Max Flow Issue has New Algorithm Solution

Algorithm promises to greatly streamline solutions to the 'max flow' problem. Research could boost the performance even of large systems like the Internet. Finding the most effective way to transportation products across a network like the US road system or the Internet is an issue that has subject to taxes specialized mathematicians and computer scientists for decades.

To deal with the issue, researchers have trusted a maximum-flow algorithm, also known as "max flow," in which a network is showed as a chart with a sequence of nodes, known as vertices, and linking lines between them, called edges.

Given that each advantage has a highest possible potential -- just like the streets or the fiber-optic wires used to deliver information around the Internet -- such algorithms look for the most effective way to deliver products from one node in the chart to another, without going above these restrictions.

However, as the dimension systems like the Internet has grown significantly, it is often excessively time-consuming to fix these problems using traditional processing techniques, according to Jonathan Kelner, an associate professor of applied mathematics at MIT and a member of MIT's Computer Science and Artificial Intelligence Laboratory (CSAIL).

So in a document to be provided at the ACM-SIAM Symposium on Discrete Algorithms in Portland, Oregon, this week, Kelner and his colleague Lorenzo Orecchia, an applied mathematics instructor, together with students Yin Tat Lee and Aaron Sidford, will explain a new theoretical algorithm that can considerably decrease the variety of operations needed to fix the max-flow issue, making it possible to deal with even large networks like the Internet or the human genome.

"There has recently been a blast in the sizes of charts being analyzed," Kelner says. "For example, if you wanted to route traffic on the Internet, study all the relationships on Facebook or analyze genomic data, you could quickly end up with charts with large numbers, enormous amounts or even immeasurable sides."

Previous max-flow algorithms have come at the issue one advantage, or direction, at some factor, Kelner says. So for example, when delivering items from node A to node B, the algorithms would transfer some of the products down one direction, until they achieved its highest possible potential, and then begin delivering some down the next direction.

"Many past algorithms," Kelner says, "would discover a direction from point A to point B, deliver some flow along it, and then say, 'Given what I've already done, can I find another direction along which I can deliver more?' When one needs to deliver flow at the same time along many different tracks, this leads to an important restriction on the speed of the algorithm."

But in 2011 Kelner, CSAIL graduate student Aleksander Madry, mathematics undergraduate Paul Christiano, and colleague at Yale University and the University of Southern California designed a strategy to evaluate all of the tracks at the same time.

The researchers considered the chart as a collection of electric resistors, and then thought linking a battery to node A and a ground to node B, and enabling the current to flow through the network. "Electrical current doesn't pick just one direction, it will deliver a little bit of current over every resistor on the network," Kelner says. "So it probes the whole chart worldwide, learning many tracks at the same time."

This permitted the new algorithm to fix the max-flow issue considerably quicker than past initiatives.

Now the MIT team has designed a strategy to decrease the running time even further, making it possible to evaluate even enormous networks, Kelner says.

Unlike past algorithms, which have considered all the tracks within a chart as is equal to, the new strategy recognizes those tracks that make a bottleneck within the network. The team's algorithm separates each chart into groups of well-connected nodes, and the tracks between them that make bottlenecks, Kelner says.

"Our algorithm figures out which areas of the chart can quickly path what they need to, and which areas are the bottlenecks. This allows you to focus on the problems areas and the high-level framework, instead of spending lots of your energy and effort creating insignificant choices, so that you can use your time and effort a lot more effectively," he says.

The outcome is an almost linear algorithm, Kelner says, meaning, how long it takes to fix an issue is very near to being directly proportionate to the variety of nodes on the network. So if the number of nodes on the chart is increased by 10, how long would be increased by something very near to 10, in contrast to being increased by 100 or 1,000, he says. "This indicates that it scales basically as well as you could hope for with the dimension the feedback," he says.

Shanghua Teng, a professor of computer science at the University of Southeast California who was not involved in the latest document, says it symbolizes a major cutting-edge in chart algorithms and optimization software.

"This document, which is the champion of the best document prize at the [ACM-SIAM] meeting, is a consequence of continual initiatives by Kelner and his colleagues in applying electric flows to design effective chart algorithms," Teng says. "The document contains an amazing array of technical initiatives."

Source: The above story is based on materials provided by Massachusetts Institute of Technology