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R. Rao, CSE 326 1
Lecture 20: Topo-Sort and Dijkstra’s Greedy Idea
✦ Items on Today’s Lunch Menu:
➭ Topological Sort (ver. 1 & 2): Gunning for linear time… ➭ Finding Shortest Paths ➧ Breadth-First Search ➧ Dijkstra’s Method: Greed is good!
✦ Covered in Chapter 9 in the textbook
Some slides based on: CSE 326 by S. Wolfman, 2000
Graph Algorithm #1: Topological Sort
(^143321)
142
322
326 (^370341)
378
401
421
Problem: Find an order in which all these courses can be taken. Example: 142 143 378 370 321 341 322 326 421 401
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Topological Sort Definition
Topological sorting problem : given digraph G = ( V , E ) ,
find a linear ordering of vertices such that:
for all edges ( v , w ) in E , v precedes w in the ordering
A
B
C
F
D E
Topological Sort
Topological sorting problem: given digraph G = ( V , E ) ,
find a linear ordering of vertices such that:
for any edge ( v , w ) in E , v precedes w in the ordering
A
B
C
F
D E
A B F C D E
Any linear ordering in which all the arrows go to the right is a valid solution
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Step 1: Identify vertices that have no incoming edge
- If no such edges , graph has cycles (cyclic graph)
A
B
C
D
Topological Sort Algorithm
Example of a cyclic graph: No vertex of in-degree 0
Step 1: Identify vertices that have no incoming edges
A
B
C
F
D E
Topological Sort Algorithm
Select
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A
B
C
F
D E
Topological Sort Algorithm
Step 2: Delete this vertex of in-degree 0 and all its
outgoing edges from the graph. Place it in the output.
A
B
C
F
D E
Topological Sort Algorithm
Repeat Steps 1 and Step 2 until graph is empty
Select
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Topological Sort Algorithm
Repeat Steps 1 and Step 2 until graph is empty
A B F C D E
Final Result:
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Summary of Topo-Sort Algorithm
- Store each vertex’s In- Degree (# of incoming edges) in an array
- While there are vertices remaining: ➭ Find a vertex with In-Degree zero and output it ➭ Reduce In-Degree of all vertices adjacent to it by 1 ➭ Mark this vertex (In- Degree = -1)
B D
E
D E
C
A B C D E F
In-Degree array
Adjacency list
A
B C
F
D E
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For input graph G = ( V , E ), Run Time =?
Break down into total time required to: § Initialize In-Degree array: O(| E |) § Find vertex with in-degree 0: | V | vertices, each takes O(| V |) to search In-Degree array. Total time = O(| V | 2 ) § Reduce In-Degree of all vertices adjacent to a vertex: O(| E |) § Output and mark vertex: O(| V |) Total time = O(| V | 2 + | E |) Quadratic time!
Topological Sort Algorithm #1: Analysis
Can we do better than quadratic time?
Problem: Need a faster way to find vertices with in-degree 0 instead of searching through entire in-degree array
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Topological Sort Algorithm
- Store each vertex’s In-Degree in an array
- Initialize a queue with all in-degree zero vertices
- While there are vertices remaining in the queue: ➭ Dequeue and output a vertex ➭ Reduce In-Degree of all vertices adjacent to it by 1 ➭ Enqueue any of these vertices whose In-Degree became zero
Sort this digraph!
A
B C
F
D E
For input graph G = ( V , E ), Run Time =?
Break down into total time to: Initialize In-Degree array: O(| E |) Initialize Queue with In-Degree 0 vertices: O(| V |) Dequeue and output vertex: | V | vertices, each takes only O(1) to dequeue and output. Total time = O(| V |) Reduce In-Degree of all vertices adjacent to a vertex and Enqueue any In-Degree 0 vertices: O(| E |) Total time = O(| V | + | E |) Linear running time!
Topological Sort Algorithm #2: Analysis
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Paths
✦ Recall definition of a path in a tree – same for graphs ✦ A path is a list of vertices {v 1 , v 2 , …, v (^) n } such that (v (^) i , v (^) i+1 ) is in E for all 0 ≤≤≤≤ i < n.
Seattle
San Francisco Dallas
Chicago
Salt Lake City
Example of a path:
p = {Seattle, Salt Lake City, Chicago, Dallas, San Francisco, Seattle}
Simple Paths and Cycles
✦ A simple path repeats no vertices (except the 1 st^ can be the last): ➭ p = {Seattle, Salt Lake City, San Francisco, Dallas} ➭ p = {Seattle, Salt Lake City, Dallas, San Francisco, Seattle} ✦ A cycle is a path that starts and ends at the same node: ➭ p = {Seattle, Salt Lake City, Dallas, San Francisco, Seattle} ✦ A simple cycle is a cycle that repeats no vertices except that the first vertex is also the last ✦ A directed graph with no cycles is called a DAG (directed acyclic graph) E.g. All trees are DAGs ➭ A graph with cycles is often a drag…
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Why study shortest path problems?
✦ Plenty of applications ✦ Traveling on a “starving student” budget : What is the cheapest multi-stop airline schedule from Seattle to city X? ✦ Optimizing routing of packets on the internet : ➭ Vertices = routers, edges = network links with different delays ➭ What is the routing path with smallest total delay? ✦ Hassle-free commuting : Finding what highways and roads to take to minimize total delay due to traffic ✦ Finding the fastest way to get to coffee vendors on campus from your classrooms
Unweighted Shortest Paths Problem
Problem: Given a “source” vertex s in an unweighted graph G = ( V , E ), find the shortest path from s to all vertices in G
A
C
B
D
F H
G
E
Find the shortest path from C to: A B C D E F G H
Source
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Solution based on Breadth-First Search
✦ Basic Idea: Starting at node s, find vertices that can be reached using 0, 1, 2, 3, …, N-1 edges (works even for cyclic graphs!)
Find the shortest path from C to: A B C D E F G H
On-board example:
A
C
B
D
F H
G
E
Breadth-First Search (BFS) Algorithm
✦ Uses a queue to store vertices that need to be expanded
✦ Pseudocode (source vertex is s):
**1. Dist[s] = 0
- Enqueue(s)
- While queue is not empty** 1. X = dequeue 2. For each vertex Y adjacent to X and not previously visited $ Dist[Y] = Dist[X] + 1 $ Prev[Y] = X $ Enqueue Y
✦ Running time (same as topological sort) = O(| V | + | E |) (why?)
( Prev allows paths to be reconstructed)
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Dijkstra to the rescue…
✦ Legendary figure in computer science ✦ Some rumors collected from previous classes…
✦ Rumor #1: Supported teaching introductory computer courses without computers (pencil and paper programming)
✦ Rumor #2: Supposedly wouldn’t read his e-mail; so, his staff had to print out his e-mails and put them in his mailbox
E. W. Dijkstra (1930-2002)
An Aside: Dijsktra on GOTOs
Opening sentence of: “ Go To Statement Considered Harmful ” by Edsger W. Dijkstra, Letter to the Editor, Communications of the ACM, Vol. 11, No. 3, March 1968, pp. 147-148.
“For a number of years I have been familiar
with the observation that the quality of
programmers is a decreasing function of the
density of go to statements in the programs
they produce.”
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Dijkstra’s Algorithm for Weighted Shortest Path
✦ Classic algorithm for solving shortest path in weighted graphs (without negative weights)
✦ Example of a greedy algorithm ➭ Irrevocably makes decisions without considering future consequences ➭ Sound familiar? Not necessarily the best life strategy… but works in some cases (e.g. Huffman encoding)
Dijkstra’s Algorithm for Weighted Shortest Path
✦ Basic Idea: ➭ Similar to BFS ➧ Each vertex stores a cost for path from source ➧ Vertex to be expanded is the one with least path cost seen so far $ Greedy choice – always select current best vertex $ Update costs of all neighbors of selected vertex ➭ But unlike BFS, a vertex already visited may be updated if a better path to it is found
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Dijkstra’s Algorithm (greed in action)
A
C
B
D (^) E
2
2
(^11) 9 3 8
3
Initial Final
E No ∞ -
D No ∞ -
C Yes 0 -
B No ∞ -
A No ∞ -
vertex known cost Prev
E Yes 2 C
D Yes 5 E
C Yes 0 -
B Yes 10 A
A Yes 8 D
vertex known cost Prev
Questions for Next Time:
Does Dijkstra’s method always work?
How fast does it run?
Where else in life can I be greedy?
To Do:
Start Homework Assignment
(Don’t wait until the last few days!!!)
Continue reading and enjoying chapter 9
