Graphs and Genetics. Outline. Computational Biology IST. Ana Teresa Freitas 2015/2016. Slides source: AED (MEEC/IST); Jones and Pevzner (book)

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1 raphs and enetics Computational Biology IST Ana Teresa Freitas / Slides source: AED (MEEC/IST); Jones and Pevzner (book) Outline l Motivacion l Introduction to raph Theory l Eulerian & Hamiltonian Cycle Problems l Benzer Experiment and Internal raphs

2 Sequencing Clone-by-clone shotgun sequencing Human enome Project Whole-genome shotgun sequencing Celera enomics (BACs) E-learning examples

3 The Bridge Obsession Problem Find a tour crossing every bridge just once Leonhard Euler, 7 Bridges of Königsberg Eulerian Cycle Problem l Find a cycle that visits every edge exactly once l Linear time

4 Applications l Maps l Hypertexts l Circuits l Schedules l Networks l Fragment Assembly in DNA Sequencing l Protein Sequencing and Identification l Biological networks l Brain connectome Introduction to raph Theory l Abstract object l Entities l Edges l Vertices or nodes l Vertices represent l People, places, genes, sequences l Edges represent l connections

5 Definitions l A graph is a set of vertices plus a set of edges that connect pairs of distinct vertices (with at most one edge connecting any pair of vertices) l When there is an edge connecting two vertices, we say that the vertices are adjacent to one another and that the edge is incident on both vertices Definitions l The degree of a vertex is the number of edges incident to the vertex l A subgraph of a graph is a graph whose vertex and edge sets are subsets of those of l A supergraph of a graph is a graph that contains as a subgraph

6 Example: 8 7 l l l Vertices and 7 are adjacent Vertices and are non-adjacent Vertex 7 has degree four Definitions l A walk is an alternating sequence of vertices and edges, beginning and ending with a vertex, in which each vertex is incident to the two edges that precede and follow it in the sequence, and the vertices that precede and follow an edge are the endvertices of that edge l A walk is closed if its first and last vertices are the same, and open if they are different (path) l The length L of a walk is the number of edges that it uses

7 Definitions l A simple path is a walk in which all vertices and edges are distinct l The closed equivalent to this type of walk is called a cycle l Two paths are internally disjoint if they do not have any vertex in common, except the first and last ones Example: 8 7 l Path:

8 Example: 8 7 l Cycle: Definitions l The graph is said to be connected if it is possible to establish a path from any vertex to any other vertex of a graph l A component is a maximally connected subgraph l A spanning graph H of a graph, is a graph that has the same vertex set as 8

9 Definitions l A tree is a connected acyclic simple graph. A vertex of degree is called a leaf l A set of trees is called a forest l A spanning tree is a spanning subgraph that is a tree l A k-ary tree is a rooted tree in which every internal vertex has k children Example: raph 8 7 l is a subgraph of l is a connected graph but is not l Subgraph has a component and a tree 9

10 Example: 8 7 l : Spanning tree of. raph properties l raphs with all edges present are called complete graphs l A complete subgraph is called a clique l We define the complement of a graph by starting with a complete graph that has the same set of vertices as the original graph, and removing the edges of

11 raph properties l A graph with V vertices has at most V(V-)/ edges l A dense graph is a graph whose average vertex degree is proportional to V l A sparse graph is a graph whose complement is dense l The density of a graph is the average degree, or E/V. E is the number of edges and V the number of vertices raph properties l A bipartide graph is a graph whose vertices we can divide into two sets such that all edges connect a vertex in one set with a vertex in the other set l raphs as defined to this point are called undirected graphs l In directed graphs, also known as digraphs, edges are one-way

12 raph properties l The first vertex in a directed edge is called the source, the second vertex is called the destination l The indegree of a vertex is the number of edges where it is the destination l The outdegree of a vertex is the number of edges where it is the source raph properties l A directed cycle in a digraph is a cycle in which all adjacent vertex pairs appear in the order indicated by (directed) graph edges l A directed acyclic graph (DA), is a digraph that has no directed cycles l A DA (an acyclic digraph) is not the same as a tree (na acyclic undirected graph)

13 raph properties l In weighted graphs, we associate numbers (weights) with each edge, which generally represents a distance or a cost l We also might associate a weight with each vertex, or multiple weights with each vertex and edge l Weighted digraphs are also refered as networks Adjacency-matrix l raph Matrix Size: V V symmetric

14 Adjacency-lists raph Lists Array of size V Linked list that is associated with that vertex Adj. Matrix main advantages l Should be used when: l there is space available; l for dense graphs; l the algorithms performs more than V operations. l Insert and remove is very efficient; l Simple to avoid parallel edges; l Simple to check if two vertices are connected

15 Adj. Lists main advantages l Initialization proportional to V. l Space proportional to V+E l Useful to represent sparse graphs l Edges insertion is very efficient raphs - First Problems l Simple Path l iven two vertices of a graph, identify if they are connected l Hamilton Path l iven two vertices of a graph, identify if there is a path that visits all the vertices exactly once l Euler Path l iven two vertices of a graph, identify if there is a path that visits all the edges exactly once

16 raphs Simple path l Call the algorithm with the following parameters: l, the graph l v, the initial vertex l w, the end vertex l Based in a depth first search. l From vertex v, find the first adjacent vertex, t. Call the algorithm recursively to find w from t. l The visited vector is used to verify that each vertex is visited only once. raphs Simple path int RAPHpath(raph, int v, int w) { int t; for (t = ; t < ->V; t++) visited[t] = ; return pathr(, v, w); }

17 raphs Simple path static int visited[maxv]; int pathr(raph, int v, int w) { int t; if (v == w) return ; visited[v] = ; for (t = ; t < ->V; t++) if (->adj[v][t] == ) if (visited[t] == ) if (pathr(, t, w)) return ; return ; } raphs Simple path 7 Call the algorithm to find a path from vertex to vertex The graph is represented by an Adjacency-matrix RAPHpath(,, ) 7

18 raphs Simple path l Calls of function pathr pathr(,, ) 7 raphs Simple path l Calls of function pathr 7 - pathr(,, ) pathr(,, ) -; ( and visited) - (not adjacent) 8

19 raphs Simple path l Calls of function pathr 7 pathr(,, ) - pathr(,, ) -; ( and visited) - (not adjacent) - pathr(,, ) -; ( and visited) raphs Simple path l Calls of function pathr 7 pathr(,, ) - pathr(,, ) -; ( and visited) - (not adjacent) - pathr(,, ) -; ( and visited) - pathr(,, ) -; (not adjacent) - (visited)... 9

20 raphs Simple path l Call of functions pathr 7. - ( visited). -; (not adjacent). - pathr(,, ) raphs Simple path l Calls of functions pathr 7 Nodes visited:. - ( visited). -; (not adjacent). - pathr(,, ) -; (not adjacent) - ( visited) - (not adjacent) - pathr(,, )

21 Exercice simple path 7 Call the algorithm to find a path from vertex to vertex. Present the list of visited nodes. The graph is represented by an Adjacency-matrix RAPHpath(,, ) Hamilton path l Hamilton path l iven two vertices, is there a simple path connecting them that visits every vertex in the graph exactly once? l If the path is from a vertex back to itself, this problem is known as the Hamilton tour problem. l Property: A recursive search for a Hamilton tour could take exponential time.

22 Hamilton path l This algorithm is very similar to the algorithm for finding simple paths. l Why are the running times dramatically different? l The simple path algorithm is guaranteed to finish quickly because it sets at least one element of the visited array to each time pathr is called l The Hamilton path algorithm can set visited elements back to. We cannot guarantee that it will finish quickly Hamilton path static int visited[maxv]; int pathr(raph, int v, int w, int int d d) {int t; if (v == w) { return if (d == ; ) return ; else return ;} visited[v] = ; for (t = ; t < ->V; t++) if (->adj[v][t] == ) if (visited[t] == ) if (pathr(, t, w, d-)) d- return ; visited[v] = ; visited[v] = ; return ;} int RAPHpathH(raph H, int v, int w) { int t; for (t = ; t < ->V; t++) visited[t] = ; return pathr(, v, w, ->V- ->V-); }

23 Hamilton path Compute a simple path connecting between and that visits every vertex in the graph exactly once RAPHpathH(,, ) Hamilton path l Calls of function pathr pathr(,,,) - ( visited)

24 Hamilton path l Calls of function pathr pathr(,,,) - ( visited) - pathr(,,,) -; ( and visited) Hamilton path l Calls of function pathr pathr(,,,) - ( visited) - pathr(,,,) -; ( and visited) - pathr(,,,) -;; (,, and visited) -; (not adjacent)

25 Hamilton path l Calls of function pathr pathr(,,,) - ( visited) - pathr(,,,) -; ( and visited) - pathr(,,,) -;; (, and visited) -; (not adjacent) (return with vertex as not visited) Hamilton path l Calls of function pathr pathr(,,,) - ( visited) - pathr(,,,) -; ( and visited) - pathr(,,,) -;; (, and visited) -; (not adjacent) (return with vertex as not visited) - pathr(,,,) - (not adjacent) (...)

26 Hamilton path l Calls of function pathr (...) - ( visited) - (not adjacent) - ( visited) - pathr(,,,) Hamilton path l Calls of function pathr (...) - ( visited) - (not adjacent) - ( visited) - pathr(,,,) (d >, return e clean )

27 Hamilton path l Calls of function pathr (...) - ( visited) - (not adjacent) - ( visited) - pathr(,,,) (d >, return e clean ) (return cleaning ) - (not adjacent) Hamilton path l Calls of function pathr (...) - ( visited) - (not adjacent) - ( visited) - pathr(,,,) (d >, return e clean ) (return cleaning ) - (not adjacent) (return cleaning ) 7

28 Hamilton path l Calls sequence: pathr (...) - ( visited) - (non adjacent) - ( visited) - pathr(,,,) (d >, return and remove ) (return removing ) - (non adjacent) (return removing ) - pathr(,,,) (...) Hamilton path l Calls sequence: pathr (...) - ( visited) - pathr(,,,) -;; (, and visited) 8

29 Hamilton path l Calls sequence: pathr (...) - ( visited) - pathr(,,,) -;; (, and visited) - pathr(,,,) - (non adjacent) -;; (, and visited) - pathr(,,,) (d ==, path found) (...) (Path found!) Euler path and tour l Euler path l Is there a path connecting two given vertices that uses each edge in the graph exactly once? l The path need not to be simple vertices may be visited multiple times l If the path is from a vertex back to itself, we have the Euler tour problem 9

30 raphs Euler path in linear time Adj. List #include STACK.h int path(raph, int v) { int w; for (; ->adj[v]!= NULL; v = w) { STACKpush(v); To STACK w = ->adj[v]->v; RAPHremoveE(, EDE(v,w)); } return v; } int raphpathe(raph,int v,int w) { STACKinit(->E); printf( %d, w); while ( (path(,v) == v && (!STACKempty()) ) printf( -%d, v = STACKpop()); printf( \n ); return (->E == ); } Create STACK of size E From STACK Euler path and tour l Property: A graph has a Euler tour if and only if it is connected and all its vertices are of even degree l Corollary: A graph has a Euler path if and only if it is connected and exactly two of its vertices are of odd degree

31 Euler path Adjacency list l : l : - l : l : - l : l : - l : - RAPHpathE(,, ) write Euler path l Call RAPHremoveE path(, ) - push() - (it was removed)

32 Euler path l Call RAPHremoveE path(, ) - push() - (it was removed) - push() Euler path l Call RAPHremoveE path(, ) - push() - (it was removed) - push() - push() -; (it was removed)

33 Euler path l Call RAPHremoveE path(, ) - push() - (it was removed) - push() - push() -; (it was removed) - push() - (it was removed) Euler path l Call RAPHremoveE path(, ) - push() - (it was removed) - push() - push() -; (it was removed) - push() - (it was removed) - push()

34 Euler path l Call RAPHremoveE path(, ) - push() - (it was removed) - push() - push() -; (it was removed) - push() - (it was removed) - push() -* push() * Assume vertex in the list before vertices and Euler path Call RAPHremoveE - push() isolated return()

35 Euler path Call RAPHremoveE - push() isolated return() pop() write - path(, ) isolated return() pop() write - path(, ) Euler path Call RAPHremoveE - push() isolated return() pop() write - path(, ) isolated return() pop() write - path(, ) - push()

36 Euler path Call RAPHremoveE - push() isolated return() pop() write - path(, ) isolated return() pop() write - path(, ) - push() - push() Euler path Call RAPHremoveE - push() isolated return() pop() write - path(, ) isolated return() pop() write - path(, ) - push() - push() - push() isolated return() pop() write -

37 Euler path Call RAPHremoveE path(, ) isolated return() pop() write - path(, ) isolated return() pop() write - path(, ) isolated return() pop() write - path(, ) isolated return() pop() write - Euler path Call RAPHremoveE path(, ) isolated return() pop() write - path(, ) isolated return() pop() write - path(, ) isolated return() pop() write - The path