ATLAS of Finite Groups

In computational complexity and communication complexity theories the decision tree model is the model of computation or communication in which an algorithm or communication process is considered to be basically a decision tree, i.e., a sequence of branching operations based on comparisons of some quantities, the comparisons being assigned the unit computational cost.

The branching operations are called "tests" or "queries". In this setting the algorithm in question may be viewed as a computation of a Boolean function $f:\{0,1\}^{n}\rightarrow \{0,1\}$ where the input is a series of queries and the output is the final decision. Every query is dependent on previous queries.

Several variants of decision tree models may be considered, depending on the complexity of the operations allowed in the computation of a single comparison and the way of branching.

Decision trees models are instrumental in establishing lower bounds for computational complexity for certain classes of computational problems and algorithms: the lower bound for worst-case computational complexity is proportional to the largest depth among the decision trees for all possible inputs for a given computational problem. The computation complexity of a problem or an algorithm expressed in terms of the decision tree model is called decision tree complexity or query complexity.

Classification by query computational complexity

Simple decision tree

The model in which every decision is based on the comparison of two numbers within constant time is called simply a decision tree model. It was introduced to establish computational complexity of sorting and searching.

The simplest illustration of this lower bound technique is for the problem of finding the smallest number among n numbers using only comparisons. In this case the decision tree model is a binary tree. Algorithms for this searching problem may result in n different outcomes (since any of the n given numbers may turn out to be the smallest one). It is known that the depth of a binary tree with n leaves is at least $\log n$, which gives a lower bound of $\Omega (\log n)$ for the searching problem. However this lower bound is known to be slack, since the following simple argument shows that at least n - 1 comparisons are needed: Before the smallest number can be determined, every number except the smallest must "lose" (compare greater) in at least one comparison.

Along the same lines the lower bound of $\Omega (n\log n)$ for sorting may be proved. In this case, the existence of numerous comparison-sorting algorithms having this time complexity, such as mergesort and heapsort, demonstrates that the bound is tight.

Linear decision tree

Linear decision trees, just like the simple decision trees, make a branching decision based on a set of values as input. As opposed to binary decision trees, linear decision trees have three output branches. A linear function $f(x_{1},\dots ,x_{i})$ is being tested and branching decisions are made based on the sign of the function (negative, positive, or 0).

Geometrically, $f(x)$ defines a hyperplane in Rⁿ. A set of input values reaching any particular nodes represents the intersection of the half-planes defined by the nodes.

Algebraic decision tree

Algebraic decision trees are a generalization of linear decision trees to allow test functions to be polynomials of degree d. Geometrically, the space is divided into semi-algebraic sets (a generalization of hyperplane). The evaluation of the complexity is more difficult.

Classification by query computational model

Deterministic decision tree

If the output of a decision tree is $f(x)$, for all $x\in \{0,1\}^{n}$, the decision tree is said to "compute" $f$. The depth of a tree is the maximum number of queries that can happen before a leaf is reached and a result obtained. $D(f)$, the deterministic decision tree complexity of $f$ is the smallest depth among all deterministic decision trees that compute $f$.

Randomized decision tree

One way to define a randomized decision tree is to add additional nodes to the tree, each controlled by a probability $p_{i}$. Another equivalent definition is to select a whole decision tree at the beginning from a set of decision trees based on a probability distribution. Based on this second definition, the complexity of the randomized tree is defined as the greatest depth among all the trees associated with probabilities greater than 0. $R_{2}(f)$ is defined as the complexity of the lowest-depth randomized decision tree whose result is $f(x)$ with probability at least $2/3$ for all $x\in \{0,1\}^{n}$ (i.e., with bounded 2-sided error).

$R_{2}(f)$ is known as the Monte Carlo randomized decision-tree complexity, because the result is allowed to be incorrect with bounded two-sided error. The Las Vegas decision-tree complexity $R_{0}(f)$ measures the expected depth of a decision tree that must be correct (i.e., has zero-error). There is also a one-sided bounded-error version known as $R_{1}(f)$.

Nondeterministic decision tree

The nondeterministic decision tree complexity of a function is known more commonly as the certificate complexity of that function. It measures the number of input bits that a nondeterministic algorithm would need to look at in order to evaluate the function with certainty.

Quantum decision tree

The quantum decision tree complexity $Q_{2}(f)$ is the depth of the lowest-depth quantum decision tree that gives the result $f(x)$ with probability at least $2/3$ for all $x\in \{0,1\}^{n}$. Another quantity, $Q_{E}(f)$, is defined as the depth of the lowest-depth quantum decision tree that gives the result $f(x)$ with probability 1 in all cases (i.e. computes $f$ exactly). $Q_{2}(f)$ and $Q_{E}(f)$ are more commonly known as quantum query complexities, because the direct definition of a quantum decision tree is more complicated than in the classical case. Similar to the randomized case, we define $Q_{0}(f)$ and $Q_{1}(f)$.

Relationship between different models

It follows immediately from the definitions that for all $n$-bit Boolean functions $f$, $Q_{2}(f)\leq R_{2}(f)\leq R_{1}(f)\leq R_{0}(f)\leq D(f)\leq n$, and $Q_{2}(f)\leq Q_{E}(f)\leq D(f)\leq n$.

Blum and Impagliazzo, Hartmanis and Hemachandra, and Tardos independently discovered that $D(f)\leq R_{0}(f)^{2}$. Noam Nisan found that the Monte Carlo randomized decision tree complexity is also polynomially related to deterministic decision tree complexity: $D(f)=O(R_{2}(f)^{3})$. (Nisan also showed that $D(f)=O(R_{1}(f)^{2})$.) A corollary of this result is that $R_{0}(f)=O(R_{2}(f)^{3})$. This inequality may be loose, however; no example is known of even a super-linear separation between $R_{0}(f)$ and $R_{2}(f)$.

The quantum decision tree complexity $Q_{2}(f)$ is also polynomially related to $D(f)$. Midrijanis showed that $D(f)=O(Q_{E}(f)^{3})$, improving a quartic bound due to Beals et al. Beals et al. also showed that $D(f)=O(Q_{2}(f)^{6})$, and this is still the best known bound. However, the largest known gap between deterministic and quantum query complexities is only quadratic. A quadratic gap is achieved for the OR function; $D(OR_{n})=n$ while $Q_{2}(OR_{n})=\Theta ({\sqrt {n}})$.

It is important to note that these polynomial relationships are valid only for total Boolean functions. For partial Boolean functions, that have a domain a subset of $\{0,1\}^{n}$, an exponential separation between $Q_{E}(f)$ and $D(f)$ is possible; the first example of such a problem was discovered by Deutsch and Jozsa. The same example also gives an exponential separation between $R_{2}(f)$ and $D(f)$.

These relationships can be summarized by the following inequalities, which are true up to constant factors:

• $D(f)\leq R_{0}(f)^{2}$
• $D(f)\leq R_{1}(f)R_{2}(f)$
• $D(f)\leq R_{2}(f)^{3}$
• $R_{0}(f)\leq R_{2}(f)^{2}\log N$
• $D(f)\leq Q_{2}(f)^{6}$
• $D(f)\leq Q_{E}(f)^{2}Q_{2}(f)^{2}$
• $D(f)\leq Q_{1}(f)^{2}Q_{2}(f)^{2}$
• $R_{0}(f)\leq Q_{1}(f)Q_{2}(f)^{2}\log N$
• $D(f)\leq Q_{1}(f)Q_{2}(f)^{2}$

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Surveys

• 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.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  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