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§2 Topological Sort

§2 Topological Sort. 〖 Example 〗 Courses needed for a computer science degree at a hypothetical university. How shall we convert this list into a graph?. 1/17. §2 Topological Sort.

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§2 Topological Sort

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  1. §2 Topological Sort 〖Example〗 Courses needed for a computer science degree at a hypothetical university How shall we convert this list into a graph? 1/17

  2. §2 Topological Sort  AOV Network ::= digraph G in which V( G ) represents activities ( e.g. the courses ) and E( G ) represents precedence relations ( e.g. means that C1 is a prerequisite course of C3 ). C1 C3  i is a predecessor of j ::= there is a path from i to j i is an immediate predecessor of j ::= < i, j >  E( G ) Then j is called a successor ( immediate successor ) of i  Partial order ::= a precedence relation which is both transitive ( i k, k j  i j ) and irreflexive ( i i is impossible ). Note: If the precedence relation is reflexive, then there must be an i such that i is a predecessor of i. That is, i must be done before i is started. Therefore if a project is feasible, it must be irreflexive. Feasible AOV network must be a dag (directed acyclic graph). 2/17

  3. §2 Topological Sort 【Definition】A topological order is a linear ordering of the vertices of a graph such that, for any two vertices, i, j, if i is a predecessor of j in the network then i precedes j in the linear ordering. 〖Example〗One possible suggestion on course schedule for a computer science degree could be: 3/17

  4. §2 Topological Sort Note: The topological orders may not be unique for a network. For example, there are several ways (topological orders) to meet the degree requirements in computer science. Test an AOV for feasibility, and generate a topological order if possible. Goal void Topsort( Graph G ) { int Counter; Vertex V, W; for ( Counter = 0; Counter < NumVertex; Counter ++ ) { V = FindNewVertexOfDegreeZero( ); if ( V == NotAVertex ) { Error ( “Graph has a cycle” ); break; } TopNum[ V ] = Counter; /* or output V */ for ( each W adjacent to V ) Indegree[ W ] – – ; } } /* O( |V| ) */  T = O( |V|2 ) 4/17

  5. §2 Topological Sort v1 v2 v3 v4 v5 Indegree v1 0 v2 1 v6 v7 v3 2 v4 3 v5 1 v6 3 v7 2 Improvement: Keep all the unassigned vertices of degree 0 in a special box (queue or stack). Mistakes in Fig 9.4 on p.289 void Topsort( Graph G ) { Queue Q; int Counter = 0; Vertex V, W; Q = CreateQueue( NumVertex ); MakeEmpty( Q ); for ( each vertex V ) if ( Indegree[ V ] == 0 ) Enqueue( V, Q ); while ( !IsEmpty( Q ) ) { V = Dequeue( Q ); TopNum[ V ] = ++ Counter; /* assign next */ for ( each W adjacent to V ) if ( – – Indegree[ W ] == 0 ) Enqueue( W, Q ); } /* end-while */ if ( Counter != NumVertex ) Error( “Graph has a cycle” ); DisposeQueue( Q ); /* free memory */ } T = O( |V| + |E| ) Home work: p.339 9.2 What if a stack is used instead of a queue? v6 v7 0 v3 1 0 v4 2 0 1 v5 0 2 1 0 v2 1 0 v1 5/17

  6. Given a digraph G = ( V, E ), and a cost function c( e ) for e E( G ). The length of a path P from source to destination is (also called weighted path length). 2 v1 v2 2 v1 v2 4 1 –10 4 1 3 10 3 2 2 v3 v4 v5 2 2 v3 v4 v5 8 4 8 6 5 4 6 5 1 v6 v7 1 v6 v7 §3 Shortest Path Algorithms 1. Single-Source Shortest-Path Problem Given as input a weighted graph, G = ( V, E ), and a distinguished vertex, s, find the shortest weighted path from s to every other vertex in G. Negative-cost cycle Note: If there is no negative-cost cycle, the shortest path from s to s is defined to be zero. 6/17

  7. §3 Shortest Path Algorithms v1 v2 v3 v4 v5 v6 v7  Unweighted Shortest Paths  Sketch of the idea 0: v3 1 2 Breadth-first search 1:  v1 and v6 2 0 3 2:  v2 and v4 3:  1 3 v5 and v7  Implementation Table[ i ].Dist ::= distance from s to vi/* initialized to be  except for s */ Table[ i ].Known ::= 1 if vi is checked; or 0 if not Table[ i ].Path ::= for tracking the path /* initialized to be 0 */ 7/17

  8. §3 Shortest Path Algorithms v8 v7 v6 v5 v3 v2 v1 v9 v4 void Unweighted( Table T ) { int CurrDist; Vertex V, W; for ( CurrDist = 0; CurrDist < NumVertex; CurrDist ++ ) { for ( each vertex V ) if ( !T[ V ].Known && T[ V ].Dist == CurrDist ) { T[ V ].Known = true; for ( each W adjacent to V ) if ( T[ W ].Dist == Infinity ) { T[ W ].Dist = CurrDist + 1; T[ W ].Path = V; } /* end-if Dist == Infinity */ } /* end-if !Known && Dist == CurrDist */ } /* end-for CurrDist */ } If V is unknown yet has Dist < Infinity, then Dist is either CurrDist or CurrDist+1.  T = O( |V|2 ) The worst case: 8/17

  9. §3 Shortest Path Algorithms v1 v2 v3 v4 v5 v6 v7 DistPath v1  0 v2  0 v3 0 0 v4  0 v5  0 v6  0 v7  0  Improvement 1 2 void Unweighted( Table T ) { /* T is initialized with the source vertex S given */ Queue Q; Vertex V, W; Q = CreateQueue (NumVertex ); MakeEmpty( Q ); Enqueue( S, Q ); /* Enqueue the source vertex */ while ( !IsEmpty( Q ) ) { V = Dequeue( Q ); T[ V ].Known = true; /* not really necessary */ for ( each W adjacent to V ) if ( T[ W ].Dist == Infinity ) { T[ W ].Dist = T[ V ].Dist + 1; T[ W ].Path = V; Enqueue( W, Q ); } /* end-if Dist == Infinity */ } /* end-while */ DisposeQueue( Q ); /* free memory */ } 0 3 2 3 1 1 v3 v7 2 v1 v5 v4 2 v1 v2 3 v2 1 v3 v6 3 v4 v3 v1 T = O( |V| + |E| ) 9/17

  10. §3 Shortest Path Algorithms Dijkstra’s Algorithm (for weighted shortest paths) Let S = { s and vi’s whose shortest paths have been found } For any u S, define distance [ u ] = minimal length of path { s ( vi S )  u }. If the paths are generated in non-decreasing order, then  the shortest path must go through ONLYvi S ;  u is chosen so that distance[ u ] = min{ wS | distance[ w ] } (If u is not unique, then we may select any of them) ; /* Greedy Method */ Why? If it is not true, then there must be a vertex w on this path that is not in S. Then ...  if distance [ u1 ] < distance [ u2 ] and we add u1 into S, then distance [ u2 ] may change. If so, a shorter path from s to u2 must go through u1 and distance’ [ u2 ] = distance [ u1 ] + length(< u1, u2>). 10/17

  11. §3 Shortest Path Algorithms Dist Path v1 0 0 6 8 9 5 1 3 3 2 v3 v7 v4 v1 v4 v4 v4 v1 v2  0 v3  0 v4  0 v5  0 2 v1 v2 v6  0 4 1 3 10 v7  0 2 2 v3 v4 v5 8 4 6 5 1 v6 v7 void Dijkstra( Table T ) { /* T is initialized by Figure 9.30 on p.303 */ Vertex V, W; for ( ; ; ) { V = smallest unknown distance vertex; if ( V == NotAVertex ) break; T[ V ].Known = true; for ( each W adjacent to V ) if ( !T[ W ].Known ) if ( T[ V ].Dist + Cvw < T[ W ].Dist ) { Decrease( T[ W ].Dist to T[ V ].Dist + Cvw ); T[ W ].Path = V; } /* end-if update W */ } /* end-for( ; ; ) */ } /* O( |V| ) */ /* not work for edge with negative cost */ Please read Figure 9.31 on p.304 for printing the path. 11/17

  12. §3 Shortest Path Algorithms  Implementation 1 V = smallest unknown distance vertex; /* simply scan the table – O( |V| ) */ Good if the graph is dense T = O( |V|2 + |E| )  Implementation 2 Home work: p.339 9.5 Find the shortest paths p.340 9.10 Modify Dijkstra’s algorithm V = smallest unknown distance vertex; /* keep distances in a priority queue and call DeleteMin – O( log|V| ) */ Decrease( T[ W ].Dist to T[ V ].Dist + Cvw ); Good if the graph is sparse /* Method 1: DecreaseKey – O( log|V| ) */ T = O( |V| log|V| + |E| log|V| ) = O( |E| log|V| ) /* Method 2: insert W with updated Dist into the priority queue */ /* Must keep doing DeleteMin until an unknown vertex emerges */ T = O( |E| log|V| ) but requires |E| DeleteMin with |E| space  Other improvements: Pairing heap (Ch.12) and Fibonacci heap (Ch. 11) 12/17

  13. §3 Shortest Path Algorithms Too simple, and naïve… Try this one out: 2 1 2 – 2 1 3 4 2  Graphs with Negative Edge Costs Hey I have a good idea: why don’t we simply add a constant  to each edge and thus remove negative edges? void WeightedNegative( Table T ) { /* T is initialized by Figure 9.30 on p.303 */ Queue Q; Vertex V, W; Q = CreateQueue (NumVertex ); MakeEmpty( Q ); Enqueue( S, Q ); /* Enqueue the source vertex */ while ( !IsEmpty( Q ) ) { V = Dequeue( Q ); for ( each W adjacent to V ) if ( T[ V ].Dist + Cvw < T[ W ].Dist ) { T[ W ].Dist = T[ V ].Dist + Cvw; T[ W ].Path = V; if ( W is not already in Q ) Enqueue( W, Q ); } /* end-if update */ } /* end-while */ DisposeQueue( Q ); /* free memory */ } T = O( |V|  |E| ) /* each vertex can dequeue at most |V| times */ /* no longer once per edge */ /* negative-cost cycle will cause indefinite loop */ 13/17

  14. §3 Shortest Path Algorithms Signals the completion ofai ai ::= activity vj Index of vertex EC Time Lasting Time Slack Time LC Time  Acyclic Graphs If the graph is acyclic, vertices may be selected in topological order since when a vertex is selected, its distance can no longer be lowered without any incoming edges from unknown nodes. T = O( |E| + |V| ) and no priority queue is needed. Application: AOE ( Activity On Edge ) Networks —— scheduling a project  EC[ j ] \ LC[ j ] ::= the earliest \ latest completion time for node vj  CPM ( Critical Path Method ) 14/17

  15. §3 Shortest Path Algorithms 6 6 16 16 a0=6 a3=1 0 0 a6=9 a9=2 4 8 5 1 7 3 6 2 0 start 7 7 a1=4 18 18 a7=7 finish a10=4 4 6 a4=1 14 14 a2=5 a8=4 5 5 7 7 a5=2  Slack Time of <v,w> = 〖Example〗 AOE network of a hypothetical project 2 2 a11=0 3 Dummy activity  Calculation of EC: Start from v0, for any ai = <v, w>, we have  Calculation of LC: Start from the last vertex v8, for any ai = <v, w>, we have  Critical Path ::= path consisting entirely of zero-slack edges. 15/17

  16. §3 Shortest Path Algorithms 2. All-Pairs Shortest Path Problem For all pairs of vi and vj( i j ), find the shortest path between. Method 1 Use single-source algorithmfor |V| times. T = O( |V|3 ) – works fast on sparse graph. Method 2 O( |V|3 ) algorithm given in Ch.10, works faster on dense graphs. 16/17

  17. §3 Shortest Path Algorithms Laboratory Project 5 Saving James Bond Due: Thursday, December 7th, 2006 at 10:00pm Detailed requirements can be downloaded from http://10.71.45.99/list.asp?boardid=47 Courseware Download Don’t forget to sign you names and duties at the end of your report. 17/17

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