Change in Schedule. I'm changing the schedule around a bit...
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- Angelina Moore
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1 Graphs
2 Announcements() {
3 Change in Schedule I'm changing the schedule around a bit... Today: Graphs Tuesday: Shortest Path Algorithms Wednesday: Minimum Spanning Trees Tursday: Review Session for Midterm II
4 Final Final: Monday, August 12th, 7-10pm Location: Cubberly Auditorium Cumulative (but weighted towards post midterm material) Covers material up through Tuesday SCPD students and students who require special arrangements should me in the next couple days We do the final early so we have time to grade it, get it back to you and resolve any grading issues.
5 } //Announcements
6 Data Structures Cheat Sheet Vector, Stack: dynamic array Queue: linked list or dynamic array Set, Map: Binary Search Tree HashSet, HashMap: Hash Table Lexicon: Trie
7 Graphs
8 A Social Network
9
10
11 A graph is a mathematical structure for representing relationships.
12 A graph is a mathematical structure for representing relationships. A graph consists of a set of nodes connected by edges.
13 A graph is a mathematical structure for representing relationships. A graph consists of a set of nodes connected by edges.
14 A graph is a mathematical structure for representing relationships. A graph consists of a set of nodes connected by edges.
15 A graph is a mathematical structure for representing relationships. Nodes A graph consists of a set of nodes connected by edges.
16 A graph is a mathematical structure for representing relationships. Edges A graph consists of a set of nodes connected by edges.
17 Some graphs are directed.
18 Some graphs are undirected. CAN MAN RAN MAT CAT SAT RAT
19 Some graphs are undirected. CAN MAN RAN MAT CAT SAT RAT You can think of them as directed graphs with edges both ways.
20 Graphs Yo Teach, why are we studying graphs? We study graphs because a lot of problems can be modeled in terms of graphs Also, there are many off-the-shelf graph algorithms that we apply if we're able to formulate a problem as a graph problem.
21 Pathfinding Each intersection is a node Each street connecting intersections is an edge Find paths between intersections that minimize distance or travel time
22 Content-Aware Resizing
23 Content-Aware Resizing Each pixel is a node in a graph Each pixel is connected to adjacent pixels Find paths from the top of the image to the bottom that minimize the energy function
24 The Wikipedia Graph Wikipedia (and the web in general) is a graph! Each page is a node. There is an edge from one page to another if the first page links to the second.
25 Social Networks and Epidemics %2Finfo2040%2F2011%2F09%2F17%2Fhow-network-structure-can-provide-advanced-warning-about-epidemics %2F&ei=rrP9Ufa8G6n0iwK9q4CoBQ&bvm=bv ,d.cGE&psig=AFQjCNHljXC1ZggH6hQDrKxPM1CV7ocVXQ&ust=
26 How can we represent graphs in C++?
27 Representing Graphs We Wecan canrepresent representaagraph graphas as aamap mapfrom fromnodes nodesto tothe thelist list of ofnodes nodeseach eachnode nodeisis connected connectedto. to. Map<Node*, Vector<Node*>> Node* Vector<Node*> Node Connected To
28 What interesting things can we do with graphs?
29 Connected Components A connected component is a subset of the nodes in a graph such that: For every pair of nodes in the subset there exists a path between them No node in the subset is not connected any node not in the subset
30 Connected Components 1 Connected Component
31 Connected Components 2 Connected Components
32 Connected Components 3 Connected Components
33 Connected Components 6 Connected Components
34 Connected Components Detecting connected components in a graph is important because it can provide useful insights into the structure of graph e.g. How do people in a community separate themselves into separate groups. In order to detect connected components we first need to be able to iterate over nodes in a graph. We'll come back to connected components later.
35 Iterating over a Graph Given a linked list, there was just one way to traverse the list. Keep going forward. In a binary search tree, there are many traversal strategies: An inorder traversal that produces all the elements in sorted order. A postorder traversal used to delete all the nodes in the BST. There are many ways to iterate over a graph.
36 Iterating over a Graph All methods of iterating over a graph involve keeping track of 3 sets of nodes: Set of Nodes already visited Set of Nodes to look at next Everything else
37 Iterating over a Graph
38 Iterating over a Graph
39 Iterating over a Graph
40 Iterating over a Graph
41 Iterating over a Graph
42 Iterating over a Graph
43 Iterating over a Graph
44 Iterating over a Graph
45 Iterating over a Graph
46 Iterating over a Graph
47 Iterating over a Graph
48 Iterating over a Graph
49 Iterating over a Graph
50 Iterating over a Graph
51 Iterating over a Graph
52 Iterating over a Graph
53 Iterating over a Graph
54 Iterating over a Graph
55 Iterating over a Graph
56 Iterating over a Graph
57 Iterating over a Graph
58 Iterating over a Graph
59 Iterating over a Graph
60 Iterating over a Graph
61 Iterating over a Graph
62 Iterating over a Graph
63 Iterating over a Graph
64 Iterating over a Graph
65 Iterating over a Graph
66 Iterating over a Graph
67 Iterating over a Graph
68 Iterating over a Graph
69 Iterating over a Graph
70 Iterating over a Graph
71 Iterating over a Graph All methods of iterating over a graph involve keeping track of 3 sets of nodes: Set of Nodes already visited Set of Nodes to look at next Everything else Methods of iterating over nodes differ in how they choose which node to look at next
72 Recursive Depth First Search To detect connected components we just want to see whether or not some path exists between them (we don't care about finding the shortest path). One way to detect connectivity would be to just pick an arbitrary direction and keep following it. When you run out of room to go in one direction, just go back and look in a different direction
73 Recursive Depth-First Search
74 Recursive Depth-First Search
75 Recursive Depth-First Search
76 Recursive Depth-First Search
77 Recursive Depth-First Search
78 Recursive Depth-First Search
79 Recursive Depth-First Search
80 Recursive Depth-First Search
81 Recursive Depth-First Search
82 Recursive Depth-First Search
83 Recursive Depth-First Search
84 Recursive Depth-First Search
85 Recursive Depth-First Search
86 Recursive Depth-First Search
87 Recursive Depth-First Search
88 Recursive Depth-First Search
89 Recursive Depth-First Search
90 Recursive Depth-First Search
91 Recursive Depth-First Search
92 Recursive Depth-First Search
93 Recursive Depth-First Search
94 Recursive Depth-First Search
95 Recursive Depth-First Search
96 Recursive Depth-First Search
97 Recursive Depth-First Search
98 Recursive Depth-First Search
99 Recursive Depth-First Search
100 Recursive Depth-First Search
101 Recursive Depth-First Search
102 Recursive Depth-First Search
103 Recursive Depth-First Search
104 Recursive Depth-First Search
105 Recursive Depth-First Search
106 Recursive Depth-First Search
107 Recursive Depth-First Search
108 Recursive Depth-First Search
109 Recursive Depth-First Search
110 Recursive Depth-First Search
111 Recursive Depth-First Search
112 Recursive Depth-First Search
113 Recursive Depth-First Search
114 Recursive Depth-First Search
115 Recursive Depth-First Search
116 Recursive Depth-First Search To do a depth-first search (DFS) from a node u, do the following: If u is already marked, stop. Mark u. For each neighbor v of u: Recursively run DFS from v. The backtracking here is similar to the backtracking done in standard recursion.
117 Iterative Depth-First Search DFS is most commonly implemented iteratively using a Stack
118 Depth-first search A B C D E F
119 Depth-first search A B C D E F Set Stack
120 Depth-first search A B C D E F A Set Stack
121 Depth-first search A B C D E F Set Stack
122 Depth-first search A B C D E F Set A Stack
123 Depth-first search A B C D E F E B Set A Stack
124 Depth-first search A B C D E F E B Set A Stack
125 Depth-first search A B C D E F B Set A Stack
126 Depth-first search A B C D E F E Set A B Stack
127 Depth-first search A B C D E F C F E Set A D B Stack
128 Depth-first search A B C D E F C F E Set A D B Stack
129 Depth-first search A B C D E F F E Set A D B Stack
130 Depth-first search A B C D E F F D E B Set A Stack C
131 Depth-first search A B C D E F B F D E B Set A Stack C
132 Depth-first search A B C D E F B F D E B Set A Stack C
133 Depth-first search A B C D E F B F D E B Set A Stack C
134 Depth-first search A B C D E F F D E B Set A Stack C
135 Depth-first search A B C D E F F D E B Set A B Stack C
136 Depth-first search A B C D E F F D E B Set A B Stack C
137 Depth-first search A B C D E F D E B Set A B Stack C
138 Depth-first search A B C D E F E D F Set A B B Stack C
139 Depth-first search A B C D E F E D F Set A B B Stack C
140 Depth-first search A B C D E F E F Set A B B Stack C
141 Depth-first search A B C D E F D E F Set A B B Stack C
142 Depth-first search A B C D E F D E F Set A B B Stack C
143 Depth-first search A B C D E F D E F Set A B Stack C
144 Coding Depth-First Search
145 DFS and Connected Components Detecting connected components becomes relatively straightforward once we have Depth First Search. Not going to code it up, but I encourage you to!
146 Problems with DFS Useful when trying to explore everything. Not good at finding shortest paths.
147 Problems with DFS Useful when trying to explore everything. Not good at finding shortest paths. Stack
148 Problems with DFS Useful when trying to explore everything. Not good at finding shortest paths. A B Stack
149 Problems with DFS Useful when trying to explore everything. Not good at finding shortest paths. A B A B Stack
150 Breadth-First Search
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159 Breadth-First Search Specialization of the general search algorithm where nodes to visit are put into a queue. Explores nodes one hop away, then two hops away, etc. Finds path with fewest edges from start node to all other nodes.
160 Note: The following animation has been simplified for pedagogic purposes. In reality there would be a Set keeping track of visited nodes and redundant adds to the Queue
161 Breadth-first search A B C D E F
162 Breadth-first search Queue A B C D E F
163 Breadth-first search Queue A A B C D E F
164 Breadth-first search Queue A B C D E F
165 Breadth-first search Queue B A B C D E F E
166 Breadth-first search Queue B A B C D E F E
167 Breadth-first search Queue E A B C D E F
168 Breadth-first search Queue E A B C D E F C
169 Breadth-first search Queue E A B C D E F C
170 Breadth-first search Queue A B C D E F C
171 Breadth-first search Queue A B C D E F C D F
172 Breadth-first search Queue A B C D E F C D F
173 Breadth-first search Queue D A B C D E F F
174 Breadth-first search Queue D A B C D E F F
175 Breadth-first search Queue F A B C D E F
176 Breadth-first search Queue F A B C D E F
177 Breadth-first search Queue A B C D E F
178 Breadth-first search Queue A B C D E F
179 Coding Breadth-First Search
180 CAN MAN RAN MAT CAT SAT RAT
181 CAN MAN RAN MAT CAT SAT RAT
182 CAN MAN RAN MAT CAT SAT RAT
183 CAN MAN RAN MAT CAT SAT RAT
184 CAN MAN RAN MAT CAT SAT RAT
185 CAN CAT RAN
186 Next Time Shortest Paths Dijkstra's Algorithm. A* Search.
Introduction to Graphs. CS2110, Spring 2011 Cornell University
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