Transcript Graph Theory for Bioinformatics

BIO/CS 471 – Algorithms for bioinformatics
Graph Theoretic
Concepts and Algorithms
for Bioinformatics
Intro. to Graph Theory
1
What is a “graph”
• Formally: A finite graph G(V, E) is a pair (V, E),
where V is a finite set and E is a binary relation on V.
– Recall: A relation R between two sets X and Y is a subset of
X x Y.
– For each selection of two distinct V’s, that pair of V’s is
either in set E or not in set E.
• The elements of the set V are called vertices (or
nodes) and those of set E are called edges.
• Undirected graph: The edges are unordered pairs of b
V (i.e. the binary relation is symmetric).
a
c
– Ex: undirected G(V,E); V = {a,b,c}, E = {{a,b}, {b,c}}
• Directed graph (digraph):The edges are ordered
pairs of V (i.e. the binary relation is not necessarily
symmetric).
– Ex: digraph G(V,E); V = {a,b,c}, E = {(a,b), (b,c)}
Intro. to Graph Theory
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Why graphs?
• Many problems can be stated in terms of a graph
• The properties of graphs are well-studied
– Many algorithms exists to solve problems posed as graphs
– Many problems are already known to be intractable
• By reducing an instance of a problem to a standard graph
problem, we may be able to use well-known graph algorithms
to provide an optimal solution
• Graphs are excellent structures for storing, searching, and
retrieving large amounts of data
– Graph theoretic techniques play an important role in increasing the
storage/search efficiency of computational techniques.
• Graphs are covered in section 2.2 of Setubal & Meidanis
Intro. to Graph Theory
3
Graphs in bioinformatics
• Sequences
– DNA, proteins, etc.
R
Y
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Chemical compounds
Intro. to Graph Theory
Metabolic pathways
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Graphs in bioinformatics
Intro. to Graph Theory
Phylogenetic trees
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Basic definitions
Undirected graph
Directed graph
loop
loop
G=(V,E)
isolated vertex
adjacent
multiple
edges
• incidence: an edge (directed or undirected) is incident to a vertex
that is one of its end points.
• degree of a vertex: number of edges incident to it
– Nodes of a digraph can also be said to have an indegree and an outdegree
• adjacency: two vertices connected by an edge are adjacent
Intro. to Graph Theory
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“Travel” in graphs
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path: no vertex can be repeated
example path: a-b-c-d-e
trail: no edge can be repeated
example trail: a-b-c-d-e-b-d
walk: no restriction
example walk: a-b-d-a-b-c
e
b
d
c
closed: if starting vertex is also ending vertex
length: number of edges in the path, trail, or walk
circuit: a closed trail (ex: a-b-c-d-b-e-d-a)
cycle: closed path (ex: a-b-c-d-a)
Intro. to Graph Theory
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Types of graphs
• simple graph: an undirected graph with no loops or multiple edges between
the same two vertices
• multi-graph: any graph that is not simple
• connected graph: all vertex pairs are joined by a path
• disconnected graph: at least one vertex pairs is not joined by a path
• complete graph: all vertex pairs are adjacent
– Kn: the completely connected graph with n vertices
Simple graph
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K5
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Intro. to Graph Theory
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Disconnected graph
with two components
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Types of graphs
• acyclic graph (forest): a graph with no cycles
• tree: a connected, acyclic graph
• rooted tree: a tree with a “root” or “distinguished” vertex
– leaves: the terminal nodes of a rooted tree
• directed acyclic graph (DAG): a digraph with no cycles
• weighted graph: any graph with weights associated with the edges (edgeweighted) and/or the vertices (vertex-weighted)
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Intro. to Graph Theory
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Digraph definitions
• for digraphs only…
Directed graph
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• Every edge has a head (starting point) and a
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tail (ending point)
• Walks, trails, and paths can only use edges in
the appropriate direction
• In a DAG, every path connects an
c
predecessor/ancestor (the vertex at the head
of the path) to its successor/descendents
d
(nodes at the tail of any path).
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• parent: direct ancestor (one hop)
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w
• child: direct descendent (one hop)
• A descendent vertex is reachable from any of
v
u
its ancestors vertices
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Intro. to Graph Theory
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Computer representation
• undirected graphs: usually represented as digraphs with two
directed edges per “actual” undirected edge.
• adjacency matrix: a |V| x |V| array where each cell i,j contains
the weight of the edge between vi and vj (or 0 for no edge)
• adjacency list: a |V| array where each cell i contains a list of all
vertices adjacent to vi
• incidence matrix: a |V| by |E| array where each cell i,j contains
a weight (or a defined constant HEAD for unweighted graphs)
if the vertex i is the head of edge j or a constant TAIL if vertex I
is the tail of edge j
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Intro. to Graph Theory
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10 2
adjacency
matrix
a c (8), d (4)
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c b (6)
d c (2), b (10)
adjacency
list
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incidence
matrix
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Computer representation
• Linked list of nodes: Node is a defined data object with labels which
include a list of pointers to its children and/or parents
Class Node:
label = NIL;
parents = []; # list of nodes coming into this node
children = []; # list of nodes coming out of this node
childEdgeWeights = []; # ordered list of edged weights
• Graph = [] # list of nodes
Intro. to Graph Theory
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Subgraphs
• G’(V’,E’) is a subgraph of G(V,E) if V’  V and E’  E.
• induced subgraph: a subgraph that contains all possible edges
in E that have end points of the vertices of the selected V’
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G(V,E)
Intro. to Graph Theory
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Induced subgraph of
G’({a,c,d},{{c,d}}) G with V’ = {b,c,d,e}
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Complement of a graph
• The complement of a graph G (V,E) is a graph with the same
vertex set, but with vertices adjacent only if they were not
adjacent in G(V,E)
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G
G
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Intro. to Graph Theory
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Famous problems: Shortest path
• Consider a weighted connected directed graph with a distinguished vertex
source: a distinguished vertex with zero in-degree
• What is the path of total minimum weight from the source to any other
vertex?
• Greedy strategy works for simple problems (no cycles, no negative weights)
• Longest path is a similar problem (complement weights)
• We will see this again soon for fragment assembly!
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Intro. to Graph Theory
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Dijkstra’s Algorithm
•
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2.
D(x) = distance from s to x (initially all )
Select the closest vertex to s, according to the current estimate
(call it c)
Recompute the estimate for every other vertex, x, as the
MINIMUM of:
1.
2.
The current distance, or
The distance from s to c , plus the distance from c to x – D(c) + W(c,
x)
Intro. to Graph Theory
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Dijkstra’s Algorithm Example
Initial
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D
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

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
B
Process A
Process C
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
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Process B
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Process D
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Process E
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Intro. to Graph Theory
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A
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D
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E
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Famous problems: Isomorphism
• Two graphs are isomorphic if a 1-to-1 correspondence between
their vertex sets exists that preserve adjacencies
• Determining to two graphs are isomorphic is NP-complete
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Intro. to Graph Theory
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Famous problems: Maximal clique
• clique: a complete subgraph
• maximal clique: a clique not contained in any other clique; the largest
complete subgraph in the graph
• Vertex cover: a subset of vertices such that each edge in E has at least one
end-point in the subset
• clique cover: vertex set divided into non-disjoint subsets, each of which
induces a clique
• clique partition: a disjoint clique cover
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Intro. to Graph Theory
Maximal cliques: {1,2,3},{1,3,4}
Vertex cover: {1,3}
Clique cover: { {1,2,3}{1,3,4} }
Clique partition: { {1,2,3}{4} }
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Famous problems: Coloring
• vertex coloring: labeling the vertices such that no edge in E has two endpoints with the same label
• chromatic number: the smallest number of labels for a coloring of a graph
• What is the chromatic number of this graph?
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• Would you believe that this problem (in general) is intractable?
Intro. to Graph Theory
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Famous problems: Hamilton & TSP
• Hamiltonian path: a path through a graph which contains
every vertex exactly once
• Finding a Hamiltonian path is another NP-complete problem…
• Traveling Salesmen Problem (TSP): find a Hamiltonian path
of minimum cost
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Intro. to Graph Theory
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Famous problems: Bipartite graphs
• Bipartite: any graph whose vertices can be partitioned into two
distinct sets so that every edge has one endpoint in each set.
• How colorable is a bipartite graph?
• Can you come up with an algorithm to determine if a graph is
bipartite or not?
• Is this problem tractable or intractable?
K4,4
Intro. to Graph Theory
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Famous problems: Minimal cut set
• cut set: a subset of edges whose remove causes the number of
graph components to increase
• vertex separation set: a subset of vertices whose removal
causes the number of graph components to increase
• How would you determine the minimal cut set or vertex
separation set?
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Intro. to Graph Theory
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cut-sets: {(a,b),(a,c)},
{(b,d),(c,d)},{(d,f)},...
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Famous problem: Conflict graphs
• Conflict graph: a graph where each vertex represents a concept or resource
and an edge between two vertices represents a conflict between these two
concepts
• When the vertices represents intervals on the real line (such as time) the
conflict graph is sometimes called an interval graph
• A coloring of an interval graph produces a schedule that shows how to best
resolve the conflicts… a minimal coloring is the “best” schedule”
• This concept is used to solve problems in the physical mapping of DNA
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Intro. to Graph Theory
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Colors?
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Famous problems: Spanning tree
• spanning tree: A subset of edges that are sufficient to keep a
graph connected if all other edges are removed
• minimum spanning tree: A spanning tree where the sum of the
edge weights is minimum
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Intro. to Graph Theory
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Famous problems: Euler circuit
• G is said to have a Euler circuit if there is a circuit in G that traverses every
edge in the graph exactly once
• The seven bridges of Konigsberg: Find a way to walk about the city so as to
cross each bridge exactly once and then return to the starting point.
area b
a
area d
area c
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Intro. to Graph Theory
This one is in P!
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Famous problems: Dictionary
• How can we organize a dictionary for fast lookup?
a b c … y z
a b c … y z
a b c … y z
a b c … y z
a b c … y z
26-ary “trie”
a b c … y z
Intro. to Graph Theory
“CAB”
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Graph traversal
• There are many strategies for solving graph problems… for
many problems, the efficiency and accuracy of the solution boil
down to how you “search” the graph.
• We will consider a “travel” problem for example:
• Given the graph below, find a path from vertex a to vertex d.
Shorter paths (in terms of edge weight sums) are desirable.
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Intro. to Graph Theory
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A greedy approach
• greedy traversal: Starting with the “root” node, take the edge
with smallest weight. Mark the edge so that you never attempt
to use it again. If you get to the end, great! If you get to a dead
end, back up one decision and try the next best edge.
• Advantages: Fast! Drawbacks: Answer is usually non-optimal
• For some problems, greedy approaches are optimal, for others
the answer may usually be close to the best answers, for yet
other problems, the greedy strategy is a poor choice.
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Intro. to Graph Theory
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Start node: a
End node: d
Traversal order: a, c, f, e, b, d
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Exhaustive search: Breadth-first
• For the current node, do any necessary work
– In this case, calculate the cost to get to the node by the current path; if the cost is
better than any previous path, update the “best path” and “lowest cost”.
• Place all adjacent unused edges in a queue (FIFO)
• Take an edge from the queue, mark it as used, and follow it to the new
current node
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Intro. to Graph Theory
Traversal order: a, b, c, d, e, f
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Exhaustive search: Depth-first
• For each current node
– do any necessary work
– Pick one unused edge out and
follow it to a new current
node
– If no unused edges exist,
unmark all of your edges an
go back from whence you
came!
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a
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}
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Intro. to Graph Theory
V.state = “visited”
Process vertex v
Foreach edge (v,w) {
if w.state = “unseen” {
DFS (G, w)
process edge (v,w)
}
}
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DFS (G, v)
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Traversal order: a, b, d, e, f, c
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Branch and Bound
• Begin a depth-first search (DFS)
• Once you achieve a successful result, note the result as our initial “best
result”
• Continue the DFS; if you find a better result, update the “best result”
• At each step of the DFS compare your current “cost” to the cost of the
current “best result”; if we already exceed the cost of the best result, stop the
downward search! Mark all edges as used, and head back up.
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Intro. to Graph Theory
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Traversal order:
Path Current Best
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AEB 6
AEBD 11
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AEF 9
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AEFC 15
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Path Current Best
ACF 7
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ACFE 15
11 < prune
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ABD 8
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ABE 7
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ABEF 14
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Binary search trees
• Binary trees have at two children per node (the child may be null)
• Binary search trees are organized so that each node has a label.
• When searching or inserting a value, compare the target value to each node;
one out-going edge corresponds to “less than” and one out-going edge
corresponds to “greater than”.
• On the average, you eliminate 50% of the search space per node… if the tree
is balanced
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Intro. to Graph Theory
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