Topological Sort. CSE 373 Data Structures Lecture 19

Transcription

1 Topological Sort S 373 ata Structures Lecture 19

2 Readings and References Reading Section 9.2 Some slides based on: S 326 by S. Wolfman, /27/02 Topological Sort - Lecture 19 2

3 Topological Sort 142 Problem: ind an order in which all these courses can be taken xample: In order to take a course, you must take all of its prerequisites first /27/02 Topological Sort - Lecture 19 3

4 Topological Sort Given a digraph G = (V, ), find a linear ordering of its vertices such that: for any edge (v, w) in, v precedes w in the ordering 11/27/02 Topological Sort - Lecture 19 4

5 Topo sort - good example ny linear ordering in which all the arrows go to the right is a valid solution Note that can go anywhere in this list because it is not connected. 11/27/02 Topological Sort - Lecture 19 5

6 Topo sort - bad example ny linear ordering in which an arrow goes to the left is not a valid solution NO! 11/27/02 Topological Sort - Lecture 19 6

7 Not all can be Sorted directed graph with a cycle cannot be topologically sorted. 11/27/02 Topological Sort - Lecture 19 7

8 ycles Given a digraph G = (V,), a cycle is a sequence of vertices v 1,v 2,,v k such that k < 1 v 1 = v k (v i,v i+1 ) in for 1 < i < k. G is acyclic if it has no cycles. 11/27/02 Topological Sort - Lecture 19 8

9 Topo sort algorithm - 1 Step 1: Identify vertices that have no incoming edges The in-degree of these vertices is zero 11/27/02 Topological Sort - Lecture 19 9

10 Topo sort algorithm - 1a Step 1: Identify vertices that have no incoming edges If no such vertices, graph has only cycle(s) (cyclic graph) Topological sort not possible Halt. xample of a cyclic graph 11/27/02 Topological Sort - Lecture 19 10

11 Topo sort algorithm - 1b Step 1: Identify vertices that have no incoming edges Select one such vertex Select 11/27/02 Topological Sort - Lecture 19 11

12 Topo sort algorithm - 2 Step 2: elete this vertex of in-degree 0 and all its outgoing edges from the graph. Place it in the output. 11/27/02 Topological Sort - Lecture 19 12

13 Repeat Step 1 and Step 2 until graph is empty Select ontinue until done 11/27/02 Topological Sort - Lecture 19 13

14 Select. opy to sorted list. elete and its edges. 11/27/02 Topological Sort - Lecture 19 14

15 Select. opy to sorted list. elete and its edges. 11/27/02 Topological Sort - Lecture 19 15

16 Select. opy to sorted list. elete and its edges. 11/27/02 Topological Sort - Lecture 19 16

17 , Select. opy to sorted list. elete and its edges. Select. opy to sorted list. elete and its edges. 11/27/02 Topological Sort - Lecture 19 17

18 one Remove from algorithm and serve. 11/27/02 Topological Sort - Lecture 19 18

19 Implementation ssume adjacency list representation Translation array value next 11/27/02 Topological Sort - Lecture 19 19

20 alculate In-degrees In-egree array /27/02 Topological Sort - Lecture 19 20

21 alculate In-degrees for i = 1 to n do [i] := 0; endfor for i = 1 to n do x := [i]; while x null do [x.value] := [x.value] + 1; x := x.next; endwhile endfor 11/27/02 Topological Sort - Lecture 19 21

22 Maintaining egree 0 Vertices Key idea: Initialize and maintain a queue (or stack) of vertices with In-egree 0 Queue /27/02 Topological Sort - Lecture 19 22

23 Topo Sort using a Queue 1 fter each vertex is output, when updating In-egree array, enqueue any vertex whose In-egree becomes zero 2 Queue 6 dequeue Output enqueue 6 11/27/02 Topological Sort - Lecture

24 Topological Sort lgorithm 1. Store each vertex s In-egree in an array 2. Initialize queue with all in-degree=0 vertices 3. While there are vertices remaining in the queue: (a) equeue and output a vertex (b) Reduce In-egree of all vertices adjacent to it by 1 (c) nqueue any of these vertices whose In-egree became zero 4. If all vertices are output then success, otherwise there is a cycle. 11/27/02 Topological Sort - Lecture 19 24

25 Some etail Main Loop while notmpty(q) do x := equeue(q) Output(x) y := [x]; while y null do [y.value] := [y.value] 1; if [y.value] = 0 then nqueue(q,y.value); y := y.next; endwhile endwhile 11/27/02 Topological Sort - Lecture 19 25

26 Topological Sort nalysis Initialize In-egree array: O( V + ) Initialize Queue with In-egree 0 vertices: O( V ) equeue and output vertex: V vertices, each takes only O(1) to dequeue and output: O( V ) Reduce In-egree of all vertices adjacent to a vertex and nqueue any In-egree 0 vertices: O( ) or input graph G=(V,) run time = O( V + ) Linear time! 11/27/02 Topological Sort - Lecture 19 26