Graph coloring involves assigning colors to objects in a graph subject to constraints. Vertex coloring assigns colors to vertices such that no adjacent vertices share the same color. The chromatic number is the minimum number of colors needed. Graph coloring has applications in scheduling, frequency assignment, and register allocation. Register allocation with chordal graphs can be done in polynomial time rather than NP-complete, since programs in SSA form have chordal interference graphs. A greedy algorithm can color chordal graphs in linear time, enabling simple and optimal register allocation.
Hello all, This is the presentation of Graph Colouring in Graph theory and application. Use this presentation as a reference if you have any doubt you can comment here.
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.
Hello all, This is the presentation of Graph Colouring in Graph theory and application. Use this presentation as a reference if you have any doubt you can comment here.
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.
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 share 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 share 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 mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
Graph coloring is an important concept in graph theory. It is a special kind of problem in which we have assign colors to certain elements of the graph along with certain constraints. Suppose we are given K colors, we have to color the vertices in such a way that no two adjacent vertices of the graph have the same color, this is known as vertex coloring, similarly we have edge coloring and face coloring. The coloring problem has a huge number of applications in modern computer science such as making schedule of time table , Sudoku, Bipartite graphs , Map coloring, data mining, networking. In this paper we are going to focus on certain applications like Final exam timetabling, Aircraft Scheduling, guarding an art gallery.
This presentation is about applications of graph theory applications....it is updated version it was given at international conference at applications of graph theory at KAULALAMPUR MALYSIA 2OO7
Talk given at neo4j conference "Graph Connect" - discussing some graph theory (old and new), and why knowing your stuff can come in handy on a software project.
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 share 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 share 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 mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
Graph coloring is an important concept in graph theory. It is a special kind of problem in which we have assign colors to certain elements of the graph along with certain constraints. Suppose we are given K colors, we have to color the vertices in such a way that no two adjacent vertices of the graph have the same color, this is known as vertex coloring, similarly we have edge coloring and face coloring. The coloring problem has a huge number of applications in modern computer science such as making schedule of time table , Sudoku, Bipartite graphs , Map coloring, data mining, networking. In this paper we are going to focus on certain applications like Final exam timetabling, Aircraft Scheduling, guarding an art gallery.
This presentation is about applications of graph theory applications....it is updated version it was given at international conference at applications of graph theory at KAULALAMPUR MALYSIA 2OO7
Talk given at neo4j conference "Graph Connect" - discussing some graph theory (old and new), and why knowing your stuff can come in handy on a software project.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
Initial Graphulo Graph Analytics Expressed in GraphBLAS:
GraphBLAS is an effort to define standard building blocks for graph algorithms in the language of linear algebra. Graphulo is a project to implement the GraphBLAS using Accumulo.
Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can
divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The
condition which we follow in graph coloring is that the incident vertices/edges have not the same color.
There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and
others are online algorithm. Where offline means the graph is known in advance and the online means that
the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is
added to the graph and the main issue is that we want to minimize the number of colors. We cannot change
the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm
for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is Δ, then it
is possible to color its edges, in polynomial time, using at most Δ+ 1 color. The algorithm provided by
Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one
by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of
Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors
online in multiple rounds.
3. 3
Basics
• Assignment of "colors" to certain objects in a
graph subject to certain constraints
– Vertex coloring (the default)
– Edge coloring
– Face coloring (planar)
4. 4
Not Graph Labeling
• Graph coloring
– Just markers to keep track of adjacency or
incidence
• Graph labeling
– Calculable problems that satisfy a numerical
condition
5. 5
Vertex coloring
• In its simplest form, it is a way of coloring the
vertices of a graph such that no two adjacent
vertices share the same color
• Edge and Face coloring can be transformed
into Vertex version
6. 6
Vertex Color example
• Anything less results in adjacent vertices with
the same color
– Known as “proper”
• 3-color example
9. 9
Chromatic Number
• χ - least number of colors needed to color a
graph
– Chromatic number of a complete graph:
χ(Kn) = n
10. 10
Properties of χ(G)
• χ(G) = 1 if and only if G is totally disconnected
• χ(G) ≥ 3 if and only if G has an odd cycle (equivalently, if G is
not bipartite)
• χ(G) ≥ ω(G) (clique number)
• χ(G) ≤ Δ(G)+1 (maximum degree)
• χ(G) ≤ Δ(G) for connected G, unless G is a complete graph or
an odd cycle (Brooks' theorem).
• χ(G) ≤ 4, for any planar graph
– The “four-color theorem”
11. 11
Four-color Theorem
• Dates back to 1852 to Francis Guthrie
• Any given plane separated into regions may be
colored using no more than 4 colors
– Used for political boundaries, states, etc
– Shares common segment (not a point)
• Many failed proofs
12. 12
Algorithmic complexity
• Finding minimum coloring: NP-hard
• Decision problem:
“is there a coloring which uses at most k colors?”
• Makes it NP-complete
13. 13
Coloring a Graph - Applications
• Sudoku
• Scheduling
• Mobile radio frequency assignment
• Pattern matching
• Register Allocation
14. 14
Register Allocation with Graphs
Coloring
• Register pressure
– How determine what should be stored in registers
– Determine what to “spill” to memory
• Typical RA utilize graph coloring for underlying
allocation problem
– Build graph to manage conflicts between live
ranges
15. 15
Chordal Graphs
• Each cycle of four or more nodes has a chord
• Subset of perfect graphs
• Also known as triangulated graphs
17. 17
RA with Chordal Graphs
• Normal register allocation was an NP-
complete problem
– Graph coloring
• If program is in SSA form, it can be
accomplished in polynomial time with chordal
graphs!
– Thereby decreasing need for registers
18. 18
Quick Static Single Assignment Review
• SSA Characteristics
– Basic blocks
– Unique naming for variable assignments
– Φ-functions used at convergence of control flows
• Improves optimization
– constant propagation
– dead code elimination
– global value numbering
– partial redundancy elimination
– strength reduction
– register allocation
19. 19
RA with Chordal Graphs
• SSA representation needs fewer registers
• Key insight: a program in SSA form has a
chordal interference graph
– Very Recent
20. 20
RA with Chordal Graphs cont
• Result is based on the fact that in strict-
SSA form, every variable has a single
contiguous live range
• Variables with overlapping live ranges
form cliques in the interference graph
21. 21
RA with Chordal Graphs cont
• Greedy algorithm can color a chordal graph in
linear time
• New SSA-elimination algorithm done without
extra registers
• Result:
– Simple, optimal, polynomial-time algorithm for
the core register allocation problem
22. 22
Chordal Color Assignment
• Algorithm: Chordal Color Assignment
• Input: Chordal Graph G = (V, E), PEO σ
• Output: Color Assignment f : V → {1… χG}
For Integer : i ← 1 to |V| in PEO order
Let c be the smallest color not assigned to a vertex
in Ni(vi)
f(vi) ← c
EndFor
24. 24
References
• http://en.wikipedia.org
• Engineering a Compiler, Keith D. Cooper and Linda Torczon,
2004
• http://www.math.gatech.edu/~thomas/FC/fourcolor.html
• An Optimistic and Conservative Register Assignment Heuristic
for Chordal Graphs, Philip Brisk, et. Al., CASES 07
• Register Allocation via Coloring of Chordal Graphs, Jens
Palsberg, CATS2007