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Transcript of Announcements Exam Next week Review on Monday in class. Review on Wednesday in lab. DisjointSets lab...
AnnouncementsAnnouncementsExam Next weekReview on
Monday in class.Review on
Wednesday in lab.DisjointSets lab
program due tonight.
BucketSort lab due next Wednesday.
On Monday…On Monday…Radix Sort2 proofs:
◦ A Lower bound for comparison based sorting.
◦ A lower bound for swapping adjacent elements.
Graphs◦ Definitions◦ DFS◦ BFS
TodayTodayQuick review of DFS and BFSTopological SortIntro to Dijkstra’s
Depth First SearchDepth First Search
Start with 1 2,4,3
◦ Stop at 3 Backtrack to 4, can we search? 7,6
◦ Stop at 2 Backtrack to 6, can we search? 10,13
◦ Stop at 13
1 2 5
10
6
11
15
3 4 7 12
9
13
14
8
Backtrack to 10,6,79, 11, 8, 5
◦ Stop at 5Backtrack to 8,1115,14,12
◦ STOP – All nodes are marked!!
OutputList:
1 2 4 3 7 610
13
911
8 515
14
12
1 2
3
5
6
11
4
10
15
7 9
8
12
13
14
Depth First Search – Depth First Search – Real Life Real Life ApplicationApplication
From xkcd.com
Breadth First SearchBreadth First Search
L0: 1L1: 2, 3L2: 4,5,6,11L3: 7,8,10,9,15L4: 13,12,14
1 2 5
10
6
11
15
3 4 7 12
9
13
14
1 2
3
5
6
11
4
8
10
15
7 9
8
12
13
14
Final output: 1, 2, 3, 4, 5, 6, 11, 7, 8, 10, 9, 15, 13, 12, 14
Topological SortTopological Sort
Topological SortTopological SortThe goal of a Topological Sort:
◦ When given a list of items with dependencies (i.e. item 5 must be completed before item 3, etc.)
◦ Produce an ordering of the items that satisfies the given constraints.
In order for the problem to be solvable, there can’t be a cyclic set of constraints.◦ We can’t have item 5 must be completed
before item 3, item 3 must be completed before item 7, and item 7 must be completed before item 5. Since that would be IMPOSSIBLE!!
Topological SortTopological Sort We can model dependencies
(or constraints) like these using
◦ A Directed Acyclic Graph
Given a set of items and constraints, we create the corresponding graph as follows:1)Each item corresponds to a vertex
in the graph.
2)For each constraint where item A must finish before item B, place a directed edge A B.
A stands for waking up,
B stands for taking a shower,
C stands for eating breakfast
D leaving for work◦ The constraints would be
A before B, A before C, B before D, and C before D.
A B
C D
A topological sort would give A, B, C, D.
Topological SortTopological SortConsider the following CS classes and their
prereq’s:Prereq’s: COP 3223 before COP 3330 and
COP 3502. COP 3330 before COP 3503. COP 3502 before COT 3960, COP
3503, CDA 3103. COT 3100 before COT 3960. COT 3960 before COP 3402 and
COT 4210. COP 3503 before COT 4210.
The goal of a topological sort would be to find an ordering of these classes that you can take.
How Convenient!!How Convenient!!
CS classes:◦ COP 3223◦ COP 3502◦ COP 3330◦ COT 3100,◦ COP 3503◦ CDA 3103◦ COT 3960 (Foundation
Exam)◦ COP 3402◦ COT 4210.
Topological Sort AlgorithmTopological Sort AlgorithmWe can use a DFS in our algorithm to
determine a topological sort!◦ The idea is:
When you do a DFS on a directed acyclic graph, eventually you will reach a node with no outgoing edges. Why? Because if this never happened, you hit a cycle, because the
number of nodes if finite. This node that you reach in a DFS is “safe” to place at
the end of the topological sort. Think of leaving for work!
◦ Now what we find is If we have added each of the vertices “below” a vertex
into our topological sort, it is safe then to add this one in. If we added in Leaving for work at the end, then we can
surely add taking a shower.
Topological SortTopological Sort
If we explore from A◦ And if we have
marked B,C,D as well as anything connected And added all of them
into our topological sort in backwards order.
Then we can add A!
So basically all you have to do is run a DFS◦ But add the fact
that add the end of the recursive function, add the node the DFS was called with to the end of the topological sort.
A
B C D
Topological Sort Step-by-Topological Sort Step-by-Step DescriptionStep Description1) Do a DFS starting with some node.
2) The “last” node you reach before you are forced to backtrack.
◦ Goes at the end of the topological sort.
3) Continue with your DFS, placing each “dead end” node into the topo sort in backwards order.
Example on the BoardExample on the Board
Topological Sort CodeTopological Sort Codetop_sort(Adjacency Matrix adj, Array ts)
{
n = adj.last
k = n //assume k is global
for i=1 to n
visit[i] = false
for i=1 to n
if (!visit[i])
top_sort_rec(adj, i, ts)
}
Topological Sort CodeTopological Sort Codetop_sort_rec(Adjacency Matrix adj, Vertex start, Array ts)
{
visit[start] = true
trav = adj[start] //trav points to a LL
while (trav != null)
{
v = trav.ver
if (!visit[v])
top_sort_recurs(adj, v, ts);
trav = trav.next
}
ts[k] = start, k=k-1
}
Topological Sort ExampleTopological Sort ExampleConsider the following items of
clothing to wear:
Shirt Slacks Shoes Socks Belt Undergarments
1 2 3 4 5 6
There isn't exactly one order to put these items on, but we must adhere to certain restrictions
◦ Socks must be put on before Shoes◦ Undergarments must be put on before Slacks and
Shirt◦ Slacks must be put on before Belt◦ Slacks must be put on before Shoes
Finish Example on the Finish Example on the BoardBoard
ReferencesReferencesSlides adapted from Arup Guha’s
Computer Science II Lecture notes: http://www.cs.ucf.edu/~dmarino/ucf/cop3503/lectures/
Additional material from the textbook:Data Structures and Algorithm Analysis in
Java (Second Edition) by Mark Allen WeissAdditional images:
www.wikipedia.comxkcd.com