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Retrieve Least Element, Elements are Dynamically-Ordered

I have collection of elements from which I need to retrieve the least/minimum element.
Normally I would use a PriorityQueue as they are designed specifically for this purpose, and offer O(log(n)) time for dequeing methods.
However, the elements in my array have a dynamic order, ie there natural order changes unpredictably over time. I assume PriorityQueue and other such Sorted collections sort an element when inserted, and then leave it. If this is so PriorityQueue wouldn't work for dynamically-ordered elements. Am I correct in my assumption? Or would PriorityQueue still be appropriate in this situation?
If I can't use PriorityQueue, Collections.min would be my next instinct. However this iterates over the entire collection, which presumably gives O(n) time. Is this the next best solution?
What is the best collection/method to use to retrieve the least element from a collection, given that the natural order of the elements may change unpredictably over time?
Edit:
The order of several elements changes per retrieval operation
Edit 2:
The compare algorithm remains constant, however the values of the fields which it assesses vary unpredictably between retrievals.

I think if the change is truly "unpredictable" you may be stuck with Collections.min(). However, maybe for some other collections like PriorityQueue you could try, before calling for the min.
Add something that you KNOW is the min.
Remove that
Then ask again for the "real" min and hope that your little kludge resorted things...
Alternatively, do you know if the order has changed over time? e.g. some OrderChangedEvent can be fired? If so, recreate the sorted whatever as needed.

A possible way to do this would be to extend PriorityQueue that contains a list as one of the fields. This list will store the java.lang.Object.hashCode() of each object. Whenever an add, peek, poll, offer, etc. is called on the PriorityQueue, the queue will check the hash codes of each element and make see if any element changed. If they have, it will re-order the elements that have changed. Then, it will replace the hashcodes of the changed elements in the list. I don't know how fast this will be, but I suspect it will be faster than O(n).

Without any further assumption on the operations you are going to do, you can't achieve better performance than with a PriorityQueue or another O(log(n))-insert collection (TreeSet , for example, but you lose the O(1)-peek).
As you correctly assumed Collections.min(Collection, Comparator) is a linear operation.
But it depends on how often you need to change the ordering: for example if you only need to change it once in a while and still keep a "standard" ordering, min() is a viable option, but if you need to switch ordering completely then you will probably be better off with reordering the queue/set (that is, traversing and adding all the elements in a new one), tough at a O(nlog(n)) cost. Using Collections.sort(List, Comparator) may be effective if you need a lot of reordering compared to inserts, but requires you to use a List.
Of course if you can make somewhat strong assumptions on the types of sorting you will need (for example, if it can be restricted to a part of the data) you could write your own collection.
Edit:
So you have a (more or less) finite number of orderings (never mind that it's the same type of comparison over different fields, it's different Comparators and that's what matters)? If that's the case, you can probably achieve best performance by using m queues that reference the same objects, each using a different comparator (the simplest method, really). This way you have:
constant time access
O(m*logn(n)) inserts (to insert in every queue)
O(m*n) removals (to remove from every queue)
no ordering costs (as it's handled by the inserts)
slightly larger memory cost (probably negligible)
additional O(n*log(n)) cost the first time a particolar ordering is requested
Supposing a value of m orders of magnitude smaller than n, this is comparable to optimal (single-ordering PriorityQueue) performance. For convenience, you can wrap this into a custom collection that takes a Comparator parameter on retrieval operations, and use it as a key for an HashMap of all the PriorityQueues.
Edit #2:
In that case, there is no better solution than running min() on every retrieval (unless you can make assumptions on the changes of the data); this also means that it's better to just use an ArrayList as the collection, since it has basically the lowest possible cost on every operation and you will not benefit from PriorityQueue's natural ordering anyway. You will end up with linear cost on retrieval (for min) and constant on insertion and deletion: this is optimal as there is no sorting algorithm that has less than Ω(n) and Θ(nlog n) anyway.
As a side note, ordered collections work on the assumption that values will not change after insertion; this is because there is no cost-effective way to monitor the changes nor to reorder them "in place".

Can't you use a java TreeSet which keeps the collection sorted at all times. You need to implement the Comparable interface on your objects to do so. Checkout http://docs.oracle.com/javase/1.4.2/docs/api/java/util/TreeSet.html

Java collection insertion: Set vs. List

I'm thinking about filling a collection with a large amount of unique objects.
How is the cost of an insert in a Set (say HashSet) compared to an List (say ArrayList)?
My feeling is that duplicate elimination in sets might cause a slight overhead.

There is no "duplicate elimination" such as comparing to all existing elements. If you insert into hash set, it's really a dictionary of items by hash code. There's no duplicate checking unless there already are items with the same hash code. Given a reasonable (well-distributed) hash function, it's not that bad.
As Will has noted, because of the dictionary structure HashSet is probably a bit slower than an ArrayList (unless you want to insert "between" existing elements). It also is a bit larger. I'm not sure that's a significant difference though.

You're right: set structures are inherently more complex in order to recognize and eliminate duplicates. Whether this overhead is significant for your case should be tested with a benchmark.
Another factor is memory usage. If your objects are very small, the memory overhead introduced by the set structure can be significant. In the most extreme case (TreeSet<Integer> vs. ArrayList<Integer>) the set structure can require more than 10 times as much memory.

If you're certain your data will be unique, use a List. You can use a Set to enforce this rule.
Sets are faster than Lists if you have a large data set, while the inverse is true for smaller data sets. I haven't personally tested this claim.
Which type of List?
Also, consider which List to use. LinkedLists are faster at adding, removing elements.
ArrayLists are faster at random access (for loops, etc), but this can be worked around using the Iterator of a LinkedList. ArrayLists are are much faster at: list.toArray().

You have to compare concrete implementations (for example HashSet with ArrayList), because the abstract interfaces Set/List don't really tell you anything about performance.
Inserting into a HashSet is a pretty cheap operation, as long as the hashCode() of the object to be inserted is sane. It will still be slightly slower than ArrayList, because it's insertion is a simple insertion into an array (assuming you insert in the end and there's still free space; I don't factor in resizing the internal array, because the same cost applies to HashSet as well).

If the goal is the uniqueness of the elements, you should use an implementation of the java.util.Set interface. The class java.util.HashSet and java.util.LinkedHashSet have O(alpha) (close to O(1) in the best case) complexity for insert, delete and contains check.
ArrayList have O(n) for object (not index) contains check (you have to scroll through the whole list) and insertion (if the insertion is not in tail of the list, you have to shift the whole underline array).
You can use LinkedHashSet that preserve the order of insertion and have the same potentiality of HashSet (takes up only a bit more of memory).

I don't think you can make this judgement simply on the cost of building the collection. Other things that you need to take into account are:
Is the input dataset ordered? Is there a requirement that the output data structure preserves insertion order?
Is there a requirement that the output data structure is ordered (or reordered) based on element values?
Will the output data structure be subsequently modified? How?
Is there a requirement that the output data structure is duplicate free if other elements are added subsequently?
Do you know how many elements are likely to be in the input dataset?
Can you measure the size of the input dataset? (Or is it provided via an iterator?)
Does space utilization matter?
These can all effect your choice of data structure.

Java List:
If you don't have such requirement that you have to keep duplicate or not. Then you can use List instead of Set.
List is an interface in Collection framework. Which extends Collection interface. and ArrayList, LinkedList is the implementation of List interface.
When to use ArrayList or LinkedList
ArrayList: If you have such requirement that in your application mostly work is accessing the data. Then you should go for ArrayList. because ArrayList implements RtandomAccess interface which is Marker Interface. because of Marker interface ArrayList have capability to access the data in O(1) time. and you can use ArrayList over LinkedList where you want to get data according to insertion order.
LinkedList: If you have such requirement that your mostly work is insertion or deletion. Then you should use LinkedList over the ArrayList. because in LinkedList insertion and deletion happen in O(1) time whereas in ArrayList it's O(n) time.
Java Set:
If you have requirement in your application that you don't want any duplicates. Then you should go for Set instead of List. Because Set doesn't store any duplicates. Because Set works on the principle of Hashing. If we add object in Set then first it checks object's hashCode in the bucket if it's find any hashCode present in it's bucked then it'll not add that object.

When do you know when to use a TreeSet or LinkedList?

What are the advantages of each structure?
In my program I will be performing these steps and I was wondering which data structure above I should be using:
Taking in an unsorted array and
adding them to a sorted structure1.
Traversing through sorted data and removing the right one
Adding data (never removing) and returning that structure as an array

When do you know when to use a TreeSet or LinkedList? What are the advantages of each structure?
In general, you decide on a collection type based on the structural and performance properties that you need it to have. For instance, a TreeSet is a Set, and therefore does not allow duplicates and does not preserve insertion order of elements. By contrast a LinkedList is a List and therefore does allow duplicates and does preserve insertion order. On the performance side, TreeSet gives you O(logN) insertion and deletion, whereas LinkedList gives O(1) insertion at the beginning or end, and O(N) insertion at a selected position or deletion.
The details are all spelled out in the respective class and interface javadocs, but a useful summary may be found in the Java Collections Cheatsheet.
In practice though, the choice of collection type is intimately connected to algorithm design. The two need to be done in parallel. (It is no good deciding that your algorithm requires a collection with properties X, Y and Z, and then discovering that no such collection type exists.)
In your use-case, it looks like TreeSet would be a better fit. There is no efficient way (i.e. better than O(N^2)) to sort a large LinkedList that doesn't involve turning it into some other data structure to do the sorting. There is no efficient way (i.e. better than O(N)) to insert an element into the correct position in a previously sorted LinkedList. The third part (copying to an array) works equally well with a LinkedList or TreeSet; it is an O(N) operation in both cases.
[I'm assuming that the collections are large enough that the big O complexity predicts the actual performance accurately ... ]

The genuine power and advantage of TreeSet lies in interface it realizes - NavigableSet
Why is it so powerfull and in which case?
Navigable Set interface add for example these 3 nice methods:
headSet(E toElement, boolean inclusive)
tailSet(E fromElement, boolean inclusive)
subSet(E fromElement, boolean fromInclusive, E toElement, boolean toInclusive)
These methods allow to organize effective search algorithm(very fast).
Example: we need to find all the names which start with Milla and end with Wladimir:
TreeSet<String> authors = new TreeSet<String>();
authors.add("Andreas Gryphius");
authors.add("Fjodor Michailowitsch Dostojewski");
authors.add("Alexander Puschkin");
authors.add("Ruslana Lyzhichko");
authors.add("Wladimir Klitschko");
authors.add("Andrij Schewtschenko");
authors.add("Wayne Gretzky");
authors.add("Johann Jakob Christoffel");
authors.add("Milla Jovovich");
authors.add("Taras Schewtschenko");
System.out.println(authors.subSet("Milla", "Wladimir"));
output:
[Milla Jovovich, Ruslana Lyzhichko, Taras Schewtschenko, Wayne Gretzky]
TreeSet doesn't go over all the elements, it finds first and last elemenets and returns a new Collection with all the elements in the range.

TreeSet:
TreeSet uses Red-Black tree underlying. So the set could be thought as a dynamic search tree. When you need a structure which is operated read/write frequently and also should keep order, the TreeSet is a good choice.

If you want to keep it sorted and it's append-mostly, TreeSet with a Comparator is your best bet. The JVM would have to traverse the LinkedList from the beginning to decide where to place an item. LinkedList = O(n) for any operations, TreeSet = O(log(n)) for basic stuff.

The most important point when choosing a data structure are its inherent limitations. For example if you use TreeSet to store objects and during run-time your algorithm changes attributes of these objects which affect equal comparisons while the object is an element of the set, get ready for some strange bugs.
The Java Doc for Set interface state that:
Note: Great care must be exercised if mutable objects are used as set elements. The behavior of a set is not specified if the value of an object is changed in a manner that affects equals comparisons while the object is an element in the set. A special case of this prohibition is that it is not permissible for a set to contain itself as an element.
Interface Set Java Doc

What are the pros and cons of a TreeSet [closed]

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Just wondering what the pros and cons of a TreeSet is, if anyone could tell me please? Thanks!

One of the Collection classes. It lets you access the elements in your collection by key, or sequentially by key. It has considerably more overhead than ArrayList or HashMap. Use HashSet when you don’t need sequential access, just lookup by key. Use an ArrayList and use Arrays. sort if you just want the elements in order. TreeSet keeps the elements in order at all times. With ArrayList you just sort when you need to.
With TreeSets the key must be embedded in the object you store in the collection. Often you might have TreeSet of Strings. All you can do then is tell if a given String is in the Set. It won’t find you an associated object he way a Treemap will. With a TreeMap the keys and the objects they are associated with are separate.
TreeSet and its brother TreeMap oddly have nothing to do with representing trees. Internally they use a tree organisation to give you an alphabetically sorted Set/Map, but you have no control over links between parents and children.
Internally TreeSet uses red-black trees. There is no need to presort the data to get a well-balanced tree. On the other hand, if the data are sorted (ascending or descending), it won’t hurt as it does with some other types of tree.
If you don’t supply a Comparator to define the ordering you want, TreeSet requires a Comparable implementation on the item class to define the natural order.

Cons: One pitfall with TreeSet is that it implements the Set interface in an unexpected way.
If a TreeSet contains object a, then object b is considered part of the set if a.compareTo(b) returns 0, even if a.equals(b) is false, so if compareTo and equals isn't implemented in a consistent way, you are in for a bad ride.
This is especially a problem when a method returns a Set, and you don't know if the implementation is a TreeSet or, for instance, a HashSet.
The lesson to learn here is, always avoid implementing compareTo and equals inconsistently. If you need to order objects in a way that is inconsistent with equals, use a Comparator.

TreeSet:
Pros: sorted, based on a red/black tree algorithm, provides O(log(N)) complexity for operations.
Cons: value must either be Comparable or you need to provide Comparator in the constructor. Moreover, the HashSet implementation provides better performance as it provides ~O(1) complexity.

TreeSet fragments memory and has additional memory overheads. You can look at the sources and calculate amount of additional memory and amount of additional objects it creates. Of course it depends on the nature of stored objects and you also can suspect me to be paranoiac about memory :) but it's better to not spend it here and there - you have GC, you have cache misses and all of these things are slooow.
Often you can use PriorityQueue instead of TreeSet. And in your typical use case it's better just to sort the array of strings.

I guess this datastructure would be using binary tree to maintain data so that ascending order retrieval is possible. In that case, if it tries to keep the tree in balance then the remove operation would be bit costly.

Which Java collection's Linked list adds in O(1) when given an element?

i need to insert into a very large LinkedList, whose elements i hold in a fast-access HashMap.
it's important to keep the list in order (which is not the natural order of the keys).
i thought it would be possible to hash the linked list nodes, then insert directly on the node (getting the node from the map + insert in a linked list == constant time).
however, i couldn't find any Java collection that would do that or similar...
i'm currently using LinkedHashMap, which doesn't meet the requirements above.
thanks, asaf :-)

If the LinkedList should be sorted after each insertion, I doubt you will be able to find such a data structure, as it implies that you would get a sorting algorithm with time complexity O(n), which has been proven impossible. (The lowest bound on sorting is O(n log n).) The best you could get on insertion is O(log n).
Then you can use the TreeMap data structure.

Use a TreeSet or TreeMap. Inserts are O(log(n)) but remember that means LOG. So if you have 4 billion entries the runtime is O(32). If you have 264 entries, then an insert takes O(64), so it's not really a big deal.

As it's "important to keep the list in order" your effectively doing an insertion sort which has a best-case-performance of O(n). Additionally, you must not confuse hashing with sorting - as the order of a hash isn't defined as it depends on the size of the underlying hashtable. (Using a hash to insert would only be useful if you know the predecessor or successor of the node you want to insert)

When you say that you must keep your list in order - but that it is not the natural order of the keys, I hear you saying that you must preserve insertion order.
But then I don't know why LinkedHashMap doesn't meet your requirements.
Can you explain what LinkedHashMap fails to do?