The “overlays” approach exploits orchestration between several protocols relying over the intra-communication facility. Our proposal adds a new overlay that interacts with the JA-BE-JA one. First of all, the JA-BE-JA’s overlay is the actual graph topology plus some long range links that are created through the retrievement of a small sample of random nodes for each vertex.
As shown in Figure 6.1, such overlay could be thought as a set of stains which interact each other “moving” like fluids. In this metaphor, the long range links fold the stains in a N dimensional space, where N is greater than the actual dimension, allowing the interaction between areas of the stains that are not physically near each other. The JA- BE-JA algorithm orchestrates those stains in order to minimize the edge-cut. Whenever a swap succeeds, analogously in the stain metaphor, is like the stains flow rolling in some directions, as shown in Figure 6.2.
6.2.1 Our Proposal
T-MAN, described in section 4.3, is a gossip algorithm whose aim is to create and manage topology overlays. The overlays are created and maintained according to a ranking function which measures the similarities between two peers. Our proposal is to introduce an ad-hoc overlay and exploit over it better choices. More precisely, JA-BE- JA interacts with such overlay in a sense of the stain metaphor. The interaction’s type is competitive, which means JA-BE-JA takes advantage of such overlay but as side-effects it damages such topology, i.e. the 2 overlays are somehow related and a modification
(a) Topology before the swap (b) Topology after the swap
Figure 6.2: Swap in the stain metaphor of the JA-BE-JA’s topology
on one of them implies the transformation of the other. Thus, the T-MAN’s ability to maintain the topology against dynamic changes is crucial. Indeed, the balance between the JA-BE-JA ruination and T-MAN repairing has to be tuned.
The T-MAN algorithm relies over the definition of a ranking function. Hence, the ad-hoc topology has to be defined through that function. The idea is to abstract the property of “borderness” of each peer from the actual topology, i.e. the idea is to point out the peer’s property to be on a border of some stains. Whenever 2 peers are on the borders the swap could take place and so figuratively the peers cross the boundaries. Moreover, the boundaries where the swaps take place do not have to be the same where the swap equation is evaluated on. At the end, we exploit the T-MAN overlay for the equation evaluation and perform the swap over the actual JA-BE-JA one. More precisely, the equation evaluation is executed firstly over the T-MAN overlay and in case no partner is found the algorithm falls back to the original evaluation, i.e. the neighbourhood and the random sample are taken into account. The side-effect of such orchestration is the damages that the T-MAN overlays is subjected to, i.e. the T-MAN overlay is defined starting from the JA-BE-JA one.
The ranking function using the JA-BE-JA notation is defined as:
rankF unction(peerA, peerB) = +∞, if peerA.colour == peerB.colour |HA− HB|, otherwise where HP = dp(πp) |Np|
We recall that two peers A and B are more similar as the rankF unction(A, B) → 0. The function ranks similar two peers that have the same percentages of neighbours of the same colour. Moreover they have to be coloured differently to each other, i.e. the peers are ranked among their “borderness” property and their relative colours. As a consequence, peers that are in the middle of a partition (a stain in the metaphor
(a) Before the colour exchange (b) After the colour exchange
Figure 6.3: Example of an exchange colour using the TMAN layer over the real JA-BE-JA coloured graph.
nomenclature) are similar to the ones that are in the middle of another partition (100% case, HP = 1) and, more useful for our purposes, peers which are on “the borders” of
a partition result similar to others that are also on the same kind of border, i.e. peers that are at the same distance from the border are similar. As a result, in a figurative way as shown in Figure 6.3, the stains flow between each other like a liquid with the aim of finding a good solution mapping the actual topology to another more friendly for the purpose. Moreover, such abstraction allows to find hopefully better candidates to exchange the colour with filtering and organizing the peers among their “borderness” property.
The decisional equation is evaluated over the T-MAN overlay and in case the swap will succeed the JA-BE-JA underline topology is updated. Therefore, the T-MAN overlays will be out-of-date at least in such peers and it has to rearrange their neighbourhood in order to adhere to their new “borderness” value. Optimistically the “borderness” value is updated incrementally allowing smoothly changes to the T-MAN overlay, otherwise the protocol strongly relies over the random sample to reconstruct its topology.
The JA-BE-JA algorithm performs local search optimization and the achieved solution could be stuck in a poor local minimum edge-cut value. In order to avoid those local minima the Simulated Annealing technique is employed. A temperature T is employed in order to manage the energy of the system, which could be increased at the beginning until a conservative policy is adopted falling back to the original algorithm. More pre- cisely, JA-BE-JA introduces the temperature into the decisional equation, as shown in equation (5.2). The result of contextualizing such parameter into the stain metaphor is shown in Figure 6.4. The temperature defines an area over the border where those peers are able to interact each other, i.e. the temperature defines a stripe and peers that are into it acts like to be on the same border. The cooling process, i.e. the temperature
T
Figure 6.4: Contextualization of the Simulated Annealing into the Stain Metaphor
decreasing, corresponds to tightening the stripe. A temperature value equal to 1 corre- sponds to vanish the stripe. Intuitively at the beginning many peers are able to interact each other whenever if they are actually on the border or not. Along the execution the stripe will be tighten and tighten reducing the number of peers which are able to interact, i.e. the simulated annealing creates (biased) chaos converging eventually to the usual behaviour. This aspect advantages the T-MAN overlay because softer require- ments are requested along the computation and a less strict-timing repair capability is needed allowing the usage of more cycles to the T-MAN protocol.