Trammel of Archimedes

In graph theory, the zig-zag product of regular graphs $G,H$, denoted by $G\circ H$, takes a large graph ($G$) and a small graph ($H$), and produces a graph that approximately inherits the size of the large one but the degree of the small one. An important property of the zig-zag product is that if $H$ is a good expander, then the expansion of the resulting graph is only slightly worse than the expansion of $G$.

Roughly speaking, the zig-zag product $G\circ H$ replaces each vertex of $G$ with a copy (cloud) of $H$, and connects the vertices by moving a small step (zig) inside a cloud, followed by a big step (zag) between two clouds, and finally performs another small step inside the destination cloud.

The zigzag product was introduced by Template:Harvtxt. When the zig-zag product was first introduced, it was used for the explicit construction of constant degree expanders and extractors. Later on the zig-zag product was used in computational complexity theory to prove that symmetric logspace and logspace are equal Template:Harv.

Definition

Let $G$ be a $D$-regular graph on $[N]$ with rotation map ${\mathrm{R}\mathrm{o}\mathrm{t}}_G$ and let $H$ be a $d$-regular graph on $[D]$ with rotation map ${\mathrm{R}\mathrm{o}\mathrm{t}}_H$. The zig-zag product $G\circ H$ is defined to be the $d^2$-regular graph on $[N]\times [D]$ whose rotation map ${\mathrm{R}\mathrm{o}\mathrm{t}}_{G\circ H}$ is as follows:
${\mathrm{R}\mathrm{o}\mathrm{t}}_{G\circ H}((v,a),(i,j))$:

1. Let $(a^{\prime },i^{\prime })={\mathrm{R}\mathrm{o}\mathrm{t}}_H(a,i)$.
2. Let $(w,b^{\prime })={\mathrm{R}\mathrm{o}\mathrm{t}}_G(v,a^{\prime })$.
3. Let $(b,j^{\prime })={\mathrm{R}\mathrm{o}\mathrm{t}}_H(b^{\prime },j)$.
4. Output $((w,b),(j^{\prime },i^{\prime }))$.

Properties

Reduction of the degree

It is immediate from the definition of the zigzag product that it transforms a graph $G$ to a new graph which is $d^2$-regular. Thus if $G$ is a significantly larger than $H$, the zigzag product will reduce the degree of $G$. Roughly speaking, by amplifying each vertex of $G$ into a cloud of the size of $H$ the product in fact splits the edges of each original vertex between the vertices of the cloud that replace it.

Spectral gap preservation

The expansion of a graph can be measured by its spectral gap. An important property of the zigzag product is the preservation of the spectral gap. That is, if $H$ is a “good enough” expander (has a large spectral gap) then the expansion of the zigzag product is close to the original expansion of $G$.

Formally: Define a $(N,D,\lambda )$-graph as any $D$-regular graph on $N$ vertices, whose second largest eigenvalue (of the associated random walk) has absolute value at most $\lambda$.

Let $G_1$ be a $(N_1,D_1,{\lambda }_1)$-graph and $G_2$ be a $(D_1,D_2,{\lambda }_2)$-graph, then $G\circ zH$ is a $(N_1\cdot D_1,D_2^2,f({\lambda }_1,{\lambda }_2))$-graph, where $f({\lambda }_1,{\lambda }_2)<{\lambda }_1+{\lambda }_2+{\lambda }_2^2)$.

Connectivity preservation

The zigzag product $G\circ H$ operates separately on each connected component of $G$.

Formally speaking, given two graphs: $G$, a $D$-regular graph on $[N]$ and $H$, a $d$-regular graph on $[D]$ - if $S\subseteq [N]$ is a connected component of $G$ then $G{|}_S\circ H=G\circ H{|}_{S\times D}$, where $G{|}_S$ is the subgraph of $G$ induced by $S$ (i.e., the graph on $S$ which contains all of the edges in $G$ between vertices in $S$).

Applications

Construction of constant degree expanders

In 2002 Omer Reingold, Salil Vadhan, and Avi Wigderson have given a simple, explicit combinatorial construction of constant-degree expander graphs. The construction is iterative, and needs as a basic building block a single, expander of constant size. In each iteration the zigzag product is used in order to generate another graph whose size is increased but its degree and expansion remains unchanged. This process continues, yielding arbitrarily large expanders.

From the properties of the zigzag product mentioned above, we see that the product of a large graph with a small graph, inherits a size similar to the large graph, and degree similar to the small graph, while preserving its expansion properties from both, thus enabling to increase the size of the expander without deleterious effects.

Solving the undirected s-t connectivity problem in logarithmic space

In 2005 Omer Reingold introduced an algorithm that solves the undirected st-connectivity problem, the problem of testing whether there is a path between two given vertices in an undirected graph, using only logarithmic space. The algorithm relies heavily on the zigzag product.

Roughly speaking, in order to solve the undirected s-t connectivity problem in logarithmic space, the input graph is transformed, using a combination of powering and the zigzag product, into a constant-degree regular graph with a logarithmic diameter. The power product increases the expansion (hence reduces the diameter) at the price of increasing the degree, and the zigzag product is used to reduce the degree while preserving the expansion.

See also

• Graph operations

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.
  
  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.
• 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.