CSE 21: Mathematics for Algorithms and Systems Analysis
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1 CSE 21: Mathematics for Algorithms and Systems Analysis Week 10 Discussion David Lisuk June 4, 2014 David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
2 Agenda 1 Announcements 2 Graph Basics 3 Greedy Algorithm 4 Dijkstra s Algorithm 5 Prim s Algorithm 6 Kruskal s Algorithm iclicker Frequency: BA David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
3 Announcements ABK Review Section: June 8th 6-8pm PCYNH 106 RRR Review Section 1: June 6th 6-8pm Center 115 RRR Review Section 2: June 7th 6-8pm Center 119 Homework 9 due Tonight at Midnight TA evaluations up on CAPE, please fill it out for me if you ve been coming to my section I lied last week: RRR/ABK finals will be quite different from each other David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
4 What a Graph is Graph is a collection of vertices (V) and edges (E) Can be directed (digraph) or undirected Used to model any kind of networks David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
5 Graph Coloring The minimum number of colors needed to color a graph such that no adjacent nodes have same color David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
6 Graph Coloring The minimum number of colors needed to color a graph such that no adjacent nodes have same color A simple method is to try coloring with 2,3,4 colors until you reach a contradiction. In general it is NP-hard to find optimal coloring David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
7 Coloring iclicker Question (Frequency: BA) How many colors are needed to color this graph? (A=1,B=2,C=3,D=4,E=5) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
8 Coloring iclicker Question (Frequency: BA) How many colors are needed to color this graph? (A=1,B=2,C=3,D=4,E=5) Answer: 3 David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
9 Complete Graph David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
10 Bipartite A bipartite graph is one where nodes can be split into two sets such that only inter set connections exist. David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
11 Bipartite A bipartite graph is one where nodes can be split into two sets such that only inter set connections exist. Bipartite matching algorithm is a famous algorithm which uses this (will see in 101) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
12 Graph Isomorphism Two graphs are isomorphic iff the nodes of one graph can be relabeled such that they match the other graph/have same adjacency matrix David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
13 David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
14 Greedy Algorithms A greedy algorithm is one in which you make a locally optimal choice to try and yield a globally optimal solution. Example: Making change (Repeatedly pick the largest coin smaller than the change you have to make. Greedy algorithms are extremely useful as they are simple to design and yield optimal solutions to a surprising number of problems In particular there are several graph algorithms which solve complex sounding tasks very efficiently David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
15 Shortest Path Problem Given an undirected edge weighted graph, find the path between two nodes with the smallest total weight. David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
16 Shortest Path Problem Given an undirected edge weighted graph, find the path between two nodes with the smallest total weight. Solve with Dijkstra s algorithm David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
17 Shortest Path Problem Given an undirected edge weighted graph, find the path between two nodes with the smallest total weight. Solve with Dijkstra s algorithm A related problem is All pairs shortest path which you ll learn about in cse 101 David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
18 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
19 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
20 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes Mark all nodes as unvisited David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
21 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes Mark all nodes as unvisited Until T is marked visited or the nearest unvisited node is : David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
22 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes Mark all nodes as unvisited Until T is marked visited or the nearest unvisited node is : Let U be the nearest unvisited node David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
23 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes Mark all nodes as unvisited Until T is marked visited or the nearest unvisited node is : Let U be the nearest unvisited node Mark U as visited David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
24 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes Mark all nodes as unvisited Until T is marked visited or the nearest unvisited node is : Let U be the nearest unvisited node Mark U as visited For all unvisited neighbors V of U: David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
25 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes Mark all nodes as unvisited Until T is marked visited or the nearest unvisited node is : Let U be the nearest unvisited node Mark U as visited For all unvisited neighbors V of U: Set dist(v )=min(dist(v ), dist(u)+length(u,v ) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
26 Dijkstra s Algorithm Greedy algorithm to solve the shortest path problem from S to T Assign initial distances S = 0 and for all other nodes Mark all nodes as unvisited Until T is marked visited or the nearest unvisited node is : Let U be the nearest unvisited node Mark U as visited For all unvisited neighbors V of U: Set dist(v )=min(dist(v ), dist(u)+length(u,v ) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
27 Dijkstra s Complexity Every edge will be visited at most once David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
28 Dijkstra s Complexity Every edge will be visited at most once Every vertex will be visited at most once David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
29 Dijkstra s Complexity Every edge will be visited at most once Every vertex will be visited at most once Looking up an edge is constant time (just look at adjacency list of current node) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
30 Dijkstra s Complexity Every edge will be visited at most once Every vertex will be visited at most once Looking up an edge is constant time (just look at adjacency list of current node) Looking up next node can be done in log( V ) time using heap David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
31 Dijkstra s Complexity Every edge will be visited at most once Every vertex will be visited at most once Looking up an edge is constant time (just look at adjacency list of current node) Looking up next node can be done in log( V ) time using heap Thus O( E + log( V ) V ) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
32 Dijkstra s Example On board David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
33 Dijkstra s iclicker iclicker Question (Frequency: BA) What is the sequence of Current Nodes when finding the shortest path from A to B. Assume nearest node ties goto node with lowest label. (a) A, B (b) A,C,D,E,B (c) A,C,B, (d) A,C,D,B David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
34 Dijkstra s iclicker iclicker Question (Frequency: BA) What is the sequence of Current Nodes when finding the shortest path from A to B. Assume nearest node ties goto node with lowest label. (a) A, B (b) A,C,D,E,B (c) A,C,B, (d) A,C,D,B David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
35 Minimum Spanning Tree A spanning tree is a tree (graph with no loops) such that every vertex in a graph is connected The minimum spanning tree is the spanning tree of a weighted undirected graph such that no spanning tree has a lower sum of edge weights We consider two algorithms for this: Prim s and Kruskal s algorithm David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
36 Prim s Algorithm Greedy algorithm to find the minimum spanning tree for a weighted undirected graph Choose a single vertex to be the root of your tree While there are vertexes not in the tree: Let E be the set of edges which connect vertexes in the tree to those not in the tree Add the edge from E which has minimum weight David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
37 Prim s Complexity Every vertex will be added once David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
38 Prim s Complexity Every vertex will be added once Every edge will be considered at most once David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
39 Prim s Complexity Every vertex will be added once Every edge will be considered at most once Can put vertexes into binary heap sorted by edge to it with shortest length, log( V ) time per edge/vertex David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
40 Prim s Complexity Every vertex will be added once Every edge will be considered at most once Can put vertexes into binary heap sorted by edge to it with shortest length, log( V ) time per edge/vertex Thus O(( E + V ) log( V )), since E = O( V 2 ), algorithm is O( E log( V )) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
41 Prim s Example On board David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
42 Prim s iclicker iclicker Question (Frequency: BA) What is the sequence of nodes added to the MST using Prim s starting at G (a) C,D,E,B,A,F (b) B,A,E,F,D,C (c) C,D,E,F,A,B (d) D,C,E,F,A,B David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
43 Prim s iclicker iclicker Question (Frequency: BA) What is the sequence of nodes added to the MST using Prim s starting at G (a) C,D,E,B,A,F (b) B,A,E,F,D,C (c) C,D,E,F,A,B (d) D,C,E,F,A,B David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
44 Kruskal s Algorithm Greedy algorithm to find the minimum spanning tree for a weighted undirected graph Create a forest of trees F by making individual trees for every vertex Put all edges into a set S of edges not yet considered While S is non empty and F contains more than 1 tree: Remove the edge (u, v) with minimum weight from S If that edge connects two different trees inf F, remove the trees from F and add the connected tree to F. David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
45 Kruskal s Complexity Every edge will be considered at most once David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
46 Kruskal s Complexity Every edge will be considered at most once Finding minimum length edge in heap is log( E ) = O(log( V 2 )) = O(log( V ) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
47 Kruskal s Complexity Every edge will be considered at most once Finding minimum length edge in heap is log( E ) = O(log( V 2 )) = O(log( V ) Thus O( E log( V )) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
48 Kruskal s Example On board David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
49 Kruskal s iclicker iclicker Question (Frequency: BA) What is the set of forests after 4 steps of Kruskal s algorithm (a) (A)(B)(C,D,G)(E,F) (b) (A,B)(C,D)(E,F)(G) (c) (A,B)(C,D,G)(E,F) (d) (A,B,C)(D)(E,F)(G) (e) (A,B,C,D)(E,F,G) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
50 Kruskal s iclicker iclicker Question (Frequency: BA) What is the set of forests after 4 steps of Kruskal s algorithm (a) (A)(B)(C,D,G)(E,F) (b) (A,B)(C,D)(E,F)(G) (c) (A,B)(C,D,G)(E,F) (d) (A,B,C)(D)(E,F)(G) (e) (A,B,C,D)(E,F,G) David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
51 Questions for Finals Remaining time will be for questions regarding the final exams David Lisuk CSE 21: Mathematics for Algorithms and Systems Analysis June 4, / 26
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