Graph coloring and_applications
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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.
Graph coloring and_applications
- 1. 1 Graph Coloring and Applications
- 2. 2 Overview • Graph Coloring Basics • Planar/4-color Graphs • Applications • Chordal Graphs • New Register Allocation Technique
- 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
- 7. 7 Vertex Color Example 1 2 4 5 3
- 8. 8 Vertex Color Example 1 2 4 5 3
- 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
- 16. 16 Chordal Graph Example • Removing a green edge will make it non-chordal
- 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
- 23. 23 Conclusion • Graph Coloring • Chordal Graphs • New Register Allocation Technique – Polynomial time
- 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