I want to implement a Set in C.
Is it OK to use a linked list, when creating the SET, or should I use another approach ?
How do you usually implement your own set (if needed).
NOTE:
If I use the Linked List approach, I will probably have the following complexities for Set my operations:
init : O(1);
destroy: O(n);
insert: O(n);
remove: O(n);
union: O(n*m);
intersection: O(n*m);
difference: O(n*m);
ismember: O(n);
issubset: O(n*m);
setisequal: O(n*m);
O(n*m) seems may be a little to big especially for huge data... Is there a way to implement my Set more efficient ?
Sets are typically implemented either as red-black trees (which requires the elements to have a total order), or as an automatically-resizing hashtable (which requires a hash function).
The latter is typically implemented by having the hashtable double in size and reinserting all elements when a certain capacity threshold (75% works well) is exceeded. This means that inidividual insert operations can be O(n), but when amortized over many operations, it's actually O(1).
std::set is often implemented as a red black tree: http://en.wikipedia.org/wiki/Red-black_tree
This approach will give you much better complexity on all the listed operations.
I have used Red-Black trees in the past to build sets.
Here are the time complexities from the Wikipedia article.
Space O(n)
Search O(log n)
Insert O(log n)
Delete O(log n)
There are lot of ways for set implementation. Here are some of them. Besides MSDN have very good article on it.
Since you already have a linked list implemented, the easiest is a skip list. If you want to use balanced trees, the easiest in my opinion is a treap. These are randomized data structures, but generally they are just as efficient as their deterministic counterparts, if not more (and a skip list can be made deterministic).
Related
PREMISE
So lately i have been thinking of a problem that is common to databases: trying to optimize insertion, search, deletion and update of data.
Usually i have seen that most of the databases nowadays use the BTree or B+Tree to solve such a problem, but they are usually used to store data inside the disk and i wanted to work with in-memory data, so i thought about using the AVLTree (the difference should be minimal because the purpose of the BTrees is kind of the same of the AVLTree but the implementation is different and so are the effects).
Before continuing with the reasoning behind this i would like to get in a deeper level of what i am trying to solve.
So in a modern database data stored in a table with a PRIMARY KEY which tends to be INDEXED (i am not very experienced in indexing so what i will say is basic reasoning i put into this problem), usually the PRIMARY KEY is an increasing number (even though nowadays is a bad practice) starting from 1.
Using normally an AVLTree should be more then enough to solve the problem cause this particular tree is always balanced and offers O(log2(n)) operations, BUT i wanted to reach this on a deeper level trying to optimize it even more then needed.
THEORY
So as the title of the question suggests i am trying to optimize the AVLTree merging it with a Btree.
Basically every node of this new Tree is lets say an array of ten elements every node as also the corresponding height in the tree and every element of the array is ordered ascending.
INSERTION
The insertion initally fills the array of the root node when the root node is full it generates the left and right children which also contains an array of 10 elements.
Whenever a new node is added the Tree autorebalances the nodes based on the first key of the vectors of the left and right child using also their height (note that this is actually how the AVLTree behaves but the AVLTree only has 2 nodes and no vector just the values).
SEARCH
Searching an element works this way: staring from the root we compare the value we are searching K with the first and last key of the array of the current node if the value is in between, we know that it surely will be in the array of the current node so we can start using a binarySearch with O(log2(n)) complexity into this array of ten elements, otherise we go on the left if the key we are searcing is smaller then the first key or we go to the right if it is bigger.
DELETION
The same of the searching but we delete the value.
UPDATE
The same of the searching but we update the value.
CONCLUSION
If i am not wrong this should have a complexity of O(log10(log2(10))) which is always logarithmic so we shouldn't care about this optimization, but in my opinion this could make the height of the tree so much smaller while providing also quick time on the search.
B tree and B+ tree are indeed used for disk storage because of the block design. But there is no reason why they could not be used also as in-memory data structure.
The advantages of a B tree include its use of arrays inside a single node. Look-up in a limited vector of maybe 10 entries can be very fast.
Your idea of a compromise between B tree and AVL would certainly work, but be aware that:
You need to perform tree rotations like in AVL in order to keep the tree balanced. In B trees you work with redistributions, merges and splits, but no rotations.
Like with AVL, the tree will not always be perfectly balanced.
You need to describe what will be done when a vector is full and a value needs to be added to it: the node will have to split, and one half will have to be reinjected as a leaf.
You need to describe what will be done when a vector gets a very low fill-factor (due to deletions). If you leave it like that, the tree could degenerate into an AVL tree where every vector only has 1 value, and then the additional vector overhead will make it less efficient than a genuine AVL tree. To keep the fill-factor of a vector above a minimum you cannot easily apply the redistribution mechanism with a sibling node, as would be done in B-trees. It would work with leaf nodes, but not with internal nodes. So this needs to be clarified...
You need to describe what will be done when a value in a vector is updated. Of course, you would insert it in its sorted position: but if it becomes the first or last value in that vector, this may violate the order with regards to left and right children, and so also there you may need to define more precisely the algorithm.
Binary search in a vector of 10 may be overkill: a simple left-to-right scan may be faster, as CPUs are optimised to read consecutive memory. This does not impact the time complexity, since we set that the vector size is limited to 10. So we are talking about doing either at most 4 comparisons (3-4 on average depending on binary search implementation) or at most 10 comparisons (5 on average).
If I am not wrong this should have a complexity of O(log10(log2(n))) which is always logarithmic
Actually, if that were true, it would be sub-logarithmic, i.e. O(loglogn). But there is a mistake here. The binary search in a vector is not related to n, but to 10. Also, this work comes in addition to finding the node with that vector. So it is not a logarithm of a logarithm, but a sum of logarithms:
O(log10n + log210) = O(log n)
Therefore the time complexity is no different than the one for AVL or B-tree -- provided that the algorithm is completed with the missing details, keeping within the logarithmic complexity.
You should maybe also consider to implement a pure B tree or B+ tree: that way you also benefit from some of the advantages that neither the AVL, nor the in-between structure has:
The leaves of the tree are all at the same level
No rotations are needed
The tree height only changes at one spot: the root.
B+ trees provide a very fast mean for iterating all values in their order.
I have an array of structs, where each struct is a 2D position (pair of 32-bit values). This array is for tracking points of interest on a map.
struct Point {
int x;
int y;
};
// ...
struct Point pointsOfInterest[1024];
The problem is, those points of interest are constantly changing, meaning the entries in the array are very frequently being added or removed. On top of that, each reported point of interest can possibly already exist in the array, so I can't blindly add new ones without checking whether they already exist.
At the moment the array is unsorted (new entries added to the end, swap and pop to remove), and I iterate over the entire list to find entries for removal or duplication check. I'd like to know what my options are for speeding up this process.
In other languages, this is where I break out a dictionary or hash set. Neither exist in C, so I have to weigh the complexity of adding something like that.
I've considered sorting the list (i.e. first by X, then Y). But given the frequency of updates, I feel like I'll be thrashing the table far more than when iterating. But my knowledge of sorting algorithms is minimal.
Would a binary tree of some sort be any better here? Or would I again be spending all of my time re-balancing the tree?
Theoretically, given the (perceived) complexity of these algorithms, is there a threshold below which a linear search remains a viable option?
I'm assuming this is a known solved problem, so I'm hoping to get pointed in the right direction before I spend a lot of time reinventing the wheel and testing possible solutions.
Apart from trivial cases, it's often extremely hard to predict where the performance gains are. That's why you should benchmark your code before and after changes. Also profile your code to find where it spends most time.
In other languages, this is where I break out a dictionary or hash set. Neither exist in C, so I have to weigh the complexity of adding something like that.
TBH, it's not that complicated to implement. If you need the performance, it's a no brainer. But it's not guaranteed that it will be faster.
I've considered sorting the list (i.e. first by X, then Y). But given the frequency of updates, I feel like I'll be thrashing the table far more than when iterating. But my knowledge of sorting algorithms is minimal.
It's very likely that this is not optimal. But you can try it out. And you don't need to do a complete sort. Just do a binary search and move everything that comes after.
Would a binary tree of some sort be any better here? Or would I again be spending all of my time re-balancing the tree?
Only one way to find out. Try it and benchmark.
Theoretically, given the (perceived) complexity of these algorithms, is there a threshold below which a linear search remains a viable option?
I'm sure there are, but these always have to be balanced with reality. Like cache misses that can have a great impact on performance. One thing that might improve cache-friendlyness could be changing
struct Point {
int x;
int y;
};
struct Point pointsOfInterest[1024];
to
int pointsOfInterest[2][1024];
And use the first index for x or y. Might work, depending on what you're doing with the data. I guess it would not work in your case, but it could speed up a function that's only loops over one dimension.
Do any of the sort algorithms sort a given array of integers such that in case a[i]=a[j], it is considered that a[i]>a[j] if i>j?
You are probably looking for stable sorting algorithms. Stable sorting algorithms maintain the relative order of records with equal keys (i.e. values).
Concept explanation -
https://en.wikipedia.org/wiki/Sorting_algorithm#Stability
Algorithms which are stable -
https://en.wikipedia.org/wiki/Sorting_algorithm#Comparison_of_algorithms
Sorting algorithms defines the general steps to sort the input. Such edge cases are handled by actual implementation depending on the requirement of the user.
I have coded a few sorting methods in C and I would like to find the input size at which the program is optimal (i.e.) profiling each algorithm. But how do I do this? I know to time each method, but I don't know how I can find the size at which it is 'optimal'.
It depends on some factors:
Data behaviour: is your data already partially sorted? or it is very random?
Data size: for a big input (say 1 thousand or more) you can assure that O(N^2) sorting methods will lose to O(N*log(N)) methods..
Data structure of the data: is it array or list or ?. Sorting method with non sequential access to data will be slower for something like list
So the answer is by empirically running your program with some real data you will likely handle combined by varying in the input size.
When a slower method (like O(N^2)) gets beaten by some faster method (like O(N*log(N))) when input size is > X then you can say that the slower method is 'empirically optimal' for input size <= X (the value depends on the characteristics of the input data).
Sort algorithms do not have a single number at which they are optimal.
For pure execution time, almost every sort algorithm will be fastest on a set of 2 numbers, but that it not useful in most cases.
Some sort algorithms may work more efficiently on smaller data sets, but that does not mean they are 'optimal' at that size.
Some sorts may also work better on other characteristics of the data. There are sorts that can be extremely efficient if the data is almost sorted already, but may be very slow if it is not. Others will run the same on any set of a given size.
It is more useful to look at the Big O of the sort (such as O(n^2), O(n log n) etc) and any special properties the sort has, such as operating on nearly sorted data.
To find the input size at which the program is optimal (by which I assume you mean the fastest, or for which the sorting algorithm requires the fewest comparisons) you will have to test it against various inputs and graph the independent axis (input size) against the dependent axis (runtime) and find the minimum.
After implementing most of the common and needed functions for my Graph implementation, I realized that a couple of functions (remove vertex, search vertex and get vertex) don't have the "best" implementation.
I'm using adjacency lists with linked lists for my Graph implementation and I was searching one vertex after the other until it finds the one I want. Like I said, I realized I was not using the "best" implementation. I can have 10000 vertices and need to search for the last one, but that vertex could have a link to the first one, which would speed up things considerably. But that's just an hypothetical case, it may or may not happen.
So, what algorithm do you recommend for search lookup? Our teachers talked about Breadth-first and Depth-first mostly (and Dikjstra' algorithm, but that's a completely different subject). Between those two, which one do you recommend?
It would be perfect if I could implement both but I don't have time for that, I need to pick up one and implement it has the first phase deadline is approaching...
My guess, is to go with Depth-first, seems easier to implement and looking at the way they work, it seems a best bet. But that really depends on the input.
But what do you guys suggest?
If you’ve got an adjacency list, searching for a vertex simply means traversing that list. You could perhaps even order the list to decrease the needed lookup operations.
A graph traversal (such as DFS or BFS) won’t improve this from a performance point of view.
Finding and deleting nodes in a graph is a "search" problem not a graph problem, so to make it better than O(n) = linear search, BFS, DFS, you need to store your nodes in a different data structure optimized for searching or sort them. This gives you O(log n) for find and delete operations. Candidatas are tree structures like b-trees or hash tables. If you want to code the stuff yourself I would go for a hash table which normally gives very good performance and is reasonably easy to implement.
I think BFS would usually be faster an average. Read the wiki pages for DFS and BFS.
The reason I say BFS is faster is because it has the property of reaching nodes in order of their distance from your starting node. So if your graph has N nodes and you want to search for node N and node 1, which is the node you start your search form, is linked to N, then you will find it immediately. DFS might expand the whole graph before this happens however. DFS will only be faster if you get lucky, while BFS will be faster if the nodes you search for are close to your starting node. In short, they both depend on the input, but I would choose BFS.
DFS is also harder to code without recursion, which makes BFS a bit faster in practice, since it is an iterative algorithm.
If you can normalize your nodes (number them from 1 to 10 000 and access them by number), then you can easily keep Exists[i] = true if node i is in the graph and false otherwise, giving you O(1) lookup time. Otherwise, consider using a hash table if normalization is not possible or you don't want to do it.
Depth-first search is best because
It uses much less memory
Easier to implement
the depth first and breadth first algorithms are almost identical, except for the use of a stack in one (DFS), a queue in the other (BFS), and a few required member variables. Implementing them both shouldn't take you much extra time.
Additionally if you have an adjacency list of the vertices then your look up with be O(V) anyway. So little to nothing will be gained via using one of the two other searches.
I'd comment on Konrad's post but I can't comment yet so... I'd like to second that it doesn't make a difference in performance if you implement DFS or BFS over a simple linear search through your list. Your search for a particular node in the graph doesn't depend on the structure of the graph, hence it's not necessary to confine yourself to graph algorithms. In terms of coding time, the linear search is the best choice; if you want to brush up your skills in graph algorithms, implement DFS or BFS, whichever you feel like.
If you are searching for a specific vertex and terminating when you find it, I would recommend using A*, which is a best-first search.
The idea is that you calculate the distance from the source vertex to the current vertex you are processing, and then "guess" the distance from the current vertex to the target.
You start at the source, calculate the distance (0) plus the guess (whatever that might be) and add it to a priority queue where the priority is distance + guess. At each step, you remove the element with the smallest distance + guess, do the calculation for each vertex in its adjacency list and stick those in the priority queue. Stop when you find the target vertex.
If your heuristic (your "guess") is admissible, that is, if it's always an under-estimate, then you are guaranteed to find the shortest path to your target vertex the first time you visit it. If your heuristic is not admissible, then you will have to run the algorithm to completion to find the shortest path (although it sounds like you don't care about the shortest path, just any path).
It's not really any more difficult to implement than a breadth-first search (you just have to add the heuristic, really) but it will probably yield faster results. The only hard part is figuring out your heuristic. For vertices that represent geographical locations, a common heuristic is to use an "as-the-crow-flies" (direct distance) heuristic.
Linear search is faster than BFS and DFS. But faster than linear search would be A* with the step cost set to zero. When the step cost is zero, A* will only expand the nodes that are closest to a goal node. If the step cost is zero then every node's path cost is zero and A* won't prioritize nodes with a shorter path. That's what you want since you don't need the shortest path.
A* is faster than linear search because linear search will most likely complete after O(n/2) iterations (each node has an equal chance of being a goal node) but A* prioritizes nodes that have a higher chance of being a goal node.