Programming for Chemical and Life Science Informatics

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Programming for Chemical

and Life Science Informatics

I573 Week 3 (Graph Theory -I)

Rajarshi Guha 24th January, 2008

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Outline

 What are graphs?

 Where can we use graphs and graph theory?  Basic concepts

 Algorithms on graphs

 Toolkits for graph theory  Books / references

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What Is a Graph?

 In abstract terms, a collection of nodes (or vertices),

some or all of which are connected by edges

 Nodes and edges can have

properties associated with them

 Nodes and edges can represent

a variety of things − atoms, bonds − people, friendship − cities, roads Edge Node

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Where Do We Use Graph Theory?

 Cheminformatics − Ring perception − Substructure searching − Reaction transformations − Molecular descriptors

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Where Do We Use Graph Theory?

 Chemistry

− Stability of fullerenes

− Indications of chemical reactivity − Electronic structure

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Where Do We Use Graph Theory?

 Biology

− Metabolic networks − Ecological systems

− Analysis of gene expression − Proteomics

 Many other areas

− Wherever there's a set of objects that can be

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 Size – number of edges in a graph

 Degree – the number of edges connected to a

vertex

 Loop – an edge whose start and end

nodes are the same

 Directed graph – a graph in which

one can traverse edges in a specific direction only

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Graph Theory Concepts

 Subgraph – a graph g is a subgraph of G if all

nodes and edges of g are contained in the set of nodes and edges of G

 The red nodes and edges constitute

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Data Structures in Graph Theory

 Adjacency matrix A B C D 1 1 0 0 D 1 1 1 0 C 0 1 1 1 B 0 0 1 1 A D C B A

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Data Structures in Graph Theory

 Distance matrix A B C D 0 1 2 3 D 1 0 1 2 C 2 1 0 1 B 3 2 1 0 A D C B A

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Isomorphic Graphs

 Two graphs with the same number of nodes,

connected in the same way

 Two graphs G1 and G2 are isomorphic if

− there is a one to one correspondence between

their nodes and

− if two nodes are adjacent in G1 they are also

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Subgraph Isomorphism

 Given a graph G1, find a subgraph of G1 that is

isomorphic to a graph G2

 This is known to be NP-complete

− Not computable in polynomial time G1

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Maximum Common Subgraph

Isomorphism

 Given graphs G1 and G2, what is the largest common

subgraph?

− Not computable in polynomial time

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Maximum Common Subgraph

Isomorphism

 Given graphs G1 and G2, what is the largest

common subgraph?

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Traversing Graphs

 Walk – an alternating sequence of nodes and

edges, beginning and ending at a node

 Length – number of edges in a walk  Path – a walk with unique vertices

 Cycle – a walk with unique vertices, but starting

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Examples of Walks, Paths & Cycles

A walk of length 4

A cycle of length 6

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Algorithms

 Graph traversal − Breadth first − Depth first  Finding cycles  Shortest paths

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Breadth First Traversal

 Uninformed search  Start from a node S

 Examine all the nearest neighbors of S

 For each neighbor look at its nearest neighbors  Stop when all nodes have been visited

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Breadth First Traversal

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What Is It Good For?

 Find all connected components

− The set of nodes reached by a BFS are the largest

connected component containing the start node

 Finding the shortest path between 2 nodes  Used for path finding in computer games

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Depth First Traversal

 Start from a node S

 Follow the first neighbor till a node is reached

that has no new neighbors

 Repeat for each nearest neighbor of S  Stop when all nodes have been visited  Requires less memory than BFS

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Depth First Traversal

 The order of the nodes

visited starting from A is A, B, D, F, C, G, E

or

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What Is It Good For?

 Finding connected components  Topological sorting, used in

− instruction scheduling − Makefile dependencies

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Spanning Tree

 Subgraph of a connected, undirected graph that

is a tree and connects all the vertices

The nodes connected by the the dark edges represent a spanning tree of the original graph

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Minimum Spanning Tree

 If edges have weights, then the minimum

spanning tree is that spanning tree with the overall minimum cost

 Kruskal's algorithm or Prim's algorithm

The dark edges represent a path that has the lowest cost amongst all possible spanning trees

4 5 3 1 3 7 5 9 7 8 8 6 6 5 4 2

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What Is It Good For?

 Determining optimal paths  Laying cables for cable TV

− Cables must be laid along certain paths − Some paths are most costly than others

− Find the paths connecting all houses that is

cheapest

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Toolkits for Graph Theory

 JGraphT - Java library for graph theory algo's  Supports

− directed and undirected graphs − weighted/unweighted edges

− variety of graph types

 This is different from Jgraph which is a graph

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Toolkits for Graph Theory

 Boost graph library (BGL) – a C++ library  Based on STL so it makes heavy use of

templates and parametrization

 A wide variety of algorithms that include

− Shortest paths − Spanning trees

− Sorting and ordering

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References

 Books for graph theory in chemistry

− Bonchev − Trinajistic

 General books

− Bonchev & Rouvray − Chartrand

− Trudeau