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CE222 – Data Structures & Algorithms II Chapter 9.3

WEEKS 9 and 10 Graphs II Unweighted Shortest Paths, Dijkstra’s Algorithm Graphs with negative costs, Acyclic Graphs. CE222 – Data Structures & Algorithms II Chapter 9.3 (based on the book by M. A. Weiss, Data Structures and Algorithm Analysis in C++, 3rd edition, 2006).

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CE222 – Data Structures & Algorithms II Chapter 9.3

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  1. WEEKS 9 and 10Graphs IIUnweighted Shortest Paths, Dijkstra’s AlgorithmGraphs with negative costs, Acyclic Graphs CE222 – Data Structures & Algorithms II Chapter 9.3 (based on the book by M. A. Weiss, Data Structures and Algorithm Analysis in C++, 3rd edition, 2006)

  2. 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]  weightedpathlength • Theunweighted path length is merely the number of edges on the path, namely, N-1. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  3. Shortest-Path Algorithms • 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  4. 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. • 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  5. 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 CE 222-Data Structures & Algorithms II, Izmir University of Economics

  6. 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  7. Unweighted Shortest Paths • Algorithm: find vertices that are at distance 1, 2, ... N-1 by processing vertices in layers (breadth-first search) CE 222-Data Structures & Algorithms II, Izmir University of Economics

  8. Unweighted Shortest Paths CE 222-Data Structures & Algorithms II, Izmir University of Economics

  9. Unweighted Shortest Paths-Implementation For each vertex v • The distance from s in the entry dv • The bookkeeping variable (path) which will allow us to print the actual path pv • A boolean variable to check is the vertex is processed or not  known • classVertex { • Listadj; • boolknown; • intdist; • Vertexpath; } CE 222-Data Structures & Algorithms II, Izmir University of Economics

  10. Unweighted Shortest Paths • Complexity O(|V|2) CE 222-Data Structures & Algorithms II, Izmir University of Economics

  11. Unweighted Shortest Paths - Improvement CE 222-Data Structures & Algorithms II, Izmir University of Economics

  12. 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|) CE 222-Data Structures & Algorithms II, Izmir University of Economics

  13. 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). CE 222-Data Structures & Algorithms II, Izmir University of Economics

  14. Dijkstra’s Algorithm - Example ► ► ► CE 222-Data Structures & Algorithms II, Izmir University of Economics

  15. Dijkstra’s Algorithm - Example • A proof by contradiction will show that this algorithm always works as long as no edge has a negative cost. ► ► ► ► CE 222-Data Structures & Algorithms II, Izmir University of Economics

  16. Dijkstra’s Algorithm Example – Stages I CE 222-Data Structures & Algorithms II, Izmir University of Economics

  17. Dijkstra’s Algorithm Example – Stages II CE 222-Data Structures & Algorithms II, Izmir University of Economics

  18. 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  19. 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  20. 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  21. 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  22. 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. CE 222-Data Structures & Algorithms II, Izmir University of Economics

  23. Acyclic Graphs - Example CE 222-Data Structures & Algorithms II, Izmir University of Economics

  24. 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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