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Chapter 9 : Graphs Part I (Topological Sort & Shortest Path Algorithms)

Chapter 9 : Graphs Part I (Topological Sort & Shortest Path Algorithms). CE 221 Data Structures and Algorithms. T ext : Read Weiss, § 9.1 – 9.3. Definitions - I. A graph G=(V, E) consists of a set of vertices , V, and a set of edges , E. Each edge is a pair (v, w), where v, w є V.

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Chapter 9 : Graphs Part I (Topological Sort & Shortest Path Algorithms)

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  1. Chapter 9 : Graphs Part I(Topological Sort & Shortest Path Algorithms) CE 221 Data Structures and Algorithms Text: Read Weiss, §9.1 – 9.3 Izmir University of Economics

  2. Definitions - I • A graph G=(V, E) consists of a set of vertices, V, and a set of edges, E. • Each edge is a pair (v, w), where v, w є V. • If the pair is ordered then G is directed (digraph). • Vertex w is adjacent to v iff (v, w) є E. • In an undirected graph with edge (v, w), w is adjacent to v and v is adjacent to w. • Sometimes and edge has a third component, weight or cost. Izmir University of Economics

  3. Definitions - II • A path in a graph is w1, w2,...,wN such that (wi, wi+1) є E for 1≤i<N. The length of such a path is the number of edges on the path. If a path from a vertex to itself contains no edges, then the path length is zero. If G contains an edge (v, v), then the path v, v is called a loop. • A simple path is a path such that all vertices are distinct, except that the first and the last could be the same. Izmir University of Economics

  4. Definitions - III • A cycle in a directed graph is a path of length at least 1 such that w1=wN. This cycle is simple if the path is simple. For undirected graphs, the edges are required to be distinct (Why?). • A directed graph is acyclic if it has no cycles (DAG). • An undirected graph is connected if there is a path from every vertex to every other vertex. A directed graph with this property is called strongly connected. If directed graph is not, but underlying undirected graph is, it is weakly connected. A complete graph is a graph in which there is an edge between every pair of vertices. Izmir University of Economics

  5. Representation of Graphs - I • One simple way is to use a two-dimensional array (adjacency matrix representation). If vertices are numbered starting at 1, A[u][v]=true if (u, v) є E. Space requirement is Θ(|V|2). • If the graph is not dense (sparse), adjacency lists may be used. The space requirement is O(|E|+|V|). Izmir University of Economics

  6. Representation of Graphs - II Izmir University of Economics

  7. Topological Sort - I • A topological sort is an ordering of vertices in a DAG, such that if there is path from vi to vj, then vj appears after vi in the ordering. • A simple algorithm to find a topological ordering is first to find any vertex with no incoming edges. We can then print this vertex, and remove it, along with its edges. Then apply the same strategy to the rest of the graph. To formalize this, define the indegree of a vertex v as the number of edges (u, v). Izmir University of Economics

  8. Topological Sort – Initial Attempt • running time of the algorithm is O(|V|2). Izmir University of Economics

  9. Topological Sort – A Better Algorithm • We can remove the inefficiency by keeping all the unassigned vertices of indegree 0 in a special data structure (queue or stack). When a new vertex with degree zero is needed, it is returned by removing one from the queue, and when the indegrees of adjacent vertices are decremented, they are inserted into the queue if the indegree falls to zero. The running time is O(|E|+|V|) Izmir University of Economics

  10. Homework Assignments • 9.1, 9.2, 9.3, 9.4, 9.38 • You are requested to study and solve the exercises. Note that these are for you to practice only. You are not to deliver the results to me. Izmir University of Economics

  11. Shortest-Path Algorithms • The input is a weighted graph: associated with each edge (vi, vj) is a cost ci,j. The cost of a path v1v2...vN is ∑ci,i+1 for i in [1..N-1]. This is weighted path length. • Unweighted path lengthis merely the number of edges on the path, namely, N-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. Izmir University of Economics

  12. Negative Cost Cycles • In the graph to the left, the shortest path from v1 to v6 has a cost of 6 and the path itself is v1v4v7v6. The shortest unweighted path has 2 edges (v1 - v4) and (v4 - v6). • In the graph to the right, we have a negative cost. The path from v5 to v4 has cost 1, but a shorter path exists by following the loop v5v4v2v5v4 which has cost -5. This path is still not the shortest, because we could stay in the loop arbitrarily long. Izmir University of Economics

  13. Shortest Path Length: Problems We will examine 4 algorithms to solve four versions of the problem • Unweighted shortest path  O(|E|+|V|) • Weighted shortest path without negative edges  O(|E|log|V|) using queues • Weighted shortest path with negative edges  O(|E| . |V|) • Weighted shortest path of acyclic graphs  linear time Izmir University of Economics

  14. Unweighted Shortest Paths • Using some vertex, s, which is an input parameter, find the shortest path from s to all other vertices in an unweighted graph. Assume s=v3. Izmir University of Economics

  15. Unweighted Shortest Paths • Algorithm: find vertices that are at distance 1, 2, ... N-1 by processing vertices in layers (breadth-first search) Izmir University of Economics

  16. Unweighted Shortest Paths Izmir University of Economics

  17. Unweighted Shortest Paths • Complexity O(|V|2) Izmir University of Economics

  18. Unweighted Shortest Paths - Improvement • At any point in time there are only two types of unknown vertices that have dv≠∞. Some have dv = currDist and the rest have dv = currDist +1. • We can make use of a queue data structure. • O(|E|+|V|) Izmir University of Economics

  19. Weighted Shortest Path Dijkstra’s Algorithm • With weighted shortest path,distance dv is tentative. It turns out to be the shortest path length from s to v using only known vertices as intermediates. • Greedy algorithm: proceeds in stages doing the best at each stage. Dijkstra’s algorithm selects a vertex v with smallest dv among all unknown vertices and declares it known. Remainder of the stage consists of updating the values dw for all edges (v, w). Izmir University of Economics

  20. Dijkstra’s Algorithm - Example ► ► ► Izmir University of Economics

  21. Dijkstra’s Algorithm - Example • A proof by contradiction will show that this algorithm always works as long as no edge has a negative cost. ► ► ► ► Izmir University of Economics

  22. Dijkstra’s Algorithm - Pseudocode • If the vertices are sequentially scanned to find minimum dv, each phase will take O(|V|) to find the minimum, thus O(|V|2) over the course of the algorithm. • The time for updates is constant and at most one update per edge for a total of O(|E|). • Therefore the total time spent is O(|V|2+|E|). • If the graph is dense, OPTIMAL. Izmir University of Economics

  23. Dijkstra’s Algorithm-What if the graph is sparse? • If the graph is sparse |E|=θ(|V|), algorithm is too slow. The distances of vertices need to be kept in a priority queue. • Selection of vertex with minimum distance via deleteMin, and updates via decreaseKey operation. Hence; O(|E|log|V|+|V|log|V|) • find operations are not supported, so you need to be able to maintain locations of di in the heap and update them as they change. • Alternative: insert w and dw with every update. Izmir University of Economics

  24. Graphs with negative edge costs • Dijkstra’s algorithm does not work with negative edge costs. Once a vertex u is known, it is possible that from some other unknown vertex v, there is a path back to u that is very negative. • Algorithm: A combination of weighted and unweighted algorithms. Forget about the concept of known vertices. Izmir University of Economics

  25. Graphs with negative edge costs - I • O(|E|*|V|). Each vertex can dequeue at most O(|V|) times. (Why? Algorithm computes shortest paths with at most 0, 1, ..., |V|-1 edges in this order). Hence, the result! • If negative cost cycles, then each vertex should be checked to have been dequeued at most |V| times. Izmir University of Economics

  26. Acyclic Graphs • If the graph is known to be acyclic, the order in which vertices are declared known, can be set to be the topological order. • Running time = O(|V|+|E|) • This selection rule works because when a vertex is selected, its distance can no longer be lowered, since by topological ordering rule it has no incoming edges emanating from unknown nodes. Izmir University of Economics

  27. Dijkstra's Shortest Path Algorithm • Find shortest path from s to t. 2 24 3 9 s 18 14 6 2 6 4 19 30 11 5 15 5 6 16 20 t 7 44

  28. Dijkstra's Shortest Path Algorithm S = { } PQ = { s, 2, 3, 4, 5, 6, 7, t }   2 24 3 0 9 s 18  14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label 

  29. Dijkstra's Shortest Path Algorithm S = { } PQ = { s, 2, 3, 4, 5, 6, 7, t } Here   2 24 3 0 9 s 18  14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label 

  30. Dijkstra's Shortest Path Algorithm S = { s } PQ = { 2, 3, 4, 5, 6, 7, t } Node discovered   9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label  15 X

  31. Dijkstra's Shortest Path Algorithm S = { s } PQ = { 2, 3, 4, 5, 6, 7, t } Here   9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label  15 X

  32. Dijkstra's Shortest Path Algorithm S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t }   9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44   15 X

  33. Dijkstra's Shortest Path Algorithm S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t } Node discovered  33 X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44   15 X

  34. Dijkstra's Shortest Path Algorithm S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t }  33 X  9 X 2 24 3 0 9 Here s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44   15 X

  35. Dijkstra's Shortest Path Algorithm S = { s, 2, 6 } PQ = { 3, 4, 5, 7, t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  44 4  X 19 30 11 5 15 5 6 16 20 t 7 44   15 X

  36. Dijkstra's Shortest Path Algorithm S = { s, 2, 6 } PQ = { 3, 4, 5, 7, t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  44 4  X 19 30 11 5 15 5 6 16 20 t 7 44  Here  15 X

  37. Dijkstra's Shortest Path Algorithm S = { s, 2, 6, 7 } PQ = { 3, 4, 5, t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  35 44 X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  59 X  15 X

  38. Dijkstra's Shortest Path Algorithm S = { s, 2, 6, 7 } PQ = { 3, 4, 5, t } Here 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  35 44 X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  59 X  15 X

  39. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 6, 7 } PQ = { 4, 5, t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  35 34 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  51 59 X X  15 X

  40. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 6, 7 } PQ = { 4, 5, t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  35 34 44 X X 4  X 19 30 11 5 15 5 6 16 Here 20 t 7 44  51 59 X X  15 X

  41. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 5, 6, 7 } PQ = { 4, t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

  42. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 5, 6, 7 } PQ = { 4, t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 Here 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

  43. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 4, 5, 6, 7 } PQ = { t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

  44. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 4, 5, 6, 7 } PQ = { t } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  Here 50 51 59 X X X  15 X

  45. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 4, 5, 6, 7, t } PQ = { } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

  46. Dijkstra's Shortest Path Algorithm S = { s, 2, 3, 4, 5, 6, 7, t } PQ = { } 32  33 X X  9 X 2 24 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

  47. The Algoritm • 1. Assign to every node a distance value. Set it to zero for our initial node and to infinity (-9999) for all other nodes. • 2. Mark all nodes as unvisited. Set initial node as current. • 3. For current node, consider all its unvisited neighbors and calculate their distance (from the initial node). For example, if current node (A) has distance of 6, and an edge connecting it with another node (B) is 2, the distance to B through A will be 6+2=8. If this distance is less than the previously recorded distance (infinity in the beginning, zero for the initial node), overwrite the distance. • 4. When we are done considering all neighbors of the current node, mark it as visited. A visited node will not be checked ever again; its distance recorded now is final and minimal. • 5. If all nodes have been visited, finish. Otherwise, set the unvisited node with the smallest distance (from the initial node) as the next "current node" and continue from step 3.

  48. Pseudo Code of the Dijkstra’s Alg • Algorithm Dijkstra(Graph, source): • foreach vertex v in Graph: // Initializations • dist[v] := infinity // Unknown distance function from source to v • previous[v] := undefined // Previous node in optimal path from source • dist[source] := 0 // Distance from source to source • Q := the set of all nodes in Graph • // All nodes in the graph are unoptimized - thus are in Q • while Q is not empty: // The main loop • u:= dequeue (Q) // u := vertex in Q with smallest dist[] • if dist[u] = infinity: • break // all remaining vertices are inaccessible from source • dequeue(u,Q) // remove u from Q • for each neighbor v of u: // where v has not yet been removed from Q. • alt := dist[u] + dist_between(u, v) // Dist. from start point to neighbor • if alt < dist[v]: // Relax (u,v) • dist[v] := alt • previous[v] := u • return dist

  49. Homework Assignments • 9.5, 9.7, 9.10, 9.40, 9.42, 9.44, 9.46, 9.47, 9.52 • You are requested to study and solve the exercises. Note that these are for you to practice only. You are not to deliver the results to me. Izmir University of Economics

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