Graph Algorithms Chapter 22 1 CPTR 430 Algorithms Graph - PowerPoint PPT Presentation

Graph Algorithms Chapter 22 1 CPTR 430 Algorithms Graph Algorithms

Why Study Graph Algorithms? ■ Mathematical graphs seem to be relatively specialized and abstract ■ Why spend so much time and effort on algorithms for such an obscure area? ■ As it turns out, many practical problems can be solved by applying some basic graph algorithms 2 CPTR 430 Algorithms Graph Algorithms

☎ ✝ ☎ ☎ ☎ ☎ ✆ ☎ ✄ ☎ ✆ ✂ ☎ ✁ � ☎ ✝ ☎ Notation ■ A graph is a mathematical structure that consists of a set of vertices and a set of edges ■ G G E V is the number of vertices ■ E is the number of edges ■ ■ In the context of asymptotic notation, V means V and E means E V 2 2 ■ For example, O really means O V 3 CPTR 430 Algorithms Graph Algorithms

Graph Representation ■ Adjacency matrix ■ Adjacency list 4 CPTR 430 Algorithms Graph Algorithms

Undirected Graph Representations 0 1 2 4 3 0 1 2 3 4 1 4 0 0 0 1 0 0 1 0 4 2 3 1 1 1 0 1 1 1 2 1 3 2 0 1 0 1 0 3 1 4 2 3 0 1 1 0 1 4 3 0 1 4 1 1 0 1 0 5 CPTR 430 Algorithms Graph Algorithms

Directed Graph Representations 0 1 2 4 5 3 0 1 2 3 4 5 1 3 0 0 0 1 0 1 0 0 4 1 1 0 0 0 0 1 0 2 5 4 2 0 0 0 0 1 1 3 1 3 0 1 0 0 0 0 4 3 4 0 0 0 1 0 0 5 5 5 0 0 0 0 0 1 6 CPTR 430 Algorithms Graph Algorithms

☎ ✝ ☎ ☎ ✆ ☎ � ✝ ☎ ☎ ✆ ☎ ☎ ☎ ☎ ☎ ☎ Which Representation is Better? ■ Adjacency lists are good for sparse graphs 2 ❚ Sparse graph: E V ❚ The list nodes could instead contain pointers to vertices E ❚ For digraphs: sum of lengths of the lists = ❚ For undirected graphs: sum of lengths of the lists = 2 E ❚ Space required: Θ V E ■ Adjacency matrix is better for dense graphs 2 E V ❚ Dense graph: is not mush less than ❚ Can quickly tell if two vertices share an edge ❚ Space required: Θ V 2 , independent of the number of edges ❚ Undirected graphs can use a triangular matrix ❚ Unweighted graphs require only one bit per edge! 7 CPTR 430 Algorithms Graph Algorithms

✂ ✁ ✆ ✝ ✝ ✂ ✆ ✂ � ✝ ✂ ✆ Weighted Graphs u v E , w : E ■ For each ■ w u v u v is the weight of edge ■ Weighted graphs represent edge costs, lengths, time, etc. ■ Adjacency list: the edge weight is stored in the node for the edge in the list ■ Adjacency matrix: the edge weight is stored in matrix 12 0 1 10 8 2 10 9 8 11 3 4 8 CPTR 430 Algorithms Graph Algorithms

� Breadth-first Search ■ Used in a number of classic graph algorithms: ❚ Prim’s minimal spanning tree ❚ Dijkstra’s single source shortest path ■ Finds all the vertices reachable from a source vertex, s ■ Produces a breadth-first tree whose root is s ■ Computes the length of the path to each reachable vertex Path length = number of edges in the path ■ Breadth-first all the vertices at distance k are located before any reachable vertices farther than distance k are found 9 CPTR 430 Algorithms Graph Algorithms

✆ ✂ ✝ � � Coloring the Vertices ■ BFS marks vertices to manage its progress ■ Vertices are: ❚ Colored white initially ❚ Colored gray when first discovered ❚ Colored black when all their neighbors have been discovered u v E and u is black v is either gray or black ■ ■ Gray vertices represent the frontier between discovered and undiscovered vertices 0 1 2 3 4 10 CPTR 430 Algorithms Graph Algorithms

� Breadth-first Tree ■ BFS essentially builds a tree ❚ The tree’s root is the source vertex ❚ Vertex v discovered while scanning vertex u neighbors the tree has an edge from parent u to child v 0 1 2 3 4 11 CPTR 430 Algorithms Graph Algorithms

✁ � ✁ � Implementation Particulars ■ Augment vertices with additional information: ❚ color (black, gray, white) ( color ) ❚ predecessor (parent vertex in the BF tree), π v ( predecessor ) ❚ distance (number of edges from root [source vertex] to this vertex), d v ( distance ) ■ Use a queue to manage the gray vertices 12 CPTR 430 Algorithms Graph Algorithms

✁ � How It Works ■ First, for each vertex: ❚ set its color to white (undiscovered) ❚ set its predecessor to null (not in the BF tree) ❚ set its distance to ∞ (may not even be a path to it from the source vertex) ■ For the start vertex s : ❚ color s gray ❚ set d s to 0 ❚ place s in the queue 13 CPTR 430 Algorithms Graph Algorithms

✁ ✁ ✁ � � � � How It Works (cont.) ■ As long as the queue is not empty, extract a vertex u from the queue and: ❚ for each vertex v adjacent to u : ❙ color v gray ❙ set d v to d u 1 ❙ set π v to u ❙ Place v into the queue ❚ Color u black 14 CPTR 430 Algorithms Graph Algorithms

� � ✝ � ✝ ✆ ✆ ✝ ✆ Analysis of Running Time ■ Use aggregate analysis ■ Every vertex must be colored and properly initialized the running time for initialization is O V ■ After initialization, no vertex is ever made white again each vertex is enqueued at most once Unreachable vertices will not be enqueued ■ Recall that the time to enqueue and dequeue elements is O 1 all queue operations take O V time 15 CPTR 430 Algorithms Graph Algorithms

� � � ✆ ✝ ✝ ✆ � ✆ ✆ ✝ ✝ ✆ � ✆ � ✝ ✝ Analysis of Running Time (cont.) ■ Each time a vertex is dequeued: ❚ its adjacent vertices (via its adjacency list) are each are checked to see if they need to be enqueued (i.e., are white) ❚ a vertex’s adjacency list is checked once only ❚ the sum of the vertices in all adjacency lists is Θ E And entry in an adjacency list represents an edge The time spent scanning the adjacency lists is O E the total time for breadth-first search is ■ O V O V O E O V E ■ This means that BFS runs in time linear in the size of the (complete) adjacency list representation of the graph 16 CPTR 430 Algorithms Graph Algorithms

✆ ✂ ✆ ✂ � ✆ � ✂ ✝ ✝ ✝ ✆ ✝ ✆ ✂ ✝ ✝ � ✝ ✂ ✆ � � � ✂ ✂ ✝ ✆ � ✂ ✝ ✝ ✂ ✆ ✝ � ✁ ✂ ✂ ✂ ✄ ✆ ☎ � � ✆ ✆ ✝ ✝ Shortest Path ■ Define δ s v to be the shortest distance from vertex s to vertex v ❚ The minimum number of edges in any path from s to v ❚ δ ∞ s v no path exists between s and v δ δ u v E s v s u 1 (Lemma 22.1) ■ ❚ u is reachable from s v is reachable from s ❚ The shortest path from s to v cannot be longer than the shortest path from s to u followed by edge u v δ δ s v s u 1 δ ∞ ❚ u is unreachable from s s u δ δ s v s u 1 17 CPTR 430 Algorithms Graph Algorithms

✂ ✂ � � ✁ � ✝ ✆ ✂ ✆ ✆ ✝ ✝ � ✂ � ✆ � � Shortest Path Property of BFS , BFS assigns δ For any directed or undirected graph G V E s v the distance attribute of each vertex v V δ ■ Proof: First, show something weaker: v V d v s v Lemma 22.2 ■ Proceed by induction on the number of enqueue operations ❚ Basis case: Is it true after the first enqueue operation? ❚ Inductive step: True after k enqueue operations true after k 1 enqueue operations? 18 CPTR 430 Algorithms Graph Algorithms

✁ ✝ � ✆ ✂ ✂ ✂ ✁ � ✝ � ✆ ✆ � ✂ � ✂ � ✆ ✁ � � � ✁ ✆ � ✂ ✝ � Basis Step This is the case where s , the source vertex, is enqueued Due to the initialization part of the algorithm: δ d s 0 s s and ∞ δ v V s d v s v δ v V d v s v In either situation, 19 CPTR 430 Algorithms Graph Algorithms

✂ ✂ � ✆ � ✁ � � ✆ ✁ � ✁ � ✝ ✝ ✂ � ✆ � ✝ � � Inductive Step ■ A vertex is enqueued only if it is white ■ Let vertex v be a white vertex discovered while considering vertex u ’s neighbors I.e., u v E , and v is white ■ Vertex v must be enqueued ■ The inductive step is then d v d u 1 Assignment in BFS δ s u 1 I.H. δ s v Lemma 22.1 ■ Once enqueued, v ’s color is set to gray, so it can never be enqueued again d v is never changed again, and the I.H. is maintained 20 CPTR 430 Algorithms Graph Algorithms

� ✂ ✁ ✂ � ✂ � ✝ ✁ � � � ✂ � ✝ ✂ � ✆ ✆ ✁ ✝ ✄ � ✂ � � ✂ � ✁ ✂ ✁ ✁ � Lemma 22.3 v 1 v 2 v r ■ Let be the vertices, in order, in the queue during some V E part of the execution of BFS on graph ■ v 1 is the head of the queue ■ v r is the tail of the queue 1 ■ d v r d v 1 i 1 2 r 1 d v i d v i ■ 1 21 CPTR 430 Algorithms Graph Algorithms

� � Proof of Lemma 22.3 Proceed by induction of the number of queue operations (not just the number of enqueue operations) ■ Basis case: Does the lemma hold after the first queue operation? ■ Inductive step: True after k queue operations true after k 1 queue operations? 22 CPTR 430 Algorithms Graph Algorithms

✝ ✁ � � ✁ � ✁ ✁ � ✝ ✁ � � ✁ � � ✁ � � Basis Case ■ The first queue operation is always enqueuing s , the source vertex ■ Here, r 1 : ❚ d v r d s d v 1 d v 1 1 ❚ d v i d v i holds vacuously 1 23 CPTR 430 Algorithms Graph Algorithms