**Important Results**

There are well-known classical algorithms such as depth-first search and breadth-first search which solve USTCON in linear time and space. Their existence, shown long before **SL** was defined, proves that **SL** is contained in **P**. It's also not difficult to show that USTCON, and so **SL**, is in **NL**, since we can just nondeterministically guess at each vertex which vertex to visit next in order to discover a path if one exists.

The first nontrivial result for **SL**, however, was Savitch's theorem, proved in 1970, which provided an algorithm that solves USTCON in log2 *n* space. Unlike depth-first search, however, this algorithm is impractical for most applications because of its potentially superpolynomial running time. One consequence of this is that USTCON, and so **SL**, is in DSPACE(log2*n*). (Actually, Savitch's theorem gives the stronger result that **NL** is in DSPACE(log2*n*).)

Although there were no (uniform) *deterministic* space improvements on Savitch's algorithm for 22 years, a highly practical probabilistic log-space algorithm was found in 1979 by Aleliunas et al.: simply start at one vertex and perform a random walk until you find the other one (then accept) or until *|V|*3 time has passed (then reject). False rejections are made with a small bounded probability that shrinks exponentially the longer the random walk is continued. This showed that **SL** is contained in RLP, the class of problems solvable in polynomial time and logarithmic space with probabilistic machines that reject incorrectly less than 1/3 of the time. By replacing the random walk by a universal traversal sequence, Aleliunas et al. also showed that **SL** is contained in L/poly, a non-uniform complexity class of the problems solvable deterministically in logarithmic space with polynomial advice.

In 1989, Borodin et al. strengthened this result by showing that the complement of USTCON, determining whether two vertices are in different connected components, is also in **RLP**. This placed USTCON, and **SL**, in co-**RLP** and in the intersection of **RLP** and co-**RLP**, which is ZPLP, the class of problems which have log-space, expected polynomial-time, no-error randomized algorithms.

In 1992, Nisan, Szemerédi, and Wigderson finally found a new deterministic algorithm to solve USTCON using only log1.5 *n* space. This was improved slightly, but there would be no more significant gains until Reingold.

In 1995, Nisan and Ta-Shma showed the surprising result that **SL** is closed under complement, which at the time was believed by many to be false; that is, **SL** = co-**SL**. Equivalently, if a problem can be solved by reducing it to a graph and asking if two vertices are in the *same* component, it can also be solved by reducing it to another graph and asking if two vertices are in *different* components. However, Reingold's paper would later make this result redundant.

One of the most important corollaries of **SL** = co-**SL** is that **L****SL** = **SL**; that is, a deterministic, log-space machine with an oracle for **SL** can solve problems in **SL** (trivially) but cannot solve any other problems. This means it does not matter whether we use Turing reducibility or many-one reducibility to show a problem is in **SL**; they are equivalent.

A breakthrough October 2004 paper by Omer Reingold showed that USTCON is in fact in L. Since USTCON is **SL**-complete, this implies that **SL** = **L**, essentially eliminating the usefulness of consideration of **SL** as a separate class. A few weeks later, graduate student Vladimir Trifonov showed that USTCON could be solved deterministically using O(log *n* log log *n*) space—a weaker result—using different techniques.

Read more about this topic: SL (complexity)

### Other articles related to "important results, important":

**Important Results**

... One of the most

**important**findings was that a new measles vaccine used in low-income countries was associated with a two-fold increase in mortality among girls ...

### Famous quotes containing the words results and/or important:

“In the works of man, everything is as poor as its author; vision is confined, means are limited, scope is restricted, movements are labored, and *results* are humdrum.”

—Joseph De Maistre (1753–1821)

“We mean by “politics” the people’s business—the most *important* business there is.”

—Adlai Stevenson (1900–1965)