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Gregory John Chaitin is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy.
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
In compiler design, static single assignment form is a type of intermediate representation (IR) where each variable is assigned exactly once. SSA is used in most high-quality optimizing compilers for imperative languages, including LLVM, the GNU Compiler Collection, and many commercial compilers.
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.
In compiler optimization, register allocation is the process of assigning local automatic variables and expression results to a limited number of processor registers.
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.
In computer science, instruction scheduling is a compiler optimization used to improve instruction-level parallelism, which improves performance on machines with instruction pipelines. Put more simply, it tries to do the following without changing the meaning of the code:
In graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
In computer science, rematerialization or remat is a compiler optimization which saves time by recomputing a value instead of loading it from memory. It is typically tightly integrated with register allocation, where it is used as an alternative to spilling registers to memory. It was conceived by Gregory Chaitin, Marc Auslander, Ashok Chandra, John Cocke, Martin Hopkins and Peter Markstein and implemented in the Pl.8 compiler for the 801 Minicomputer in the late 1970s. Later improvements were made by Preston Briggs, Keith D. Cooper, and Linda Torczon in 1992.
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests. The graphs with treewidth at most 2 are the series–parallel graphs. The maximal graphs with treewidth exactly k are called k-trees, and the graphs with treewidth at most k are called partial k-trees. Many other well-studied graph families also have bounded treewidth.
In computer science, pointer analysis, or points-to analysis, is a static code analysis technique that establishes which pointers, or heap references, can point to which variables, or storage locations. It is often a component of more complex analyses such as escape analysis. A closely related technique is shape analysis.
In proof theory, the Geometry of Interaction (GoI) was introduced by Jean-Yves Girard shortly after his work on linear logic. In linear logic, proofs can be seen as various kinds of networks as opposed to the flat tree structures of sequent calculus. To distinguish the real proof nets from all the possible networks, Girard devised a criterion involving trips in the network. Trips can in fact be seen as some kind of operator acting on the proof. Drawing from this observation, Girard described directly this operator from the proof and has given a formula, the so-called execution formula, encoding the process of cut elimination at the level of operators. Subsequent constructions by Girard proposed variants in which proofs are represented as flows, or operators in von Neumann algebras. Those models were later generalised by Seiller's Interaction Graphs models.
Martin Charles Golumbic is a mathematician and computer scientist known for his research on perfect graphs, graph sandwich problems, compiler optimization, and spatial-temporal reasoning. He is a professor emeritus of computer science at the University of Haifa, and was the founder of the journal Annals of Mathematics and Artificial Intelligence.
In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not, in general, use the minimum number of colors possible.
In graph theory, a branch of mathematics, a crown graph on 2n vertices is an undirected graph with two sets of vertices {u1, u2, …, un} and {v1, v2, …, vn} and with an edge from ui to vj whenever i ≠ j.
In computer science and graph theory, the term color-coding refers to an algorithmic technique which is useful in the discovery of network motifs. For example, it can be used to detect a simple path of length k in a given graph. The traditional color-coding algorithm is probabilistic, but it can be derandomized without much overhead in the running time.
In computational complexity theory, a problem is NP-complete when:
- It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no".
- When the answer is "yes", this can be demonstrated through the existence of a short solution.
- The correctness of each solution can be verified quickly and a brute-force search algorithm can find a solution by trying all possible solutions.
- The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified.
The multi-trials technique by Schneider et al. is employed for distributed algorithms and allows breaking of symmetry efficiently. Symmetry breaking is necessary, for instance, in resource allocation problems, where many entities want to access the same resource concurrently. Many message passing algorithms typically employ one attempt to break symmetry per message exchange. The multi-trials technique transcends this approach through employing more attempts with every message exchange.
References
- Gregory Chaitin (April 2004). "Register allocation and spilling via graph coloring". ACM SIGPLAN Notices. 39 (4): 66–74. doi:10.1145/989393.989403. S2CID 23108905.
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