Programming Abstractions

Transcription

1 Programming bstractions S X ynthia ee

2 Upcoming Topics raphs! 1. asics What are they? ow do we represent them? 2. Theorems What are some things we can prove about graphs? 3. readth-first search on a graph Spoiler: just a very, very small change to tree version 4. ijkstra s shortest paths algorithm Spoiler: just a very, very small change to S 5. * shortest pathsalgorithm Spoiler: just a very, very small change to ijkstra s 6. Minimum Spanning Tree ruskal s algorithm

3 raphs What are graphs? What are they good for?

4 raph This file is licensed under the reative ommons ttribution 3.0 Unported license. fd34

5 Social Network Slide by eith Schwarz

6 hemical onds Slide by eith Schwarz

7 Slide by eith Schwarz

8 8 nternet This file is licensed under the reative ommons ttribution 2.5 eneric license. The Opte Project

9 graph is a mathematical structure for representing relationships onsists of: set V of vertices (or nodes) Often have an associated label set of edges (or arcs) onsist of two endpoint vertices Often have an associated cost or weight graph may be directed (an edge from to only allow you to go from to, not to ) or undirected (an edge between and allows travel in both directions) We talk about the number of vertices or edges as the size of the set, using the notation V and

10 oggle as a graph Vertex = letter cube; dge = connection to neighboring cube

11 Maze as graph f a maze is a graph, what is a vertex and what is an edge?

12 raphs ow do we represent graphs in code?

13 raph terminology iagram shows a graph with four vertices: pple (outgoing edges to banana and blum) anana (with outgoing edge to self) This Pear is (with a RT outgoing edges graph to banana and plum) Plum (with outgoing edge to banana) This is an UNRT graph

14 raph terminology iagrams each show a graph with four vertices: pple, anana, Pear, Plum. ach answer choice has different edge sets. Which : no edges of the following is a correct graph? : pple to Plum directed, pple to banana... undirected : pple and anana point to each other. Two edges point from Plum to Pear. None of the above/other/more than one of the above

15 Paths path: path from vertex a to b is a sequence of edges that can be followed starting from a to reach b. can be represented as vertices visited, or edges taken example, one path from V to Z: {b, h} or {V, X, Z} What are two paths from U to Y? a V b path length: Number of vertices or edges contained in the path. neighbor or adjacent: Two vertices connected directly by an edge. U c d W f e X Y g h Z example: V and X

16 Reachability, connectedness reachable: Vertex a is reachable from b if a path exists from a to b. connected: graph is connected if every vertex is reachable from every other. U a c d V W e b X g h Z complete: f every vertex has a direct edge to every other. a c b d a c f Y b d e

17 oops and cycles cycle: path that begins and ends at the same node. example: {V, X, Y, W, U, V}. example: {U, W, V, U}. acyclic graph: One that does not contain any cycles. a V b loop: n edge directly from a node to itself. Many graphs don't allow loops. U c d W e X g h Z f Y

18 Weighted graphs weight: ost associated with a given edge. Some graphs have weighted edges, and some are unweighted. dges in an unweighted graph can be thought of as having equal weight (e.g. all 0, or all 1, etc.) Most graphs do not allow negative weights. example: graph of airline flights, weighted by miles between cities: SO OR PV N X W M

19 Representing raphs: djacency Matrix We can represent a graph as a rid<bool> (unweighted) or rid<int> (weighted)

20 Representing raphs: adjacency list We can represent a graph as a map from nodes to the set of nodes each node is connected to. Map<Node*, Set<Node*>> Node onnected To Slide by eith Schwarz

21 ommon ways of representing graphs djacency list: Map<Node*, Set<Node*>> djacency matrix: rid<bool> unweighted rid<int> weighted ow many of the following are true? djacency list can be used for directed graphs djacency list can be used for undirected graphs djacency matrix can be used for directed graphs djacency matrix can be used for undirected graphs () 0 () 1 () 2 () 3 () 4

22 raphs Theorems about graphs

23 raphs lend themselves to fun theorems and proofs of said theorems! ny graph with 6 vertices contains either a triangle (3 vertices with all pairs having an edge) or an empty triangle (3 vertices no two pairs having an edge)

24 ulerian graphs et be an undirected graph graph is ulerian if it can drawn without lifting the pen iagram shows a graph with four vertices, and without which repeating are all edges fully connected with each other by undirected edges (6 edges in s this graph ulerian? total).. Yes. No

25 ulerian graphs iagram shows a graph with five vertices, four of which are all fully connected with et be an each undirected other by undirected graph edges (6 edges in total). The 5 th vertex is connected to two of graph is the ulerian remaining if it can four vertices by an edge. (6 drawn without + 2 = 8 lifting edges the in pen total) and without repeating edges What about this graph. Yes. No

26 Our second graph theorem efinition: egree of a vertex: number of edges adjacent to it uler s theorem: a connected graph is ulerian iff the number of vertices with odd degrees is either 0 or 2 (eg all vertices or all but two have even degrees) oes it work for and?

27 readth-irst Search raph algorithms

28 readth-irst Search S is useful for finding the shortest path between two nodes. xample: What is the shortest way to go from to?

29 readth-irst Search S is useful for finding the shortest path between two nodes. xample: What is the shortest way to go from to? Way 1: ->->-> 3 edges

30 readth-irst Search S is useful for finding the shortest path between two nodes. xample: What is the shortest way to go from to? Way 2: ->-> 2 edges

31 S is useful for finding the shortest path between two nodes. Map xample: What is the shortest way to go from Yoesmite () to Palo lto ()?

32 readth-irst Search Yoesmite Palo lto TO STRT: (1)olor all nodes RY (2)Queue is empty

33 readth-irst Search TO STRT (2): (1)nqueue the desired start node (2)Note that anytime we enqueue a node, we mark it YOW

34 readth-irst Search OOP PROUR: (1)equeue a node (2)Mark current node RN (3)Set current node s RY neighbors parent pointers to current node, then enqueue them (remember: set them YOW)

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52 readth-irst Search You predict the next slide!. s neighbors,, are yellow and in the queue and their parents are pointing to. s neighbors, are yellow and in the queue and their parents are pointing to. s neighbors, are yellow and in the queue. Other/none/more

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70 readth-irst Search one! Now we know that to go from Yoesmite () to Palo lto (), we should go: ->->->-> (4 edges) (note we follow the parent pointers backwards)

71 readth-irst Search TNS TO NOT: (1) We used a queue (2) What s left is a kind of subset of the edges, in the form of parent pointers (3) f you follow the parent pointers from the desired end point, you will get back to the start point, and it will be the shortest way to do that

72 Quick question about efficiency et s say that you have an extended family with somebody in every city in the western U.S.

73 Quick question about efficiency You re all going to fly to Yosemite for a family reunion, and then everyone will rent a car and drive home, and you ve been tasked with making custom Yosemite-to-home driving directions for everyone.

74 Quick question about efficiency You calculated the shortest path for yourself to return home from the reunion (Yosemite to Palo lto) and let s just say that it took time X = O(( + V )log V ) With respect to the number of cities V, and the number of edges or road segments ow long will it take you, in total, to calculate the shortest path for you and all of your relatives?. O( V *X). O( * V * X). X. Other/none/more

75 readth-irst Search TNS TO NOT: (4) We now have the answer to the question What is the shortest path to you from? for every single node in the graph!!