Katz centrality

The List Update or the List Access problem is a simple model used in the study of competitive analysis of online algorithms. Given a set of items in a list where the cost of accessing an item is proportional to its distance from the head of the list, e.g. a Linked List, and a request sequence of accesses, the problem is to come up with a strategy of reordering the list so that the total cost of accesses is minimized. The reordering can be done at any time but incurs a cost, and the standard cost model includes a free transposition of the item being accessed anywhere ahead of its current position and a paid transposition of a unit cost for exchanging any two items in the list. Performance of algorithms depend on the construction of request sequences by adversaries under various Adversary models

An online algorithm for this problem has to reorder the elements and serve requests based only on the knowledge of previously requested items and hence its strategy may not have the optimum cost as compared to an offline algorithm that gets to see the entire request sequence and devise a complete strategy before serving the first request.

Adversary Models

An adversary is an entity that gets to choose the request sequence $\sigma$ for an algorithm ALG. Depending of whether $\sigma$ can be changed based on the strategy of ALG, adversaries are given various powers, and the performance of ALG is measured against these adversaries.

An oblivious adversary has to construct the entire request sequence $\sigma$ before running ALG, and pays the optimal offline price, $OPT(\sigma )$ which is compared against $ALG(\sigma )$

An adaptive online adversary gets to make the next request based on the previous results of the online algorithm, but pays for the request optimally and online.

An adaptive offline adversary gets to make the next request based on the previous results of the online algorithm, but pays the optimal offline cost.

Offline algorithms

Competitive analysis for many list update problems were carried out without any specific knowledge of the exact nature of the optimum offline algorithm (OPT). The best known algorithm runs in O(n2^l(l-1)!) time and O(l!) space where n is the length of the request sequence and l is the length of the list.

It is interesting to note that paid transpositions are in general necessary for optimum algorithms. Consider a list (a,b,c) where a is at the head of the list, and a request sequence c,b,c,b. An optimal offline algorithm using only free exchanges would cost 9 (3+3+2+1), whereas an optimal offline algorithm using only paid exchanges would cost 7 (Swap c and a in the beginning = 1+(1+2+1+2)). So, we cannot get away with just using free transpositions for the optimum offline algorithm.

The optimum list update problem was proven to be NP-Hard by Template:Harv.

Online algorithm

An online algorithm ALG has a competitive ratio c if for any input it performs at least as good as c times worse than OPT. i.e. if there exists an $\alpha \geq 0$ such that for all finite length request sequences $\sigma$, $ALG(\sigma )-c.OPT(\sigma )\leq \alpha$. Online algorithms can either be deterministic or randomized and it turns out that randomization in this case can truly help against oblivious adversaries.

Deterministic

Most deterministic algorithms are variants of these three algorithms :

MTF (Move to front)

After accessing an item move it to the front of the list without changing the order of other items

TRANS (Transpose)

After accessing an item, transpose it with the immediately preceding item.

FC (Frequency Count)

For each item maintain a frequency count of the number of accesses to it - when an element is accessed increase its frequency count and reorder the list in the decreasing order of frequencies.

Observe that all these use just free transpositions. It turns out that both TRANS and FC are not competitive. In a classic result using Potential method analysis Template:Harv proved that MTF is 2-competitive. The proof does not require the explicit knowledge of OPT but instead counts the number of inversions i.e. elements occurring in opposite order in the lists of MTF and OPT.

Any deterministic algorithm has a lower bound of $2-{\frac {2}{l+1}}$ for a list of length l, and MTF is actually the optimum deterministic list update algorithm. The type of adversary doesn't matter in the case of deterministic algorithms, because the adversary can run a copy of the deterministic algorithm on his own to precompute the most disastrous sequence.

Randomized

Consider the following simple randomized algorithm :

BIT

For every item in the list, maintain a bit. Initialize all the bits uniformly and randomly to 0 or 1. When an item is accessed, flip the bit, and if it is 1 move it to the front, else don't.

This algorithm is barely random - it makes all its random choices in the beginning and not during the run. It turns out that BIT breaks the deterministic bound - it is better than MTF against oblivious adversaries. It is 7/4-competitive. There are other randomized algorithms, like COMB, that perform better than BIT. Boris Teia proved a lower bound of 1.5 for any randomized list update algorithm.

Related Problems

The list update problem where elements maybe inserted and deleted is called the dynamic list update problem, as opposed to the static list update problem where only accessing list elements are allowed. The upper bound of $2-{\frac {2}{l+1}}$ holds for the dynamic model as well.

There are different cost models as well. In the usual full cost model, an access to an element located at a position i costs i, but the last comparison is inevitable for any algorithm, i.e. there are i-1 elements standing in the way of i. In the partial cost model, these final comparison costs totaling to the number of elements in the request sequence are ignored. For the costs of paid transpositions other than unity, P^d models are used.

See also

• Online algorithm
• Competitive analysis
• Cache algorithms

Notes

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References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010