1/29/18 Objec&ves • Graphs • Graph Connec&vity, Traversal • BFS & DFS Implementa&ons, Analysis Jan 29, 2018 CSCI211 - Sprenkle 1 Review • What is a heap? Ø When is it useful? • What is a graph? Ø What is one way to implement a graph? Jan 29, 2018 CSCI211 - Sprenkle 2 1
1/29/18 Graph Representa&on: Adjacency Matrix • n×n matrix with A uv = 1 if (u, v) is an edge Ø Two representa&ons of each edge (symmetric matrix) Ø Space? Ø Checking if (u, v) is an edge? Ø Iden&fying all edges? 1 2 3 4 5 6 7 8 1 0 1 1 0 0 0 0 0 2 1 0 1 1 1 0 0 0 3 1 1 0 0 1 0 1 1 4 0 1 0 0 1 0 0 0 5 0 1 1 1 0 1 0 0 6 0 0 0 0 1 0 0 0 7 0 0 1 0 0 0 0 1 8 0 0 1 0 0 0 1 0 Jan 29, 2018 CSCI211 - Sprenkle 3 Graph Representa&on: Adjacency Matrix • n×n matrix with A uv = 1 if (u, v) is an edge Ø Two representa&ons of each edge (symmetric matrix) Ø Space: Θ (n 2 ) Ø Checking if (u, v) is an edge: Θ (1) &me Ø Iden&fying all edges: Θ (n 2 ) &me 1 2 3 4 5 6 7 8 1 0 1 1 0 0 0 0 0 2 1 0 1 1 1 0 0 0 3 1 1 0 0 1 0 1 1 4 0 1 0 0 1 0 0 0 5 0 1 1 1 0 1 0 0 6 0 0 0 0 1 0 0 0 7 0 0 1 0 0 0 0 1 8 0 0 1 0 0 0 1 0 Jan 29, 2018 CSCI211 - Sprenkle 4 2
1/29/18 Graph Representa&on: Adjacency List • Node indexed array of lists Ø Two representa&ons of each edge What are the Ø Space? extremes? Ø Checking if (u, v) is an edge? Ø Iden&fying all edges? edges 1 2 3 2 1 3 4 5 1 2 5 7 8 3 node 4 2 5 5 2 3 4 6 5 6 7 3 8 8 3 7 Jan 29, 2018 CSCI211 - Sprenkle 5 Graph Representa&on: Adjacency List • Node indexed array of lists degree = number of Ø Two representa&ons of each edge neighbors of u Ø Space = 2m + n = O(m + n) Ø Checking if (u, v) is an edge takes O(deg(u)) &me Ø Iden&fying all edges takes Θ (m + n) &me edges 1 2 3 2 1 3 4 5 3 1 2 5 7 8 node 4 2 5 5 2 3 4 6 6 5 7 3 8 8 3 7 Jan 29, 2018 CSCI211 - Sprenkle 6 3
1/29/18 Paths and Connec&vity • Def. A path in an undirected graph G = (V, E) is a sequence P of nodes v 1 , v 2 , …, v k-1 , v k Ø Each consecu&ve pair v i , v i+1 is joined by an edge in E • Def. A path is simple if all nodes are dis%nct • Def. An undirected graph is connected if ∀ pair of nodes u and v, there is a path between u and v • Short path • Distance Jan 29, 2018 CSCI211 - Sprenkle 7 Cycles • Def. A cycle is a path v 1 , v 2 , …, v k-1 , v k in which v 1 = v k , k > 3, and the first k-1 nodes are all dis&nct cycle C = 1-2-4-5-3-1 Jan 29, 2018 CSCI211 - Sprenkle 8 4
1/29/18 TREES Jan 29, 2018 CSCI211 - Sprenkle 9 Trees • Def. An undirected graph is a tree if it is connected and does not contain a cycle • Simplest connected graph Ø Dele&ng any edge from a tree will disconnect it Jan 29, 2018 CSCI211 - Sprenkle 10 5
1/29/18 Rooted Trees • Given a tree T, choose a root node r and orient each edge away from r • Models hierarchical structure root r parent of v v child of v a tree the same tree, rooted at 1 Why n-1 edges? Jan 29, 2018 CSCI211 - Sprenkle 11 Rooted Trees • Why n-1 edges? Ø Each non-root node has an edge to its parent root r parent of v v child of v a tree the same tree, rooted at 1 Jan 29, 2018 CSCI211 - Sprenkle 12 6
1/29/18 Trees • Theorem. Let G be an undirected graph on n nodes. Any two of the following statements imply the third: Ø G is connected Ø G does not contain a cycle Ø G has n -1 edges Jan 29, 2018 CSCI211 - Sprenkle 13 Phylogeny Trees • Describe evolu&onary history of species Ø mammals and birds share a common ancestor that they animals do not share with other species Ø all animals are descended from an ancestor not shared with mushrooms, trees, and bacteria Jan 29, 2018 CSCI211 - Sprenkle 14 7
1/29/18 GRAPH CONNECTIVITY & TRAVERSAL Jan 29, 2018 CSCI211 - Sprenkle 15 Connec&vity • s-t connec9vity problem. Given nodes s and t , is there a path between s and t ? • s-t shortest path problem. Given nodes s and t , what is the length of the shortest path between s and t ? • Applica&ons Ø Facebook Ø Maze traversal Ø Kevin Bacon number Ø Spidering the web Ø Fewest number of hops in a communica&on network Jan 29, 2018 CSCI211 - Sprenkle 16 8
1/29/18 Applica&on: Connected Component • Find all nodes reachable from s • Connected component containing node 1 is { 1, 2, 3, 4, 5, 6, 7, 8 } Jan 29, 2018 CSCI211 - Sprenkle 17 Applica&on: Flood Fill • Given lime green pixel in an image, change color of en&re blob of neighboring lime pixels to blue Ø Node: pixel Ø Edge: two neighboring lime pixels Ø Blob: connected component of lime pixels recolor lime green blob to blue Jan 29, 2018 CSCI211 - Sprenkle 18 9
1/29/18 Applica&on: Flood Fill • Given lime green pixel in an image, change color of en&re blob of neighboring lime pixels to blue Ø Node: pixel Ø Edge: two neighboring lime pixels Ø Blob: connected component of lime pixels recolor lime green blob to blue Jan 29, 2018 CSCI211 - Sprenkle 19 My Facebook Friends Created with Social Graph UDel HS Family Gburg Duke Extreme Blue Jan 29, 2018 CSCI211 - Sprenkle 20 10
1/29/18 A General Algorithm R will consist of nodes to which s has a path R = {s} while there is an edge (u,v) where u ∈ R and v ∉ R add v to R it's safe to R add v s u v • R will be the connected component containing s • Algorithm is underspecified In what order should we consider the edges? Jan 29, 2018 CSCI211 - Sprenkle 21 Possible Orders • Breadth-first • Depth-first Jan 29, 2018 CSCI211 - Sprenkle 22 11
1/29/18 Breadth-First Search • Intui9on . Explore outward from s in all possible direc&ons (edges), adding nodes one "layer" at a &me L 0 • Algorithm s L 1 L 2 L n-1 Ø L 0 = { s } Ø L 1 = all neighbors of L 0 Ø L 2 = all nodes that have an edge to a node in L 1 and do not belong to L 0 or L 1 Ø L i+1 = all nodes that have an edge to a node in L i and do not belong to an earlier layer Jan 29, 2018 CSCI211 - Sprenkle 23 Run BFS on This Graph s = 1 Jan 29, 2018 CSCI211 - Sprenkle 24 12
1/29/18 Example of Breadth-First Search s = 1 L 0 L 1 L 2 L 3 Creates a tree -- is a node in the graph that is not in the tree Jan 29, 2018 CSCI211 - Sprenkle 25 Breadth-First Search • Theorem. For each i , L i consists of all nodes at distance exactly i from s . There is a path from s to t iff t appears in some layer. s L 1 L 2 L n-1 • What does this theorem mean? • Can we determine the distance between s and t? Jan 29, 2018 CSCI211 - Sprenkle 26 13
1/29/18 Looking Ahead • Monday, 11:59 p.m.: journal - Chapter 2.4, 2.5, 3.1 • Friday: Problem Set 3 due Jan 29, 2018 CSCI211 - Sprenkle 27 14
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