Algorithms and Data Structures Lecture 10 Graph Algorithms III: - PowerPoint PPT Presentation

Algorithms and Data Structures Lecture 10 Graph Algorithms III: Shortest Paths Fabian Kuhn Algorithms and Complexity Fabian Kuhn Algorithms and Complexity

Shortest Paths Single Sourse Shortest Paths Problem • Given: weighted graph 𝐻 = 𝑊, 𝐹, 𝑥 , start node 𝑡 ∈ 𝑊 – We denote the weight of an edge 𝑣, 𝑤 by 𝑥 𝑣, 𝑤 – Assumption for now: ∀𝑓 ∈ 𝐹: 𝑥 𝑓 ≥ 0 • Goal: Find shortest paths / distances from 𝑡 to all nodes – Distance from 𝑡 to 𝑤 : 𝑒 𝐻 𝑡, 𝑤 (length of a shortest path) 1 𝒕 1 1 2 15 3 3 9 7 8 3 4 2 6 6 1 5 Distance from node 1 to node 7 : 10 8 6 7 9 Fabian Kuhn Algorithms and Complexity 2

Optimality of Subpaths Lemma: If 𝑤 0 , 𝑤 1 , … , 𝑤 𝑙 is a shortest path from 𝑤 0 to 𝑤 𝑙 , then it holds for all 0 ≤ 𝑗 ≤ 𝑘 ≤ 𝑙 that the subpath 𝑤 𝑗 , 𝑤 𝑗+1 , … , 𝑤 𝑘 is also a shortest path from 𝑤 𝑗 to 𝑤 𝑘 . Shortest path from 𝒘 𝟏 to 𝒘 𝒍 : 𝒘 𝒌 𝒘 𝟏 𝒘 𝒋 𝒘 𝒍 𝒘 𝟐 • Subpath from 𝑤 𝑗 to 𝑤 𝑘 is also a shortest path. – Otherwise, one could replace the path from 𝑤 𝑗 to 𝑤 𝑘 by the shortest path from 𝑤 𝑗 to 𝑤 𝑘 . – If by doing this, nodes are visited multiple time, one can cut out cycles and obtains an even shorter path. • Lemma also holds for negative edge weights, – as long as the graph does not contain negative cycles. Fabian Kuhn Algorithms and Complexity 3

Shortest-Path Tree • Spanning tree that is rooted at node 𝑡 and that contains shortest paths from 𝑡 to all other nodes. – Such a tree always exists (follows from the optimality of subpaths) • For unweighted graphs: BFS spanning tree • Goal: Find a shortest path tree 𝒕 1 1 2 15 3 3 9 7 8 3 4 2 6 6 1 5 8 6 7 9 Fabian Kuhn Algorithms and Complexity 4

Dijkstra’s Algorithm: Idea • Algorithm by Edsger W. Dijkstra (published in 1959) Idea: • We start at 𝑡 and build the spanning tree in a step-by-step manner. Invariant: Algorithm always has a tree rooted at 𝑡 , which is a subtree of a shortest path tree. • Goal: In each step of the algorithm, add one node – Initially: subtree only consists of 𝑡 (trivially satisfies invariant...) – 1 st step: Because of the optimality of subpaths, there must be a shortest path consisting of a single edge... – Always add the remaining node at the smallest distance from 𝑡 . Fabian Kuhn Algorithms and Complexity 5

Dijkstra’s Algorithm : One Step Given: A tree 𝑈 that is rooted in 𝑡 , such that 𝑈 is a subtree of a shortest paths tree for node 𝑡 in 𝐻 . (nodes of 𝑈 : 𝑇 ) How can we extend 𝑈 by a single node? 10 𝟏 𝑇 : nodes in the tree 𝑈 𝒕 𝟐𝟏 𝟔 1 5 4 𝟐 𝑂 𝑇 : nodes that can be added to 2 3 𝟐𝟐 𝑻 the tree directly. 3 5 𝟕 𝟑 𝟗 7 To add 𝑤 ∈ 𝑂 𝑇 it most hold that 1 𝑶(𝑻) 2 3 𝑒 𝐻 𝑡, 𝑤 = min 𝑣∈𝑇 𝑒 𝐻 𝑡, 𝑣 + 𝑥 𝑣, 𝑤 2 𝟗 5 𝟓 We will see that this always holds for 𝟗 6 𝑤 ∈ 𝑂 𝑇 with minimum distance 𝟐𝟏 𝑒 𝐻 𝑡, 𝑤 from 𝑡 . Fabian Kuhn Algorithms and Complexity 6

Dijkstra’s Algorithm : One Step Given: 𝑈 is subtree of a shortest path tree for 𝑡 in 𝐻 . Lemma: For a node 𝑤 ∈ 𝑂 𝑇 and an edge 𝑣, 𝑤 with 𝑣 ∈ 𝑇 such that 𝑒 𝐻 𝑡, 𝑣 + 𝑥 𝑣, 𝑤 is minimized, it holds that 𝒆 𝑯 𝒕, 𝒘 = 𝒆 𝑯 𝒕, 𝒗 + 𝒙 𝒗, 𝒘 Consider the 𝑡 - 𝑤 path that we obtain in this way: 𝑻 𝑶(𝑻) 𝒕 𝒗 𝒘 Assume that there is a shorter path: 𝑶(𝑻) 𝑻 𝒕 𝒚 𝒛 𝒘 – Because there are no negative edge weights, we therefore have 𝑒 𝐻 𝑡, 𝑦 + 𝑥 𝑦, 𝑧 ≤ 𝑒 𝐻 (𝑡, 𝑤) < 𝑒 𝐻 𝑡, 𝑣 + 𝑥(𝑣, 𝑤) Fabian Kuhn Algorithms and Complexity 7

Dijkstra’s Algorithm Invariant: Algorithm always has a tree 𝑈 = (𝑇, 𝐵) rooted at 𝑡 , which is a subtree of a shortest path tree of 𝐻 . • At the beginning, we have 𝑈 = 𝑡 , ∅ • For each node 𝑤 ∉ 𝑇 , one at all times computes 𝜀 𝑡, 𝑤 ≔ 𝑣∈𝑇∩𝑂 in 𝑤 𝑒 𝐻 𝑡, 𝑣 + 𝑥 𝑣, 𝑤 min – as well as the incoming neighbor 𝑣 =: 𝛽 𝑤 that minimized the expression... • 𝜀 𝑡, 𝑤 corresponds to an 𝑡 - 𝑤 path ⟹ 𝜀 𝑡, 𝑤 ≥ 𝑒 𝐻 𝑡, 𝑤 • Lemma on last slide: For minimum 𝜺 𝒕, 𝒘 , we have: 𝜺 𝒕, 𝒘 = 𝒆 𝑯 𝒕, 𝒘 Fabian Kuhn Algorithms and Complexity 8

Dijkstra’s Algorithm Initialization 𝑼 = ∅, ∅ • 𝜀 𝑡, 𝑡 = 0 , and 𝜀 𝑡, 𝑤 = ∞ for all 𝑤 ≠ 𝑡 update 𝜀 𝑡, 𝑦 • 𝛽 𝑤 = NULL for all 𝑤 ∈ 𝑊 𝒚 𝒕 Iteration Step 𝒘 • Choose a node 𝑤 with smallest 𝜀 𝑡, 𝑤 ≔ 𝑣∈𝑇∩𝑂 in 𝑤 𝑒 𝐻 𝑡, 𝑣 + 𝑥 𝑣, 𝑤 min • Go through all out-neighbors 𝑦 ∈ 𝑊 ∖ 𝑇 and set 𝜀 𝑡, 𝑦 ≔ min 𝜀 𝑡, 𝑦 , 𝜀 𝑡, 𝑤 + 𝑥 𝑤, 𝑦 – If 𝜀 𝑡, 𝑦 is decreased, set 𝛽 𝑦 = 𝑤 • Add node 𝑤 and edge 𝛽 𝑤 , 𝑤 to the tree 𝑈 . Fabian Kuhn Algorithms and Complexity 9

Dijkstra’s Algorithm: Example ∞ 1 32 ∞ ∞ 3 9 10 23 4 13 ∞ ∞ 3 2 6 2 ∞ 1 ∞ 3 ∞ 17 9 19 8 2 1 20 𝟏 ∞ 18 1 ∞ Fabian Kuhn Algorithms and Complexity 10

Dijkstra’s Algorithm: Example ∞ 1 32 ∞ ∞ 3 9 10 23 4 13 𝟐 ∞ 3 2 6 2 ∞ 1 𝟐𝟖 3 𝟐𝟘 17 9 19 8 2 1 20 𝟏 𝟑𝟏 18 1 𝟐𝟗 Fabian Kuhn Algorithms and Complexity 11

Dijkstra’s Algorithm: Example ∞ 1 32 𝟓 ∞ 3 9 10 23 4 13 𝟐 𝟐𝟓 3 2 6 2 ∞ 1 𝟖 3 𝟐𝟘 17 9 19 8 2 1 20 𝟏 𝟑𝟏 18 1 𝟐𝟗 Fabian Kuhn Algorithms and Complexity 12

Dijkstra’s Algorithm: Example 𝟔 1 32 𝟓 ∞ 3 9 10 23 4 13 𝟐 𝟐𝟒 3 2 6 2 ∞ 1 𝟖 3 𝟐𝟘 17 9 19 8 2 1 20 𝟏 𝟑𝟏 18 1 𝟐𝟗 Fabian Kuhn Algorithms and Complexity 13

Dijkstra’s Algorithm: Example 𝟔 1 32 𝟓 𝟒𝟖 3 9 10 23 4 13 𝟐 𝟐𝟒 3 2 6 2 ∞ 1 𝟖 3 𝟐𝟘 17 9 19 8 2 1 20 𝟏 𝟑𝟏 18 1 𝟐𝟗 Fabian Kuhn Algorithms and Complexity 14

Dijkstra’s Algorithm: Example 𝟔 1 32 𝟓 𝟒𝟖 3 9 10 23 4 13 𝟐 𝟘 3 2 6 2 ∞ 1 𝟖 3 𝟐𝟘 17 9 19 8 2 1 20 𝟏 𝟑𝟏 18 1 𝟐𝟔 Fabian Kuhn Algorithms and Complexity 15

Dijkstra’s Algorithm: Example 𝟔 1 32 𝟓 𝟐𝟘 3 9 10 23 4 13 𝟐 𝟘 3 2 6 2 𝟐𝟑 1 𝟖 3 𝟐𝟐 17 9 19 8 2 1 20 𝟏 𝟑𝟏 18 1 𝟐𝟔 Fabian Kuhn Algorithms and Complexity 16

Dijkstra’s Algorithm: Example 𝟔 1 32 𝟓 𝟐𝟘 3 9 10 23 4 13 𝟐 𝟘 3 2 6 2 𝟐𝟑 1 𝟖 3 𝟐𝟐 17 9 19 8 2 1 20 𝟏 𝟐𝟒 18 1 𝟐𝟑 Fabian Kuhn Algorithms and Complexity 17

Dijkstra’s Algorithm Initialization 𝑼 = ∅, ∅ • 𝜀 𝑡, 𝑡 = 0 , and 𝜀 𝑡, 𝑤 = ∞ for all 𝑤 ≠ 𝑡 update 𝜀 𝑡, 𝑦 • 𝛽 𝑤 = NULL for all 𝑤 ∈ 𝑊 𝒚 𝒕 Iteration Step 𝒘 • Choose a node 𝑤 with smallest 𝜀 𝑡, 𝑤 ≔ 𝑣∈𝑇∩𝑂 in 𝑤 𝑒 𝐻 𝑡, 𝑣 + 𝑥 𝑣, 𝑤 min • Go through all out-neighbors 𝑦 ∈ 𝑊 ∖ 𝑇 and set 𝜀 𝑡, 𝑦 ≔ min 𝜀 𝑡, 𝑦 , 𝜀 𝑡, 𝑤 + 𝑥 𝑤, 𝑦 – If 𝜀 𝑡, 𝑦 is decreased, set 𝛽 𝑦 = 𝑤 Similar to the MST algorithm • Add node 𝑤 and edge 𝛽 𝑤 , 𝑤 to the tree 𝑈 . of Prim! Fabian Kuhn Algorithms and Complexity 18

Reminder : Prim’s MST Algorithm 𝐼 = new priority queue; 𝐵 = ∅ for all 𝑣 ∈ 𝑊 ∖ {𝑡} do 𝐼 .insert( 𝑣 , ∞ ); 𝛽(𝑣) = NULL 𝐼 .insert( 𝑡 , 0 ) while 𝐼 is not empty do 𝑣 = H.deleteMin() for all unmarked neighbors 𝑤 of 𝑣 do if 𝑥 𝑣, 𝑤 < 𝑒(𝑤) then 𝐼 .decreaseKey( 𝑤 , 𝑥 𝑣, 𝑤 ) 𝛽 𝑤 = 𝑣 𝑣 .marked = true if 𝑣 ≠ 𝑡 then 𝐵 = 𝐵 ∪ 𝑣, 𝛽 𝑣 Fabian Kuhn Algorithms and Complexity 19

Dijkstra’s Algorithm : Implementation 𝐼 = new priority queue; 𝐵 = ∅ for all 𝑣 ∈ 𝑊 ∖ {𝑡} do 𝐼 .insert( 𝑣 , ∞ ); 𝜀 𝑡, 𝑣 = ∞ ; 𝛽(𝑣) = NULL 𝐼 .insert( 𝑡 , 0 ) while 𝐼 is not empty do 𝑣 = H.deleteMin() for all unmarked out-neighbors 𝑤 of 𝑣 do if 𝜀 𝑡, 𝑣 + 𝑥 𝑣, 𝑤 < 𝜀 𝑡, 𝑤 then 𝜀(𝑡, 𝑤) = 𝜀 𝑡, 𝑣 + 𝑥(𝑣, 𝑤) 𝐼 .decreaseKey( 𝑤 , 𝜀 𝑡, 𝑤 ) 𝛽 𝑤 = 𝑣 𝑣 .marked = true if 𝑣 ≠ 𝑡 then 𝐵 = 𝐵 ∪ 𝛽 𝑣 , 𝑣 Fabian Kuhn Algorithms and Complexity 20

Dijkstra’s Algorithm: Running Time • Algorithm implementation is almost identical to the implementation of Prim’s MST algorithm. • Number of heap operations: create: 1 , insert: 𝑜 , deleteMin: 𝑜 , decreaseKey: ≤ 𝑛 – Or alternatively without decrease-key: 𝑃 𝑛 insert and deleteMin Op. • Running time with binary heap: 𝑷 𝒏 𝐦𝐩𝐡 𝒐 • Running time with Fibonacci heap: 𝑷 𝒏 + 𝒐 𝐦𝐩𝐡 𝒐 Fabian Kuhn Algorithms and Complexity 21