1. 1 Intro. to Graph Theory BIO/CS 471 – Algorithms for bioinformatics Graph Theoretic Concepts and Algorithms for Bioinformatics

2. 2 Intro. to Graph Theory 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 V (i.e. the binary relation is symmetric). –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)} a b c a b c

3. 3 Intro. to Graph Theory 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

4. 4 Intro. to Graph Theory Graphs in bioinformatics Sequences –DNA, proteins, etc. Chemical compounds Metabolic pathways RYLI

5. 5 Intro. to Graph Theory Graphs in bioinformatics Phylogenetic trees

6. 6 Intro. to Graph Theory Basic definitions 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 Undirected graph Directed graph isolated vertex loop multiple edges G=(V,E) adjacent loop

7. 7 Intro. to Graph Theory xy 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 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) a b c d e “Travel” in graphs

8. 8 Intro. to Graph Theory 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 –K n : the completely connected graph with n vertices Simple graph a b c d e K5K5 a b c d e Disconnected graph with two components

9. 9 Intro. to Graph Theory 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 (edge- weighted) and/or the vertices (vertex-weighted) b a c d ef 10 5 8 -3 2 6

10. 10 Intro. to Graph Theory Digraph definitions for digraphs only… Every edge has a head (starting point) and a tail (ending point) Walks, trails, and paths can only use edges in the appropriate direction In a DAG, every path connects an predecessor/ancestor (the vertex at the head of the path) to its successor/descendents (nodes at the tail of any path). parent: direct ancestor (one hop) child: direct descendent (one hop) A descendent vertex is reachable from any of its ancestors vertices Directed graph a b c d x y z w u v

11. 11 Intro. to Graph Theory 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 v i and v j (or 0 for no edge) adjacency list: a |V| array where each cell i contains a list of all vertices adjacent to v i 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 cb ad 4 2 6 108 adjacency matrix adjacency list incidence matrix

12. 12 Intro. to Graph Theory 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 Graph = [] # list of nodes 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

13. 13 Intro. to Graph Theory 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’ Subgraphs a b c d e b c d e a c d G(V,E)G’({a,c,d},{{c,d}}) Induced subgraph of G with V’ = {b,c,d,e}

14. 14 Intro. to Graph Theory 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) Complement of a graph a b c d e G a b c d e

15. 15 Intro. to Graph Theory 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! Famous problems: Shortest path cb ad 4 2 6 108

16. 16 Intro. to Graph Theory Dijkstra’s Algorithm D(x) = distance from s to x (initially all  ) 1.Select the closest vertex to s, according to the current estimate (call it c) 2.Recompute the estimate for every other vertex, x, as the MINIMUM of: 1.The current distance, or 2.The distance from s to c, plus the distance from c to x – D(c) + W(c, x)

17. 17 Intro. to Graph Theory Dijkstra’s Algorithm Example ABCDE Initial0  Process A010320  Process C0532018 Process B0531018 Process D0531018 Process E0531018 A B CE D 10 5 20 2 3 15 11

18. 18 Intro. to Graph Theory 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 Famous problems: Isomorphism a b c d e 1 23 45

19. 19 Intro. to Graph Theory 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 12 43 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} }

20. 20 Intro. to Graph Theory Famous problems: Coloring vertex coloring: labeling the vertices such that no edge in E has two end- points 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? Would you believe that this problem (in general) is intractable? 12 43

21. 21 Intro. to Graph Theory 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 ab c d e f g h i a b c d e 3 4 1 3 54 3 2 2

22. 22 Intro. to Graph Theory 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? K 4,4

23. 23 Intro. to Graph Theory 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? a b c d e f g h 12 43 cut-sets: {(a,b),(a,c)}, {(b,d),(c,d)},{(d,f)},...

24. 24 Intro. to Graph Theory 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 ab c fe d Colors?

25. 25 Intro. to Graph Theory 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 a b c d e f g h 2 2 2 4 8 1 4 6 2 4 a b c d e f g h 2 2 2 8 1 4 6 2 4

26. 26 Intro. to Graph Theory 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. Famous problems: Euler circuit area barea d area c b c d a This one is in P!

27. 27 Intro. to Graph Theory Famous problems: Dictionary How can we organize a dictionary for fast lookup? abcyz … abcyz … abcyz … abcyz … abcyz … abcyz … “CAB” 26-ary “trie”

28. 28 Intro. to Graph Theory 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. b e a d f c 3 1 2 4 5 6 7

29. 29 Intro. to Graph Theory 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. b e a d f c 3 1 2 4 5 6 7 Start node: a End node: d Traversal order: a, c, f, e, b, d

30. 30 Intro. to Graph Theory 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 b e a d f c 3 1 2 4 5 6 7 Traversal order: a, b, c, d, e, f

31. 31 Intro. to Graph Theory 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! b e a d f c 3 1 2 4 5 6 7 Traversal order: a, b, d, e, f, c DFS (G, v) V.state = “visited” Process vertex v Foreach edge (v,w) { if w.state = “unseen” { DFS (G, w) process edge (v,w) }

32. 32 Intro. to Graph Theory 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. b e a d f c 3 1 2 4 5 6 7 Path Current Best ACF711 ACFE1511 < prune AB311 ABD88 ABE78 ABEF 148 Traversal order: Path Current Best A0- AE2- AEB6- AEBD1111 AEF 911 AEFC 1511 AC111

33. 33 Intro. to Graph Theory 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 5 46 8 1 2 3 10 9 7