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- 1. Design and Analysisof Algorithms Lecture 11 • Shortest paths in graphs with positive weights 1 CS345
- 2. Recap of lastlecture 2
- 3. based on asuitable Theorem about Opt() A generic way to design and analyse a greedy algorithm P: a given optimization problem 1. Try to establish a relation between OPT() and OPT(); 2. Try to prove the relation formally by deriving a solution of from OPT() deriving a solution of from OPT() 3 𝑨 𝑨 ′ instance of size of problem P Greedy step instance of size of problem P Opt() Opt() use Theorem If you succeed, you have a recursive algorithm from construction
- 4. Single Source ShortestPaths in Graphs with non-negative weights 4 Internalizing an algorithm is different from just knowing an algorithm. Go through these slides slowly, pondering over the observations, and attempting questions posed there. This is an opportunity for you to (1) re-invent the algorithm and (2) realize the proof of correctness.
- 5. Notations and Terminologies Adirected graph • Represented as Adjacency lists or Adjacency matrix • , Question: what is a path in ? Answer: A sequence , ,…, such that (,) for all . Length of a path = 5 𝒗 𝟏 𝒗 𝟐 𝒗 𝟑 𝒗𝒌 −𝟏 𝒗 𝒌 … 12 3 29 No vertex is repeated
- 6. Notations and Terminologies Definition:The path from to of minimum length is called the shortest path from to Definition: Distance from to is the length of the shortest path from to . Notations: : distance from to . : The shortest path from to . 6
- 7. Problem Definition Input: Adirected graph with and a source vertex Aim: • Compute for all • Compute for all This problem is simple and beautiful enough to convince anyone about the importance of designing efficient algorithms. 7
- 8. An example toget an insight into this problem Inference: The distance to any vertex depends upon global parameters. 8 𝒔 G 𝒒 12 6 16 11 9 4 7 5
- 9. An example toget an insight into this problem Question: Is there any vertex in this picture for which you are certain about the distance from ? Answer: vertex . 9 𝒔 𝒖 𝒙 𝒚 𝒛 𝒗 3 5 12 10 7 G Give reasons.
- 10. An example toget an insight into this problem The shortest path to vertex is edge (,). 10 𝒔 𝒖 𝒙 𝒚 𝒛 𝒗 3 5 12 10 7 G ¿ 𝟎 The nearest vertex of must be ’s neighbor.
- 11. Designing a greedyalgorithm for shortest paths What should be the greedy step such that “we can retrieve distance from in using distance from in ” ? 11 𝑮 𝑮 ′ instance of size Greedy step instance of size Opt() Opt() ? Using generic strategy
- 12. We know thedistance to . Question: Can you remove vertex without affecting the distance from ? An example to get an insight into this problem 12 𝒔 𝒖 𝒙 𝒚 𝒛 3 5 12 10 7 G 3 2 8 21 𝒓 𝒕 𝒗 Since we have computed distance to , perhaps we can remove . No. This is because there could be vertices whose shortest path from passes through .
- 13. An example toget an insight into this problem 13 𝒙 𝒚 𝒛 5 12 10 7 G’ 3 2 8 21 𝒓 𝒕 +3 +3 +3 +3 𝒔 𝒗
- 14. An example toget an insight into this problem Theorem: For each , 14 𝒙 𝒚 𝒛 5 12 10 7 G’ 6 5 11 24 𝒓 𝒕 𝒔 𝒗
- 15. How to computeinstance Let (,) be the least weight edge from in =(, ). Transform into as follows. 1. For each edge (,)ϵ , add edge (,); (,) ?? 2. In case of two edges from to any vertex , keep only the lighter edge. 3. Remove vertex . Theorem: For each , an algorithm for distances from with time complexity. 15 (,) + (,); Can you see some negative points of this algorithm ?
- 16. Shortcomings of thealgorithm • No insight into the (beautiful) structure of shortest paths. • Just convinces that we can solve the shortest paths problem in polynomial time. • Very few options to improve the time complexity. • Silent about a compact data structure for storing all shortest paths from the source. We shall now design a very insightful algorithm based on properties of shortest paths. 16
- 17. PROPERTIES OF ASHORTEST PATH 17
- 18. Optimal subpath property Considerany shortest path . = 51 Lemma 1: Every subpath of a shortest path is also a shortest path. 18 𝒔 ? < 27 𝒂 𝒈 𝒙 12 3 9 𝒚 𝒖 15 5 7 Can you realize an assumption made by you in this proof ? Homework: Write a complete and formal proof for Lemma 1 Proof: If there is a shorter path from to , replacing the current path using the shorter path will lead to a shorter path from to .
- 19. Exploiting the non-negativeweight on edges Consider once again a shortest path . This leads to an alternate proof to the following assertion : The nearest vertex of must be ’s neighbor. Can you realize this observation ? 19 𝒂 𝒈 𝒙 𝒚 𝒖 𝒔 ≥ 𝟎 ≥ 𝟎 ≥ 𝟎 ≥ 𝟎 ≥ 𝟎 ≥ 𝟎 (1)
- 20. More insights … Let: th nearest vertex from . = . Consider the shortest path . Lemma 2: must be of the form for some . 20 𝒔 𝒙 What picture captures all shortest paths from ? What can we say about ? must be for some . shortest
- 21. Complete picture ofall shortest paths ? Shortest paths tree 21 𝒔𝟏 𝒔𝟐 𝒔 𝒔𝟒 𝒔𝟓 𝒔𝟑 𝒔𝟕 𝒔𝟔
- 22. Designing the algorithm… Lemma 2: must be of the form for some . Question: Can we use Lemma 2 to design an algorithm ? Incremental way to compute shortest paths. Ponder over it before going to the next slide. 22 shortest
- 23. Suppose we havecomputed , ,…, . How to find out ? 23 𝒔𝟏 𝒔𝟐 𝒔 𝒔𝟒 𝒔𝟑 G 𝒗 If , How will the shortest path look like ? Or Or Using Lemma 2 We need to just focus on the edges incident on from , ,…, . How to use it to find ? Isn’t it nice ? Ponder over its usefulness before proceeding further. Let us define a function Consider any
- 24. Suppose we havecomputed , ,…, . How to find out ? For each is ?? 24 𝒔𝟏 𝒔𝟐 𝒔 𝒔𝟒 𝒔𝟑 G 𝒗 the vertex with minimum value of . min ¿ ¿ (𝜹(𝒔 ,𝒔 𝒋)+𝝎(𝒔𝒋 ,𝒗)) 51 76 101 90 82 62 88 Which vertex is ? Using Lemma 2
- 25. Dijkstra’s algorithm Dijkstra-algo(,) ; ; For = 0 to do { vertex from with minimum value of ; ; move from to ; For each do { ; For each with do min( , ) } } 25 a lot of re-computation for all ; Computing labels for the next iteration Only neighbors of What are the vertices whose value may change in this iteration ?
- 26. Dijkstra’s algorithm Dijkstra-algo(,) ; ; For = 0 to do { vertex from with minimum value of ; ; move from to ; For each with do { min( , ) } } 26 1 extract-min operation deg() Decrease-key operations for all ;
- 27. Homework • What datastructure is to be used to efficiently implement the Dijkstra’s algorithm? • Write a neat pseudocode in terms of the data structure. • What is the time complexity of the implementation ?