose from many alternatives at the final level. That is, you need all of the leaf nodes to be in large buckets, comparable to the number of replicas you'd have in consistent hashing. These large buckets should be approximately equally-loaded compared to each other, and then you're using HRW to choose the specific site. Since the bucket sizes are fixed, this is an O(n) algorithm, and we get all of the key HRW properties.
Honestly though, I think this is pretty questionable. It isn't so much an implementation of HRW, as it is just combining HRW with consistent hashing. I guess there's nothing wrong with that, and it might even be better than the usual technique of using replicas, in some cases. But I think it's misleading to state that HRW is log(n), if this is actually what the author meant.
Additionally, the original description is also questionable. You don't need to apply HRW at each level, and you shouldn't, as there is no advantage in doing so; you should do something fast (such as indexing), and just use HRW for the final choice.
Is this really the best we can do, or is there some other way to make HRW O(log(n))?
If you give each site a sufficiently long random id expressed in radix k (perhaps by hashing a non-random id) then you can associate the sites with leaves of a tree which has at most k descendants at each node. There is no need to associate any site with an internal node of the tree.
To work out where to store an item, use HRW to work out from the root of the tree down which way to branch at tree nodes, stopping when you reach a leaf, which is associated with a site. You can do this without having to communicate with any site until you work out which site you want to store the item at - all you need to know is the hashed ids of the sites to construct a tree.
Because sites are associated only with leaves there is no way an internal node of the tree can drop out, except if all of the sites associated with leaves under it drop out, at which point it will become irrelevant.
I don't buy the updated answer. There are two nice properties of HRWs that appear to get lost when you compare the weights of branches instead of all sites.
One is that you can pick the top-n sites instead of just the primary, and these should be randomly distributed. If you're descending into a single tree, the top-n sites will be near each other in the tree. This could be fixed by descending multiple times with different salts but that seems like a lot of extra work.
Two is that it is obvious what happens when a site is added or remove and only 1/|sites| of the data moves in the case of an add. If you modify the existing tree, it only affects the peer site. In the case of an add, the only data that moves is from the new peer of the added site. In the case of a delete, all the data that was at that site now moves to the former peer. If you instead recompute the tree, all of the data could move depending on the way the tree is constructed.
I think you can use the same "virtual node" approach normally used for consistent hashing. Suppose you have N physical nodes with IDs:
{n1,...,nN}.
Choose V, the number of virtual nodes per physical node, and generate a new list of IDs:
{n1v1,v1v2,...,n1vV
,n2v1,n2v2,...,n2vV
,...
,nNv1,nNv2,...,nNvV}.
Arrange these into the leaves of a fixed but randomized binary tree with labels on the internal nodes. These internal labels could be, for example, a concatenation of the labels of its child nodes.
To choose a physical node to store an object O at, start at the root and choose the branch with the higher hash H(label,O). Repeat the process until you reach a leaf. Store the object at the physical node corresponding to the virtual node at that leaf. This takes O(log(NV)) = O(log(N)+log(V)) = O(log(N)) steps (since V is constant).
If a physical node fails, the objects at that node are rehashed, skipping over subtrees with no active leaves.
One way to implement HRW rendezvous hashing in log time
One way to implement rendezvous hashing in O(log N), where N is the number of cache nodes:
Each file named F is cached in the cache node named C with the largest weight w(F,C), as is normal in rendezvous hashing.
First, we use a nonstandard hash function w() something like this:
w(F,C) = h(F) xor h(C).
where h() is some good hash function.
tree construction
Given some file named F, rather than calculate w(F,C) for every cache node -- which requires O(N) time for each file --
we pre-calculate a binary tree based only on the hashed names h(C) of the cache nodes;
a tree that lets us find the cache node with the maximum w(F,C) value in O(log N) time for each file.
Each leaf of the tree contains the name C of one cache node.
The root (at depth 0) of the tree points to 2 subtrees.
All the leaves where the most significant bit of h(C) is 0 are in the root's left subtree; all the leaves where the most significant bit of h(C) are 1 are in the root's right subtree.
The two children of the root node (at depth 1) deal with the next-most-significant bit of h(C).
And so on, with the interior nodes at depth D dealing with the D'th-most-significant bit of h(C).
With a good hash function, each step down from the root approximately halves the candidate cache nodes in the chosen subtree,
so we end up with a tree of depth roughly ln_2 N.
(If we end up with a tree with that is "too unbalanced",
somehow get everyone to agree on some different hash function from some universal hashing family rebuild the tree, before we add any files to the cache, until we get a tree that is "not too unbalanced").
Once the tree has been built, we never need to change it no matter how many file names F we later encounter.
We only change it when we add or remove cache nodes from the system.
filename lookup
For a filename F that happens to hash to h(F) = 0 (all zero bits),
we find the cache node with the highest weight (for that filename) by starting at the root and always taking the right subtree when possible.
If that leads us to an interior node that doesn't have a right subtree, then we take its left subtree.
Continue until we reach a node without a left or right subtree -- i.e., a leaf node that contains the name of the selected cache node C.
When looking up some other file named F, first we hash its name to get h(F), then
we start at the root and go right or left respectively (if possible) determined by the next bit in h(F) is 0 or 1.
Since the tree (by construction) is not "too unbalanced",
traversing the whole tree from the root to the leaf that contains the name of the chosen cache node C requires O(ln N) time in the worst case.
We expect that for a typical set of file names,
the hash function h(F) "randomly" chooses left or right at each depth of the tree.
Since the tree (by construction) is not "too unbalanced",
we expect each physical cache node to cache roughly the same number of files (within a multiple of 4 or so).
drop out effects
When some physical cache node fails,
everyone deletes the corresponding leaf node from their copy of this tree.
(Everyone also deletes every interior node that then has no leaf descendants).
This doesn't require moving around any files cached on any other cache node -- they still map to the same cache node they always did.
(The right-most leaf node in a tree is still the right-most leaf node in that tree, no matter how many other nodes in that tree are deleted).
For example,
....
\
|
/ \
| |
/ / \
| X |
/ \ / \
V W Y Z
With this O(log N) algorithm, when cache node X dies, leaf X is deleted from the tree, and all its files become (hopefully relatively evenly) distributed between Y and Z -- none of the files from X end up at V or W or any other cache node.
All the files that previously went to cache nodes V, W, Y, Z continue to go to those same cache nodes.
rebalancing after dropout
Many cache nodes failing or new cache nodes adding or both, may make the tree "too unbalanced".
Picking a new hash function is a big hassle after we've added a bunch of files to the cache, so rather than pick a new hash function like we did when initially constructing the tree, maybe it would be better to somehow rebalance the tree by remove a few nodes, rename them with some new semi-random names, and then add them back to the system.
Repeat until the system is no longer "too unbalanced".
(Start with the most unbalanced nodes -- the nodes cacheing the least amount of data).
comments
p.s.:
I think this may be pretty close to what mcdowella was thinking,
but with more details filled in to clarify that (a) yes, it is log(N) because it's a binary tree that is "not too unbalanced", (b) it doesn't have "replicas", and (c) when one cache node fails, it doesn't require any remapping of files that were not on that cache node.
p.p.s.:
I'm pretty sure that Wikipedia page is wrong to imply that typical implementations of rendezvous hashing occur in O(log N) time, where N is the number of cache nodes.
It seems to me (and I suspect the original designers of the hash as well) that the time it takes to (internally, without communicating) recalculate a hash against every node in the network is going to be insignificant and not worth worrying about compared to the time it takes to fetch data from some remote cache node.
My understanding is that rendezvous hashing is almost always implemented with a simple linear algorithm that uses O(N) time, where N is the number of cache nodes, every time we get a new filename F and want to choose the cache node for that file.
Such a linear algorithm has the advantage that it can use a "better" hash function than the above xor-based w(), so when some physical cache node dies, all the files that were cached on the now-dead node are expected to become evenly distributed among all the remaining nodes.
Relaxed Radix Balanced Trees (RRB-trees) are a generalization of immutable vectors (used in Clojure and Scala) that have 'effectively constant' indexing and update times. RRB-trees maintain efficient indexing and update but also allow efficient concatenation (log n).
The authors present the data structure in a way that I find hard to follow. I am not quite sure what the invariant is that each node maintains.
In section 2.5, they describe their algorithm. I think they are ensuring that indexing into the node will only ever require e extra steps of linear search after radix searching. I do not understand how they derived their formula for the extra steps, and I think perhaps I'm not sure what each of the variables mean (in particular "a total of p sub-tree branches").
What's how does the RRB-tree concatenation algorithm work?
They do describe an invariant in section 2.4 "However, as mentioned earlier
B-Trees nodes do not facilitate radix searching. Instead we chose
the initial invariant of allowing the node sizes to range between m
and m - 1. This defines a family of balanced trees starting with
well known 2-3 trees, 3-4 trees and (for m=32) 31-32 trees. This
invariant ensures balancing and achieves radix branch search in the
majority of cases. Occasionally a few step linear search is needed
after the radix search to find the correct branch.
The extra steps required increase at the higher levels."
Looking at their formula, it looks like they have worked out the maximum and minimum possible number of values stored in a subtree. The difference between the two is the maximum possible difference between the maximum and minimum number of values underneath a point. If you divide this by the number of values underneath a slot, you have the maximum number of slots you could be off by when you work out which slot to look at to see if it contains the index you are searching for.
#mcdowella is correct that's what they say about relaxed nodes. But if you're splitting and joining nodes, a range from m to m-1 means you will sometimes have to adjust up to m-1 (m-2?) nodes in order to add or remove a single element from a node. This seems horribly inefficient. I think they meant between m and (2 m) - 1 because this allows nodes to be split into 2 when they get too big, or 2 nodes joined into one when they are too small without ever needing to change a third node. So it's a typo that the "2" is missing in "2 m" in the paper. Jean Niklas L’orange's masters thesis backs me up on this.
Furthermore, all strict nodes have the same length which must be a power of 2. The reason for this is an optimization in Rich Hickey's Clojure PersistentVector. Well, I think the important thing is to pack all strict nodes left (more on this later) so you don't have to guess which branch of the tree to descend. But being able to bit-shift and bit-mask instead of divide is a nice bonus. I didn't time the get() operation on a relaxed Scala Vector, but the relaxed Paguro vector is about 10x slower than the strict one. So it makes every effort to be as strict as possible, even producing 2 strict levels if you repeatedly insert at 0.
Their tree also has an even height - all leaf nodes are equal distance from the root. I think it would still work if relaxed trees had to be within, say, one level of one-another, though not sure what that would buy you.
Relaxed nodes can have strict children, but not vice-versa.
Strict nodes must be filled from the left (low-index) without gaps. Any non-full Strict nodes must be on the right-hand (high-index) edge of the tree. All Strict leaf nodes can always be full if you do appends in a focus or tail (more on that below).
You can see most of the invariants by searching for the debugValidate() methods in the Paguro implementation. That's not their paper, but it's mostly based on it. Actually, the "display" variables in the Scala implementation aren't mentioned in the paper either. If you're going to study this stuff, you probably want to start by taking a good look at the Clojure PersistentVector because the RRB Tree has one inside it. The two differences between that and the RRB Tree are 1. the RRB Tree allows "relaxed" nodes and 2. the RRB Tree may have a "focus" instead of a "tail." Both focus and tail are small buffers (maybe the same size as a strict leaf node), the difference being that the focus will probably be localized to whatever area of the vector was last inserted/appended to, while the tail is always at the end (PerSistentVector can only be appended to, never inserted into). These 2 differences are what allow O(log n) arbitrary inserts and removals, plus O(log n) split() and join() operations.
I recently heard about ternary search in which we divide an array into 3 parts and compare. Here there will be two comparisons but it reduces the array to n/3. Why don't people use this much?
Actually, people do use k-ary trees for arbitrary k.
This is, however, a tradeoff.
To find an element in a k-ary tree, you need around k*ln(N)/ln(k) operations (remember the change-of-base formula). The larger your k is, the more overall operations you need.
The logical extension of what you are saying is "why don't people use an N-ary tree for N data elements?". Which, of course, would be an array.
A ternary search will still give you the same asymptotic complexity O(log N) search time, and adds complexity to the implementation.
The same argument can be said for why you would not want a quad search or any other higher order.
Searching 1 billion (a US billion - 1,000,000,000) sorted items would take an average of about 15 compares with binary search and about 9 compares with a ternary search - not a huge advantage. And note that each 'ternary compare' might involve 2 actual comparisons.
Wow. The top voted answers miss the boat on this one, I think.
Your CPU doesn't support ternary logic as a single operation; it breaks ternary logic into several steps of binary logic. The most optimal code for the CPU is binary logic. If chips were common that supported ternary logic as a single operation, you'd be right.
B-Trees can have multiple branches at each node; a order-3 B-tree is ternary logic. Each step down the tree will take two comparisons instead of one, and this will probably cause it to be slower in CPU time.
B-Trees, however, are pretty common. If you assume that every node in the tree will be stored somewhere separately on disk, you're going to spend most of your time reading from disk... and the CPU won't be a bottleneck, but the disk will be. So you take a B-tree with 100,000 children per node, or whatever else will barely fit into one block of memory. B-trees with that kind of branching factor would rarely be more than three nodes high, and you'd only have three disk reads - three stops at a bottleneck - to search an enormous, enormous dataset.
Reviewing:
Ternary trees aren't supported by hardware, so they run less quickly.
B-tress with orders much, much, much higher than 3 are common for disk-optimization of large datasets; once you've gone past 2, go higher than 3.
The only way a ternary search can be faster than a binary search is if a 3-way partition determination can be done for less than about 1.55 times the cost of a 2-way comparison. If the items are stored in a sorted array, the 3-way determination will on average be 1.66 times as expensive as a 2-way determination. If information is stored in a tree, however, the cost to fetch information is high relative to the cost of actually comparing, and cache locality means the cost of randomly fetching a pair of related data is not much worse than the cost of fetching a single datum, a ternary or n-way tree may improve efficiency greatly.
What makes you think Ternary search should be faster?
Average number of comparisons:
in ternary search = ((1/3)*1 + (2/3)*2) * ln(n)/ln(3) ~ 1.517*ln(n)
in binary search = 1 * ln(n)/ln(2) ~ 1.443*ln(n).
Worst number of comparisons:
in ternary search = 2 * ln(n)/ln(3) ~ 1.820*ln(n)
in binary search = 1 * ln(n)/ln(2) ~ 1.443*ln(n).
So it looks like ternary search is worse.
Also, note that this sequence generalizes to linear search if we go on
Binary search
Ternary search
...
...
n-ary search ≡ linear search
So, in an n-ary search, we will have "one only COMPARE" which might take upto n actual comparisons.
"Terinary" (ternary?) search is more efficient in the best case, which would involve searching for the first element (or perhaps the last, depending on which comparison you do first). For elements farther from the end you're checking first, while two comparisons would narrow the array by 2/3 each time, the same two comparisons with binary search would narrow the search space by 3/4.
Add to that, binary search is simpler. You just compare and get one half or the other, rather than compare, if less than get the first third, else compare, if less than get the second third, else get the last third.
Ternary search can be effectively used on parallel architectures - FPGAs and ASICs. For example if internal FPGA memory required for search is less than half of the FPGA resource, you can make a duplicate memory block. This would allow to simultaneously access two different memory addresses and do all comparisons in a single clock cycle. This is one of the reasons why 100MHz FPGA can sometimes outperform the 4GHz CPU :)
Here's some random experimental evidence that I haven't vetted at all showing that it's slower than binary search.
Almost all textbooks and websites on binary search trees do not really talk about binary trees! They show you ternary search trees! True binary trees store data in their leaves not internal nodes (except for keys to navigate). Some call these leaf trees and make the distinction between node trees shown in textbooks:
J. Nievergelt, C.-K. Wong: Upper Bounds for the Total Path Length of Binary Trees,
Journal ACM 20 (1973) 1–6.
The following about this is from Peter Brass's book on data structures.
2.1 Two Models of Search Trees
In the outline just given, we supressed an important point that at first seems
trivial, but indeed it leads to two different models of search trees, either of
which can be combined with much of the following material, but one of which
is strongly preferable.
If we compare in each node the query key with the key contained in the
node and follow the left branch if the query key is smaller and the right branch
if the query key is larger, then what happens if they are equal? The two models
of search trees are as follows:
Take left branch if query key is smaller than node key; otherwise take the
right branch, until you reach a leaf of the tree. The keys in the interior node
of the tree are only for comparison; all the objects are in the leaves.
Take left branch if query key is smaller than node key; take the right branch
if the query key is larger than the node key; and take the object contained
in the node if they are equal.
This minor point has a number of consequences:
{ In model 1, the underlying tree is a binary tree, whereas in model 2, each
tree node is really a ternary node with a special middle neighbor.
{ In model 1, each interior node has a left and a right subtree (each possibly a
leaf node of the tree), whereas in model 2, we have to allow incomplete
nodes, where left or right subtree might be missing, and only the
comparison object and key are guaranteed to exist.
So the structure of a search tree of model 1 is more regular than that of a tree
of model 2; this is, at least for the implementation, a clear advantage.
{ In model 1, traversing an interior node requires only one comparison,
whereas in model 2, we need two comparisons to check the three
possibilities.
Indeed, trees of the same height in models 1 and 2 contain at most approximately
the same number of objects, but one needs twice as many comparisons in model
2 to reach the deepest objects of the tree. Of course, in model 2, there are also
some objects that are reached much earlier; the object in the root is found
with only two comparisons, but almost all objects are on or near the deepest
level.
Theorem. A tree of height h and model 1 contains at most 2^h objects.
A tree of height h and model 2 contains at most 2^h+1 − 1 objects.
This is easily seen because the tree of height h has as left and right subtrees a
tree of height at most h − 1 each, and in model 2 one additional object between
them.
{ In model 1, keys in interior nodes serve only for comparisons and may
reappear in the leaves for the identification of the objects. In model 2, each
key appears only once, together with its object.
It is even possible in model 1 that there are keys used for comparison that
do not belong to any object, for example, if the object has been deleted. By
conceptually separating these functions of comparison and identification, this
is not surprising, and in later structures we might even need to define artificial
tests not corresponding to any object, just to get a good division of the search
space. All keys used for comparison are necessarily distinct because in a model
1 tree, each interior node has nonempty left and right subtrees. So each key
occurs at most twice, once as comparison key and once as identification key in
the leaf.
Model 2 became the preferred textbook version because in most textbooks
the distinction between object and its key is not made: the key is the object.
Then it becomes unnatural to duplicate the key in the tree structure. But in
all real applications, the distinction between key and object is quite important.
One almost never wishes to keep track of just a set of numbers; the numbers
are normally associated with some further information, which is often much
larger than the key itself.
You may have heard ternary search being used in those riddles that involve weighing things on scales. Those scales can return 3 answers: left is lighter, both are the same, or left is heavier. So in a ternary search, it only takes 1 comparison.
However, computers use boolean logic, which only has 2 answers. To do the ternary search, you'd actually have to do 2 comparisons instead of 1.
I guess there are some cases where this is still faster as earlier posters mentioned, but you can see that ternary search isn't always better, and it's more confusing and less natural to implement on a computer.
Theoretically the minimum of k/ln(k) is achieved at e and since 3 is closer to e than 2 it requires less comparisons. You can check that 3/ln(3) = 2.73.. and 2/ln(2) = 2.88.. The reason why binary search could be faster is that the code for it will have less branches and will run faster on modern CPUs.
I have just posted a blog about the ternary search and I have shown some results. I have also provided some initial level implementations on my git repo I totally agree with every one about the theory part of the ternary search but why not give it a try? As per the implementation that part is easy enough if you have three years of coding experience.
I found that if you have huge data set and you need to search it many times ternary search has an advantage.
If you think you can do better with a ternary search go for it.
Although you get the same big-O complexity (ln n) in both search trees, the difference is in the constants. You have to do more comparisons for a ternary search tree at each level. So the difference boils down to k/ln(k) for a k-ary search tree. This has a minimum value at e=2.7 and k=2 provides the optimal result.