![Path
• A path is a sequence of vertices such that there is an
edge from each vertex to its successor.
• A path is simple if each vertex is distinct.
1 2 3
4 5 6
Cycle
Simple path from 1 to 5
= [ 1, 2, 4, 5 ]
Our text’s alternates the vertices
and edges.
A
D E F
B C
Unreachable
Cycle
If there is path p from u to v then we
say v is reachable from u via p.](https://image.slidesharecdn.com/04basicgraphtheory-250207184812-483469d3/85/Lecture-4-Design-Analysis-Of-ALgorithms-13-320.jpg)
![Representation (Matrix)
• Incidence Matrix
– E x V
– [edge, vertex] contains the edge's data
• Adjacency Matrix
– V x V
– Boolean values (adjacent or not)
– Or Edge Weights](https://image.slidesharecdn.com/04basicgraphtheory-250207184812-483469d3/85/Lecture-4-Design-Analysis-Of-ALgorithms-26-320.jpg)
![Implementation of a Graph.
• Adjacency-list representation
– an array of |V | lists, one for each vertex in V.
– For each u V , ADJ [ u ] points to all its adjacent
vertices.](https://image.slidesharecdn.com/04basicgraphtheory-250207184812-483469d3/85/Lecture-4-Design-Analysis-Of-ALgorithms-28-320.jpg)
Lecture 4- Design Analysis Of ALgorithms
- 3. Graph Theory - History Leonhard Euler's paper on “Seven Bridges of Königsberg” (Kaliningrad ) , published in 1736.
- 4. Famous problems • “The traveling salesman problem” – A traveling salesman is to visit a number of cities; how to plan the trip so every city is visited once and just once and the whole trip is as short as possible ? • In 1852 Francis Guthrie posed the “four color problem” which asks if it is possible to color, using only four colors, any map of countries in such a way as to prevent two bordering countries from having the same color. • This problem, which was only solved a century later in 1976 by Kenneth Appel and Wolfgang Haken, can be considered the birth of graph theory.
- 5. Examples • Cost of wiring electronic components • Shortest route between two cities. • Shortest distance between all pairs of cities in a road atlas. • Matching / Resource Allocation • Task scheduling • Visibility / Coverage
- 6. Examples • Flow of material – liquid flowing through pipes – current through electrical networks – information through communication networks – parts through an assembly line • In Operating systems to model resource handling (deadlock problems) • In compilers for parsing and optimizing the code.
- 7. Basics
- 8. What is a Graph? • Informally a graph is a set of nodes joined by a set of lines or arrows. 1 1 2 3 4 4 5 5 6 6 2 3
- 9. Definition: Graph • G is an ordered triple G:=(V, E, f) – V is a set of nodes, points, or vertices. – E is a set, whose elements are known as edges or lines. – f is a function • maps each element of E • to an unordered pair of vertices in V.
- 10. Definitions • Vertex – Basic Element – Drawn as a node or a dot. – Vertex set of G is usually denoted by V(G), or V • Edge – A set of two elements – Drawn as a line connecting two vertices, called end vertices, or endpoints. – The edge set of G is usually denoted by E(G), or E.
- 12. 12 Introduction to graph theory Edge types: Undirected; E.g., distance between two cities, PPIs, friendships… Directed; ordered pairs of nodes. E.g. metabolic reactions, transcriptional regulation,… Loops; usually we assume no loops. London Paris London Paris
- 13. Path • A path is a sequence of vertices such that there is an edge from each vertex to its successor. • A path is simple if each vertex is distinct. 1 2 3 4 5 6 Cycle Simple path from 1 to 5 = [ 1, 2, 4, 5 ] Our text’s alternates the vertices and edges. A D E F B C Unreachable Cycle If there is path p from u to v then we say v is reachable from u via p.
- 14. Cycle • A path from a vertex to itself is called a cycle. • A graph is called cyclic if it contains a cycle; – otherwise it is called acyclic 1 2 3 4 5 6 Cycle A D E F B C Unreachable Cycle
- 15. Connectivity • is connected if – you can get from any node to any other by following a sequence of edges OR – any two nodes are connected by a path. • A directed graph is strongly connected if there is a directed path from any node to any other node.
- 16. Sparse/Dense • A graph is sparse if | E | | V | • A graph is dense if | E | | V |2.
- 17. A weighted graph • is a graph for which each edge has an associated weight, usually given by a weight function w: E R. 1 2 3 4 5 6 .5 1.2 .2 .5 1.5 .3 1 4 5 6 2 3 2 1 3 5
- 18. Special Types • Empty Graph / Edgeless graph – No edge • Null graph – No nodes – Obviously no edge
- 19. Complete Graph • Denoted Kn • Every pair of vertices are adjacent • Has n(n-1) edges
- 20. Planar Graph • Can be drawn on a plane such that no two edges intersect • K4 is the largest complete graph that is planar
- 21. Degree • Number of edges incident on a node A D E F B C The degree of B is 2.
- 22. Degree (Directed Graphs) • In degree: Number of edges entering • Out degree: Number of edges leaving • Degree = indegree + outdegree 1 2 4 5 The in degree of 2 is 2 and the out degree of 2 is 3.
- 23. Subgraph • Vertex and edge sets are subsets of those of G – a supergraph of a graph G is a graph that contains G as a subgraph.
- 24. Spanning subgraph • Subgraph H has the same vertex set as G. – Possibly not all the edges – “H spans G”.
- 25. 25 25 Minimum spanning tree. Given a connected graph G = (V, E) with real- valued edge weights (costs) ce, an MST is a subset of the edges T E such that T is a spanning tree (contains all nodes of G) whose sum of edge weights is minimized. Cayley's Theorem. There are nn-2 spanning trees of Kn. 5 23 10 21 14 24 16 6 4 18 9 7 11 8 5 6 4 9 7 11 8 G = (V, E) T, eT ce = 50 can't solve by brute force Minimum spanning tree
- 26. Representation (Matrix) • Incidence Matrix – E x V – [edge, vertex] contains the edge's data • Adjacency Matrix – V x V – Boolean values (adjacent or not) – Or Edge Weights
- 27. Representation (List) • Edge List – pairs (ordered if directed) of vertices – Optionally weight and other data • Adjacency List
- 28. Implementation of a Graph. • Adjacency-list representation – an array of |V | lists, one for each vertex in V. – For each u V , ADJ [ u ] points to all its adjacent vertices.
- 29. Adjacency-list representation for a directed graph. 1 5 1 2 2 5 4 4 3 3 2 5 5 3 4 4 5 5 Variation: Can keep a second list of edges coming into a vertex.
- 30. Adjacency lists • Advantage: – Saves space for sparse graphs. Most graphs are sparse. – Traverse all the edges that start at v, in (degree(v)) • Disadvantage: – Check for existence of an edge (v, u) in worst case time (degree(v))
- 31. Adjacency List • Storage – For a directed graph the number of items are (out-degree (v)) = | E | So we need ( V + E ) – For undirected graph the number of items are (degree (v)) = 2 | E | Also ( V + E ) • Easy to modify to handle weighted graphs. How? v V v V
- 32. Adjacency matrix representation • |V | x |V | matrix A = ( aij ) such that aij = 1 if (i, j ) E and 0 otherwise. We arbitrarily uniquely assign the numbers 1, 2, . . . , | V | to each vertex. 1 5 2 4 3 1 2 3 4 5 1 2 3 4 5 0 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0
- 33. Adjacency Matrix Representation for a Directed Graph 1 2 3 4 5 1 2 3 4 5 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 5 2 4 3
- 34. Adjacency Matrix Representation • Advantage: – Saves space for: • Dense graphs. • Small unweighted graphs using 1 bit per edge. – Check for existence of an edge in (1) • Disadvantage: – Traverse all the edges that start at v, in (|V|)
- 35. Adjacency Matrix Representation • Storage – ( | V |2 ) ( We usually just write, ( V 2 ) ) – For undirected graphs you can save storage (only 1/2(V2 )) by noticing the adjacency matrix of an undirected graph is symmetric. How? • Easy to handle weighted graphs. How?