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Discussion #35 Dijkstra’s Algorithm

Discussion #35 Dijkstra’s Algorithm. Topics. Shortest path Greedy algorithms Dijkstra’s algorithm. b. c. a. e. d. Shortest Path. Minimum length path from one node to another (e.g. from a to e , the shortest path is 2) Algorithms Breadth-First-Search, O( n 2 ) in the worst case

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Discussion #35 Dijkstra’s Algorithm

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  1. Discussion #35Dijkstra’s Algorithm Chapter 7, Section 5.5

  2. Topics • Shortest path • Greedy algorithms • Dijkstra’s algorithm Chapter 7, Section 5.5

  3. b c a e d Shortest Path • Minimum length path from one node to another (e.g. from a to e, the shortest path is 2) • Algorithms • Breadth-First-Search, O(n2) in the worst case • Floyd’s, O(n3); but finds all • BFS, starting at each node, O(nm), which beats Floyd’s for sparse graphs Chapter 7, Section 5.5

  4. b c a e d 5 2 2 10 1 1 1 Shortest Path in Weighted Graphs • Weights on edges: positive numbers • Algorithms • Simple BFS no longer works (e.g. the shortest path from a to b is not the edge that connects them) • Floyd’s, O(n3); finds all • Is there an algorithm that beats Floyd’s? • for the single best path? • for all paths? Chapter 7, Section 5.5

  5. b c a e d 5 2 2 10 1 1 1 Dijkstra’s Algorithm • Find the shortest weighted path between two given nodes. (Easier to find the minimum from one to all.) • Intuitively, special BFS: choose smallest accumulated path length • Algorithm • Initialize D (distance) table. • Until all nodes are settled, • 2a. find smallest distance & settle node. • 2b. adjust distances in D for unsettled nodes. Chapter 7, Section 5.5

  6. Important Aside: Greedy Algorithms • A greedy algorithm always takes the best immediate or local solution while finding an answer. • Greedy algorithms find optimal solutions for some optimization problems, but may find (far) less-than-optimal solutions for other optimization problems. • Dijkstra’s algorithm is greedy. • There is no greedy algorithm for the traveling salesman problem (find the shortest path to visit all nodes). • When greedy algorithms work, they are usually best. Chapter 7, Section 5.5

  7. 5 2 2 10 1 1 d e a c b 1 iteration distance settled a b c d e 0 0 5 2 1  a adjust w/d 1 5 2 1  a,d 2 adjust w/c 2 5 2 2 a,d,c 4 adjust w/e 3 4 2 a,d,c,e 4 4 a,d,c,e,b Trace of Dijkstra’s Algorithm 1. Initialize D (start=a) 2. Until all nodes are settled: -find smallest distance & settle node. -adjust distances in D for unsettled nodes. all nodes settled Circled numbers: shortest path lengths from a. Chapter 7, Section 5.5

  8. 6 2 4 10 1 1 d e a c b 1 iteration distance settled a b c d e 0 0 6 4 1  a adjust w/d 1 6 4 1  a,d 2 adjust w/e 2 6 4 2 a,d,e 3 5 adjust w/c 3 6 3 a,d,e,c 4 5 a,d,e,c,b Trace with Different Weights 1. Initialize D (start=a) 2. Until all nodes are settled: -find smallest distance & settle node. -adjust distances in D for unsettled nodes. all nodes settled Circled numbers: shortest path lengths from a. Chapter 7, Section 5.5

  9. Proof that Dijkstra’s Algorithm Works • Loop Invariant: For each settled node x, the minimum distance from the start node s to x is D(x). • Additional observations (which help us in the proof) are also loop invariants. • (a) The shortest path from s to any settled node v consists only of settled nodes. • (b) The shortest path from s to any unsettled node y is at least the largest settled difference so far. Chapter 7, Section 5.5

  10. Proof by Induction(on the number of iterations) • Basis: 0 iterations: x = s, D(s) = 0 (which is the minimum since all weights are positive) • For (a): initially s = v, the only settled node • For (b): initially all nodes except s are unsettled, and the shortest path to each is at least 0, indeed > 0. • Induction: Assume that the main loop invariant and the loop invariants (a) and (b) hold for k iterations and show that they hold for k+1 iterations. Chapter 7, Section 5.5

  11. k settled k+1 settled s . . . v  x s . . . v  x s . . . v  x . . . w  y Proof of Induction Part (Sketch) Assume Dijkstra’s algorithm selects x to be settled. Assume that the main loop invariant does not hold for the k+1st iteration, then there is a shorter path from s to x. By the induction hypotheses, since Dijkstra’s algorithm chooses x, the path from s to x is the shortest using settled nodes, and the path from s to y is no shorter than s to x. Since all weights are positive, the path from y to x is at least 1, so that the assumed shorter path cannot exist. Further (a) and (b) continue to hold: (a) After settling x, the path from s to x consists of only settled nodes. (b) The adjustments are only for paths from x to any unsettled nodes. Since the distance from s to any unsettled node y is at least D(x), and D(x) is the largest settled distance so far. Chapter 7, Section 5.5

  12. b c a e d Unroll the loop: n + k1 + n + k2 + … + n + k(n-1) = (n1)n = O(n2) = 2(m  # of edges in the initialization) = O(m) 5 2 2 10 1 1 1 Big-Oh for Dijkstra’s Algorithm Adjacency List: • O(n)  visit at most all edges for a node  can’t be more than n • O(n)  each node needs to be settled 2a. O(n)  look through D(x) list to find smallest 2b. O(k)  for chosen node, go through list of length k  1. Initialize D (start=a) 2. Until all nodes are settled: 2a. find smallest distance & settle node. 2b. adjust distances in D for unsettled nodes. { O(n2) dominates Over all iterations, the k’s add up to 2(m  # of edges in the initialization). Chapter 7, Section 5.5

  13. d b c a e 5 2 2 10 1 1 1 Big-Oh for Dijkstra’s Algorithm Adjacency List: Clever using POTs: • O(nlogn)  POT (Partially Ordered Tree) construction dominates • O(n)  each node needs to be settled 2a. O(logn)  swap and bubble down 2b. O(klogn)  for chosen node, go through list of length k  1. Initialize D (start=a) 2. Until all nodes are settled: 2a. find smallest distance & settle node. 2b. adjust distances in D for unsettled nodes. { O(mlogn) dominates Over all iterations, the k’s add up to 2(m  # of edges in the initialization). Chapter 7, Section 5.5

  14. b c a e d 5 2 2 10 1 1 1 POT Construction/Manipulation POT construction (start = a) (b,5) (c,2) (c,2) (d,1) (d,1) (b,5)      (c,2) (b,5) (b,5) (d,1) (b,5) (c,2) (b,5) (c,2) (e,) Find smallest  always on top; then swap and bubble down no bubbling needed swap bubble down adjust (d,1) (e,) (c,2) (c,2) (c,2) …       (e,) (b,5) (c,2) (b,5) (b,5) (e,2) (b,5) (c,2) (b,5) (e,2) (e,) (d,1) Adjust distances  POT doubly linked to adjacency list nodes  bubble up and down, as needed, when going through list of chosen node to adjust values Chapter 7, Section 5.5

  15. Dijkstra’s Algorithm vs. Floyd’s • Dijkstra’s algorithm is O(mlogn) = O(n2logn) in the worst case. • Floyd’s algorithm is O(n3), but computes all shortest paths. • Dijkstra’s algorithm can compute all shortest paths by starting it n times, once for each node. Thus, to compute all paths, Dijkstra’s algorithm is O(nmlogn) = O(n3logn) in the worst case. • Which is better depends on the application. • Dijkstra’s algorithm is better if only a path from a single node to another or to all others is needed. • For all paths, Dijkstra’s algorithm is better for sparse graphs, and Floyd’s algorithm is better for dense graphs. Chapter 7, Section 5.5

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