CS 106X, Lecture 25 Topological Sort - Stanford University€¦ · Topological Sort Given a...

This document is copyright (C) Stanford Computer Science and Nick Troccoli, licensed under Creative Commons Attribution 2.5 License. All rights reserved.Based on slides created by Keith Schwarz, Julie Zelenski, Jerry Cain, Eric Roberts, Mehran Sahami, Stuart Reges, Cynthia Lee, Marty Stepp, Ashley Taylor and others.

CS 106X, Lecture 25Topological Sort

reading:Programming Abstractions in C++, Chapter 18

Plan For This Week• Graphs: Topological Sort (HW8)• Classes: Inheritance and Polymorphism (HW8)• Sorting Algorithms

Plan For Today• Topological Sort

– Kahn’s Algorithm– Recursive DFS– Spreadsheets

• Announcements

• Learning Goal: understand how to implement topological sort, and why it and dependency graphs are useful.

Plan for Today• Topological Sort

• Announcements

Graphs So Far• Graphs as Data Structures

– Adjacency List– Adjacency Matrix– Edge List

• Searching– DFS– BFS– Dijkstra’s Algorithm– A* Search

Dependency Graphs

Dependency Graphs: TasksToast Bread

Butter Toast Sauté Veggies

Chop Veggies

Add Eggs & Cook

Prepare Egg Mixture

Plate Omelette and Toast

How can I parallelize these tasks to finish more quickly?

Dependency Graphs: TasksToast Bread

Chop Veggies

Add Eggs & Cook

Prepare Egg Mixture

Plate Omelette and Toast

What step is best to optimize in this production process?

Dependency Graphs: Orderings

CS 106A

CS 106B/X CS 103

CS 109

CS 161

CS 107

CS 110

CS 221

What are valid orderings of these tasks?

Dependency Graphs: Games

What is an effective AI strategy for resource management games (like Civilization)?

Dependency Graphs: Compilers

LoginUI

Toolbar

MainUI

Bundle

DataTypes

In what order should we recompile source files with dependencies?

Dependency Graphs: Web

LoginUI

Toolbar

MainUI

Bundle

DataTypes

How can I efficiently package up and transmit web dependencies?

Topological SortToast Bread

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate FoodWhich tasks could we complete first?

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate FoodWhich tasks could we complete first?

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Topological Sort

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Toast Bread Chop Veggies

Topological Sort

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Topological Sort

Sauté Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Butter Toast

Topological Sort

Sauté Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Butter Toast

Topological Sort

Sauté Veggies

Add Eggs & Cook

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Prepare Eggs

Butter Toast

Topological Sort

Sauté Veggies

Add Eggs & Cook

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Prepare Eggs

Butter Toast

Topological Sort

Add Eggs & Cook

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Sauté Veggies

Prepare Eggs

Butter Toast

Topological Sort

Add Eggs & Cook

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Sauté Veggies

Prepare Eggs

Butter Toast

Topological Sort

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Plate Food

Add Eggs & Cook

Sauté Veggies

Prepare Eggs

Butter Toast

Topological Sort

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Plate Food

Add Eggs & Cook

Sauté Veggies

Prepare Eggs

Butter Toast

Topological Sort

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Plate Food

Add Eggs & Cook

Sauté Veggies

Prepare Eggs

Butter Toast

Topological SortGiven a directed (acyclic!) graph G = (V, E), a topological sort is a total ordering of G's vertices such that for every edge (v, w) in E, vertex vprecedes w in the ordering.

CS 106A

CS 106B/X CS 103

CS 109

CS 161

CS 107

CS 110

CS 221

Topological Sort

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Plate Food

Add Eggs & Cook

Sauté Veggies

Prepare Eggs

Butter Toast

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Plate Food

Given a directed graph G = (V, E), a topological sort is a total ordering of G's vertices such that for every edge (v, w) in E, vertex v precedes w in the ordering.

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Plate Food

Given a directed graph G = (V, E), a topological sort is a total ordering of G's vertices such that for every edge (v, w) in E, vertex v precedes w in the ordering.

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

Make Lunch

Toast Bread

Chop Veggies

Butter Toast

Prepare Eggs

Sauté Veggies

Add Eggs & Cook

Plate Food

Topological Sort: Idea1. Find node with in-degree of 02. Add it at the end of our topological ordering so far, and remove it

and its edges from the graph3. Repeat until no nodes left

Topological Sort Algorithmordering := { }. // (empty list)Repeat until graph is empty:

Find a vertex v with in-degree of 0Delete v and its outgoing edges from graphordering += v

• This algorithm modifies the passed-in graph L• Have we handled all edge cases?• What is the runtime of this algorithm?

Topological Sort Algorithmordering := { }. // (empty list)Repeat until graph is empty:

Topological Sort Algorithm: Cycles

ordering := { }.Repeat until graph is empty:

Task 1 Task 2

Task 3

Task 1 Task 2

Task 3

Ordering: { }

Task 1 Task 2

Task 3

Ordering: { }

Task 1 Task 2

Task 3

Ordering: { }

Task 1 Task 2

Task 3

Ordering: { }

Task 1 Task 2

Task 3

Ordering: { “Task 2” }

Task 1 Task 2

Task 3

Task 1 Task 2

Task 3

Find a vertex v with in-degree of 0- if none, no valid ordering possible

Delete v and its outgoing edges from graphordering += v

Task 1 Task 2

Task 3

Topological Sort Algorithm: Runtime

For graph with V vertexes and E edges:ordering := { }.Repeat until graph is empty:

For graph with V vertexes and E edges:

ordering := { }.

Repeat until graph is empty:

Delete v and its outgoing edges from graph

ordering += v

O(1) O(V(V+E))

ordering := { }.

ordering += v

O(1) O(V(V+E))Is the worst case here really O(E) every time? For example, if one node has all the outgoing edges, then we’ll have an O(E) operation that time, but a no-op every other time.

ordering := { }.

ordering += v

O(1) O(V(V+E))Key Idea: every edge can be deleted at most once. Therefore, the total runtime of this operation is O(E), not O(V*E).

ordering := { }.

ordering += v

O(E) *total*

O(1) Tighter bound: O(V2 + E)

ordering := { }.

ordering += v

O(E) *total*

Can we make this more efficient?

Topological Sort: Take 2 !Toast Bread

Chop Veggies

Add Eggs & Cook

Prepare Eggs

Plate Food

ordering := { }. Repeat until graph is empty:

3 Key Ideas for Improvement• Keep a queue of nodes with in-degree 0 so we don’t have to search

for nodes multiple times.• A node’s in-degree only changes when one of its prerequisites is

completed. Therefore, when completing a task, check if any of its neighbors now has in-degree 0.

• Keep a map of nodes’ in-degrees so we don’t need to modify the graph.

Topological Sort: Take 2 !

0-In-Degree Queue: { }In-Degree Map: { }

0-In-Degree Queue: { “A”, “D”, “E”, “G” }In-Degree Map: { “A”:0, “B”:1, “C”:1, “D”:0, “E”: 0, “F”: 2, “G”:0, “H”:3 }

0-In-Degree Queue: { “D”, “E”, “G” }In-Degree Map: { “A”:0, “B”:1, “C”:1, “D”:0, “E”: 0, “F”: 2, “G”:0, “H”:3 }

0-In-Degree Queue: { “D”, “E”, “G”, “B” }In-Degree Map: { “A”:0, “B”:0, “C”:1, “D”:0, “E”: 0, “F”: 2, “G”:0, “H”:3 }

0-In-Degree Queue: { “E”, “G”, “B” }In-Degree Map: { “A”:0, “B”:0, “C”:1, “D”:0, “E”: 0, “F”: 2, “G”:0, “H”:3 }

0-In-Degree Queue: { “G”, “B” }In-Degree Map: { “A”:0, “B”:0, “C”:1, “D”:0, “E”: 0, “F”: 1, “G”:0, “H”:3 }

0-In-Degree Queue: { “G”, “B”, “F” }In-Degree Map: { “A”:0, “B”:0, “C”:1, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:2 }

A D E G

0-In-Degree Queue: { “B”, “F” }In-Degree Map: { “A”:0, “B”:0, “C”:1, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:2 }

A D E G

0-In-Degree Queue: { “B”, “F” }In-Degree Map: { “A”:0, “B”:0, “C”:1, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:2 }

A D E G B

0-In-Degree Queue: { “F” }In-Degree Map: { “A”:0, “B”:0, “C”:1, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:2 }

A D E G B

0-In-Degree Queue: { “F”, “C” }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:1 }

A D E G B

0-In-Degree Queue: { “F”, “C” }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:1 }

A D E G B F

0-In-Degree Queue: { “C” }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:1 }

A D E G B F

0-In-Degree Queue: { “C”, “H” }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:0 }

A D E G B F

0-In-Degree Queue: { “C”, “H” }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:0 }

A D E G B F C

0-In-Degree Queue: { “H” }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:0 }

A D E G B F C

0-In-Degree Queue: { “H” }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:0 }

A D E G B F C H

0-In-Degree Queue: { }In-Degree Map: { “A”:0, “B”:0, “C”:0, “D”:0, “E”: 0, “F”: 0, “G”:0, “H”:0 }

A D E G B F C H

• Announcements

Kahn’s Algorithmmap := {each vertex ® its in-degree}.queue := {all vertices with in-degree = 0}.ordering := { }.Repeat until queue is empty:

Dequeue the first vertex v from the queue.ordering += v.Decrease the in-degree of all v's neighbors by 1 in the map.queue += {any neighbors whose in-degree is now 0}.

• This algorithm doesn’t modify the passed-in graph! J• Have we handled all edge cases?• What is the runtime of this algorithm?

Kahn’s Algorithmmap := {each vertex ® its in-degree}.queue := {all vertices with in-degree = 0}.ordering := { }.Repeat until queue is empty:

Kahn’s Algorithm: Cyclesmap := {each vertex ® its in-degree}.queue := {all vertices with in-degree = 0}.ordering := { }.Repeat until queue is empty:

0-In-Degree Queue: { “B” }In-Degree Map: { A”:1, “B”:0, “C”: 2 }

Ordering: { }

Ordering: { “B” }

0-In-Degree Queue: { }In-Degree Map: { A”:1, “B”:0, “C”: 1 }

If all vertices processed, success! Otherwise, there is a cycle.

Kahn’s Algorithm: Runtime

map := {each vertex ® its in-degree}.queue := {all vertices with in-degree = 0}.ordering := { }.Repeat until queue is empty:

For graph with V vertexes and E edges

map := {each vertex ® its in-degree}.queue := {all vertices with in-degree = 0}.ordering := { }.

Repeat until queue is empty:

Dequeue the first vertex v from the queue.ordering += v.

Decrease the in-degree of all v's neighbors by 1 in the map.queue += {any neighbors whose in-degree is now 0}.

For graph with V vertexes and E edgesO(V)

map := {each vertex ® its in-degree}.queue := {all vertices with in-degree = 0}.ordering := { }.

Repeat until queue is empty:

Dequeue the first vertex v from the queue.ordering += v.

Decrease the in-degree of all v's neighbors by 1 in the map.queue += {any neighbors whose in-degree is now 0}.

For graph with V vertexes and E edgesO(V)

O(E) *total*

O(V + E)

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Revisiting Recursive DFS

A-D-F-G-E

• Key idea: visiting node ameans we must have visited all nodes that have a as a prerequisite.

B-C-F-G-E-D

• Key idea: visiting node ameans we must have visited all nodes that have a as a prerequisite.

• But different starting points may yield different subsets. Is there a way we can combine them into one complete topological ordering?

A D F G EB C F G E DC F G ED F G EEF GGH

A D F G EB C

H B C A D F G E

HGF GED F G EC F G EB C F G E DA D F G E

HGFEDCBA

A B C D E F G H

• function topologicalSortDFS():L = { }. // to store the sorted vertexeswhile graph contains any unmarked vertexes:

• select any unmarked vertex v.• visit(L, v).

• function visit(L, v):if v is unmarked:

• for each neighbor n of v:visit(L, n).

• mark v.• add v to front of L.

Topological Sort: DFS

• function topologicalSortDFS():L = { }. // to store the sorted vertexeswhile graph contains any unmarked vertexes:

• select any unmarked vertex v.• visit(L, v).

• function visit(L, v):if v has a temp.mark, error.if v is unmarked:

• temp.mark v.• for each neighbor n of v:

visit(L, n).• mark v.• add v to front of L.

Topological Sort: DFS

• Announcements

Topological Sort: SpreadsheetsCells with formulas referencing other cells have dependencies.

A B1 20

=B1*210

2 30=A1+B1

25=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2 B1

B2A1 B3

A B1 20

=B1*215

2 30=A1+B1

25=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 30=A1+B1

25=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 30=A1+B1

35=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 30=A1+B1

35=A1+5

3 50=SUM(A1:A2)

1.166=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 50=SUM(A1:A2)

1.166=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 75=SUM(A1:A2)

1.166=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 75=SUM(A1:A2)

1.166=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 75=SUM(A1:A2)

1.166=B2/A2 B1

B2A1 B3

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 75=SUM(A1:A2)

0.777=B2/A2 B1

B2A1 B3

A B1 20

=B1*210

2 30=A1+B1

25=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2 B1

B2A1 B3

Topological Sort: SpreadsheetsCells with formulas referencing other cells have dependencies. We can use topological sort to efficiently update cells!

A B1 20

=B1*210

2 30=A1+B1

25=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2

B1 A3B2A1 B3A2

A B1 20

=B1*215

2 30=A1+B1

25=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2

B1 A3B2A1 B3A2

A B1 30

=B1*215

2 30=A1+B1

25=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2

B1 A3B2A1 B3A2

A B1 30

=B1*215

2 30=A1+B1

35=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2

B1 A3B2A1 B3A2

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 50=SUM(A1:A2)

0.833=B2/A2

B1 A3B2A1 B3A2

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 75=SUM(A1:A2)

0.833=B2/A2

B1 A3B2A1 B3A2

A B1 30

=B1*215

2 45=A1+B1

35=A1+5

3 75=SUM(A1:A2)

0.777=B2/A2

B1 A3B2A1 B3A2

• Announcements

Announcements• Homework 8 (Excel) goes out on Wednesday 11/28, is due Friday

12/7. No late submissions will be accepted.• Guest lecture on Hashing next Monday: Zach!

Recap• Topological Sort

• Announcements

Next time: Classes – Inheritance and Polymorphism

CS 106X, Lecture 25 Topological Sort - Stanford University€¦ · Topological Sort Given a...

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