Week 9. CS 400 Programming III

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Agnes George
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1 Week 9 p3b: final submission due before 10pm on Thursday, 3/28 p4: available soon x4: due before 10pm on Monday 4/1 x5: meet with x-team coach for design review 4/1-4/8 Module: Week 9 (and start on week 10 before next week) THIS WEEK: DFS and BFS Spanning Trees Minimum Spanning Trees o Prim's o Kruskal's Topological Ordering Dijkstra's Shortest Path algorithm Set operations NEXT WEEK Linear Sorts (maybe next week) o radix o flashsort Java FX (in-class demo) Programming Project Assignment: Part 1, the Design (Start organizing your teams now or keep your x-team for final Team Project) Copyright 2018 Deb Jim Skrentny CS 400 (S18): W9-1

2 Prim's Minimum Spanning Tree Greedy algorithm (best at moment not necessarily best) If you have a connected weighted graph, you may be interested in the minimum spanning tree to reduce the cost of cables, distance travelled, Copyright 2018 Deb Jim Skrentny CS 400 (S18): W9-2

4 Topological Ordering 1. get bread 2. get jelly 3. get peanut butter 4. get butter knife 5. open jelly 6. open peanut butter 7. take bread slice 1 8. take bread slice 2 9. use knife to spread jelly on bread slice 10. use knife to spread peanut butter on bread slice 11. put slices together with spreaded sides facing each other IDEA: Copyright 2018 Deb Jim Skrentny CS 400 (S18): W9-4

6 Dijkstra's Algorithm Psuedo Code for each vertex V initialize V s visited mark to false initialize V's total weight to infinity initialize V's predecessor to null set start vertex s total weight to 0 create new priority queue pq pq.insert( [start vertex total weight,start vertex] ) while!pq.isempty() [C s total weight,c] = pq.removemin() set C s visited mark to true for each unvisited successor S adjacent to C if S's total weight can be reduced S's total weight = C's total weight + edge weight from C to S update S's predecessor to C pq.insert( [S s total weight,s] ) (if S already in pq we ll just update S's total weight) Copyright 2018 Deb Jim Skrentny CS 400 (S18): W9-6

7 Dijkstra s Practice Iteration Priority Queue (just list smallest to largest) Vertex A Visite d Total Weight Predecessor 0 B 1 2 C 3 D 4 E 5 F 6 H 7 G Reconstruct shortest path from A to F Reconstruct shortest path from C to F do not know shortest path from this work Copyright 2018 Deb Jim Skrentny CS 400 (S18): W9-7

8 Sets Definition a collection of distinct items unordered can be described by words can be of finite or infinite size Examples: s1 = { 1, 2, 3 } s2 = { "bo", "kitty", "kwan" } s3 = { -30, -29, -28,, 28, 29, 30 } s4 = { 3, 6, 9, 12, } s5 = { 1, 4, 9, 16,, 225 } letters = {A, B, C, X, Y, Z } digits = { 0,, 9 } V = { set of vertices in a graph } E = { set of edges in a graph } Terminology subset proper subset superset proper superset unit set disjoint sets Copyright 2018 Deb Jim Skrentny CS 400 (S18): W9-8

10 SetADT Collection of distinct elements Operations in java.util.set boolean add(e e) add if item is not present boolean contains(object o) true iff o is present boolean remove(object o) remove o if present boolean isempty() true if no elements int size() returns number of elements Implementation Complexity analysis, if N is number of nodes insert lookup remove iteration Array O(N) O(N) O(N) Ordered (not sorted) Sorted array O(N) O(log 2 N) O(N) Sorted Linked list O(N) O(N) O(N) Ordered (not sorted) BST O(H) O(H) O(H) In-order Balanced search tree O(log 2 N) O(log 2 N) O(log 2 N) In-order (sorted) Hash table O(1) O(1) O(1) Nondeterministic De Morgan's Laws If A and B are any two sets then: (A union B)' = A' intersection B' (A intersection B)' = A' union B' Java's Set is an interface just another collection no set operations like union or intersection Copyright 2018 Deb Jim Skrentny CS 400 (S18): W9-10

The ADT priority queue Orders its items by a priority value The first item removed is the one having the highest priority value

The ADT priority queue Orders its items by a priority value The first item removed is the one having the highest priority value 1 Possible implementations Sorted linear implementations o Appropriate if