An Early Problem in Graph Theory. Clicker Question 1. Konigsberg and the River Pregel

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1 raphs Topic " Hopefully, you've played around a bit with The Oracle of acon at Virginia and discovered how few steps are necessary to link just about anybody who has ever been in a movie to Kevin acon, but could there be some actor or actress who is even closer to the center of the Hollywood universe?. y processing all of the almost half of a million people in the Internet Movie atabase I discovered that there are currently 0 people who are better numbers we see that the average (Sean) onnery Number is about.8 making onnery a better center than acon" -Who is the enter of the Hollywood Universe?, University of Virginia n arly Problem in raph Theory Leonhard uler (0-8) One of the first mathematicians to study graphs The Seven ridges of Konigsberg Problem Konigsberg now called Kaliningrad puzzle for the residents of the city The river Pregel flows through the city bridges cross the river an you cross all bridges while crossing That was in 00. In 0 Harvey Keitel has become the center of the Hollywood Universe. onnery is th. each bridge only once? acon has moved up to 0 th. S raphs Konigsberg and the River Pregel licker Question How many solutions does the Seven ridges of Konigsberg Problem have? >= S raphs S raphs

2 rute orce? uler's Solution How to Solve Redraw the map as a graph (really a multigraph) a b c d uler's Proposal connected graph has an uler tour (possible to cross each edge only once while traversing each edge only once and returning to ing point) if and only if every vertex has an even number of edges ulerian ircuit What if we reduce the problem to only crossing each edge (bridge) exactly once? oesn't matter if we end up where we ed ulerian Trail S raphs 5 S raphs raph efinitions graph is comprised of a set of vertices (nodes) and a set of edges (links, arcs) connecting the vertices in a directed graph edges are one-way movement allowed from first node to second, but not second to first directed graphs also called digraphs in an undirected graph edges are two-way movement allowed in either direction efinitions In a weighted graph the edge has cost or weight that measures the cost of traveling along the edge path is a sequence of vertices connected by edges The path length is the number of edges The weighted path length is the sum of the cost of the edges in a path cycle is a path of length or more that s and ends at the same vertex a directed acyclic graph is a directed graph with no cycles S raphs S raphs 8

3 raphs We've Seen link link link link 9 xample raph omputer Scientists use graphs to model all kinds of things 5 5 link link link rpanet 99, 9 S raphs 9 S raphs 0 Roman Land Transportation Network xample raph Roman Land Transportation Network S raphs S raphs

4 xample raph xample raph nron s 00 US irport Network S raphs S raphs xample raph xample raph S raphs 5 "Jefferson" High School, Ohio hains of ffection: The Structure of dolescent Romantic and S Sexual Networks, 005, raphs

5 Representing raphs How to store a graph as a data structure? djacency Matrix Representation ountry rgentina razil olivia hile olumbia cuador rench uiana uyana Paraguay Peru Suriname Uruguay Venezuela ode r l h o Pa Pe S U V S raphs S raphs 8 The Map oloring Problem How many colors do you need to color a map, so that no countries that have a common border (not a point) are colored the same? How to solve using rute orce? reen Solution Yellow lue lue Yellow Red reen lue Yellow reen lue Yellow S raphs 9 S raphs 0

6 What bout the Ocean? Make the Ocean lue Red Yellow Yellow Red reen Red reen Yellow lue lue reen Red Yellow lue S raphs S raphs More efinitions dense graph is one with a large number of edges maximum number of edges? sparse graph is one in which the number of edges is much less than the maximum possible number of edges No standard cutoff for dense and sparse graphs S raphs raph Representation or dense graphs the adjacency matrix is a reasonable choice or weighted graphs change booleans to cost an the adjacency matrix handle directed graphs? Most graphs are sparse, not dense or sparse graphs an adjacency list is an alternative that uses less space ach vertex keeps a list of vertices it is connected to. S raphs

7 raph Implementation raph lass This raph class stores vertices ach vertex has an adjacency list what vertices does it connect to? shortest path method finds all paths from vertex to every other vertex in graph after shortest path method called queries can be made for path length from node to destination node S raphs Vertex lass (nested in raph) dge lass (nested in raph) S raphs S raphs 8

8 V Unweighted Shortest Path iven a vertex, S (for ) find the shortest path from S to all other vertices in the graph raph is unweighted (set all edge costs to ) V S V V V 8 V Word Ladders gree upon dictionary Start word and end word of same length hange one letter at a time to form step Step must also be a word xample: Start = silly, end = funny V S raphs 9 V 5 S raphs 0 raph Representation What are the vertices and when does an edge exist between two vertices? Vertices dges. Letters Words. Words Words that share one or more letters. Letters Words that share one or more letters. Words Words that differ by one letter. Words Letters Portion of raph S raphs S raphs

9 Size of raph Number of vertices and edges depends on dictionary Modified Scrabble dictionary, 5 letter words Words are vertices 80 words dge exists between word if they are one letter different,9 edges Is this graph sparse or dense?. Sparse. ense Max number of edges = N * (N - ) /,9,0 S raphs Unweighted Shortest Path lgorithm Problem: ind the shortest word ladder between two words if one exists What kind of search should we use?. readth irst Search. epth irst Search. ither one S raphs Unweighted Shortest Path lgorithm Set distance of to itself to 0 reate a queue and add the vertex while the queue is not empty remove front loop through all edges of current vertex get node edge connects to if this node has not been visited sets its distance to current distance + sets its previous node to current node add new node to queue Portion of raph S raphs 5 S raphs

10 Start at "" and enqueue it [] S raphs equeue (), loop through edges [] S raphs 8 equeue (), loop through edges [, ] S raphs 9 equeue (), loop through edges [,, ] S raphs 0

11 equeue (), loop through edges [,,, ] S raphs equeue (), loop through edges [,,,, ] S raphs one with, dequeue () [,,, ] S raphs loop through edges of ( already present) [,,, ] S raphs

12 swarm loop through edges of ( already present) [,,, ] S raphs 5 loop through edges of [,,,, swarm] S raphs loop through edges of [,,,, swarm, sware] sware swarm Unweighted Shortest Path Implement method demo how is path printed? The diameter of a graph is the longest shortest past in the graph How to find? How to find center of graph? vertex connected to the largest number of other vertices with the shortest average path length S raphs S raphs 8

13 Positive Weighted Shortest Path dges in graph are weighted and all weights are positive Similar solution to unweighted shortest path ijkstra's algorithhm dsger W. ijkstra (90 00) UT Professor lgorithm developed in 95 and published in 959. Vertex lass (nested in raph) S raphs 9 S raphs 50 ijkstra's lgorithm Pick the vertex Set the cost of the vertex to 0 and all other vertices to ININITY While there are unvisited vertices: Let the current vertex be the lowest cost vertex that has not yet been visited mark current vertex as visited for each edge from the current vertex if the sum of the cost of the current vertex and the cost of the edge is less than the cost of the destination vertex update the cost of the destination vertex set the previous of the destination vertex to the current vertex 5 ijkstra's lgorithm xample of a reedy lgorithm reedy lgorithm does what appears to be the best thing at each stage of solving a problem ives best solution in ijkstra's lgorithm oes NOT always lead to best answer air teams: (0, 0, 8, 8, 8), teams Making change with fewest coins (, 5, 0) 5 cents (, 5, ) 5 cents S raphs 5

14 5 What is the lowest cost path from to? is vertex 5 Set cost of to 0, all others to ININITY Place in a priority queue 5 5 [(,0)] pq dequeue (,0) Mark as visited 55 [ ] pq current vertex : loop through 's edges if sum of cost from to edge is less than current cost update cost and prev 5

15 5 5 [ ] pq ->, 0 + < ININITY [(,)] pq ->, 0 + < ININITY [(,)] pq [(,), (, )] pq [(,), (, )] pq ->, 0 + < ININITY [(,), (, ), (, )] pq current vertex : [(,), (, ), (, )] pq loop through 's edges if sum of cost from to edge is less than current cost 59 update cost and prev 0

16 5 5 [(, ), (, )] pq ->, + < update 's cost and previous [(, ), (, ), (, )] pq [(, ), (, ), (, )] pq ->, + < ININITY [(, ), (, ), (, ), (, )] pq 5 5 [(, ), (, ), (, ), (, )] pq [(, ), (, ), (, )] pq current vertex is, cost ->, already visited so skip loop through 's edges

17 5 [(, ), (, ), (, )] pq ->, already visited so skip [(, ), (, ), (, )] pq ->, + < ININITY [(, ), (, ), (, ), (, )] pq 5 5 [(, ), (, ), (, ), (, )] pq current vertex is lready visited so skip [(, ), (, ), (, )] pq current vertex is loop through 's edges 5 5 5

18 9 [(, ), (, )] pq ->, already visited so skip [(, ), (, )] pq ->, + < update 's cost and previous [(, ), (, ), (, )] pq [(, ), (, ), (, )] pq current vertex is loop through 's edges [(, ), (, )] pq ->, already visited so skip

19 [(, ), (, )] pq ->, + < ININITY update 's cost and previous [(, ), (, ), (, )] pq [ (, ), (, ), (, )] pq ->, already visited so skip [(, ), (, ), (, )] pq ->, + 5 < update 's cost and previous [(, ), (, ), (, ), (, )] pq [(, ), (, ), (, )] pq current vertex is loop though edges, already visited all neighbors

20 5 5 [(, ), (, )] pq current vertex is loop though edges, already visited all neighbors No unvisited vertices. ach Vertex stores cost (distance) of lowest cost path from Vertex to itself and previous vertex in path from vertex to itself. 8 Implementing ijkstra's reate a Path class to allow for multiple distances to a given vertex Use a priority queue of Paths to store the vertices and distances lternatives to ijkstra's lgorithm *, pronounced " Star" heuristic, goal of finding shortest weighted path from single vertex to goal vertex Uses actual distance like ijkstra's but also estimates remaining cost or distance distance is set to current distance from PLUS the estimated distance to the goal or example when finding a path between towns, estimate the remaining distance as the straight-line (as the crow files) distance between current location and goal. S raphs 9 S raphs 80

21 Spanning Tree Spanning Tree: tree of edges that connects all the vertices in a graph 5 Prim's lgorithm Pick a vertex arbitrarily from graph In other words, it doesn't matter which one dd lowest cost edge between the tree and a vertex that is not part of the tree UNTIL every vertex is part of the tree reedy lgorithm, very similar to ijkstra's Minimum Spanning Tree Minimum Spanning Tree: spanning tree in a weighted graph with the lowest total cost used in network design, taxonomy, Image registration, and more! ost of spanning tree shown? Pick as root Prim's lgorithm 8. None of These 8 S raphs 8 S raphs 8

22 Prim's lgorithm Prim's lgorithm 5 8 Lowest cost edge from tree to vertex not in Tree? from to (or ) S raphs Lowest cost edge from tree to vertex not in Tree? from to (OR from o ) S raphs 8 Prim's lgorithm Prim's lgorithm 5 8 Lowest cost edge from tree to vertex not in Tree? from to S raphs Lowest cost edge from tree to vertex not in Tree? 5 from to S raphs 88

23 5 Prim's lgorithm 8 Lowest cost edge from tree to vertex not in Tree? from to S raphs 89 5 Prim's lgorithm 8 Lowest cost edge from tree to vertex not in Tree? from to S raphs 90 Prim's lgorithm Prim's lgorithm 5 5 Pick as root S raphs 9 Lowest cost edge from tree to vertex not in Tree? from to S raphs 9

24 Prim's lgorithm Prim's lgorithm 5 5 Lowest cost edge from tree to vertex not in Tree? from to S raphs 9 Lowest cost edge from tree to vertex not in Tree? from to S raphs 9 Prim's lgorithm Prim's lgorithm 5 5 Lowest cost edge from tree to vertex not in Tree? from to S raphs 95 Lowest cost edge from tree to vertex not in Tree? 5 from to S raphs 9

25 Prim's lgorithm Prim's lgorithm 5 5 Lowest cost edge from tree to vertex not in Tree? from to S raphs 9 ost of Spanning Tree? S raphs 98 Lots! Other raph lgorithms S raphs 99