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Lecture-11-CS345A-2023 of Design and Analysis
- 1. Design and Analysisof Algorithms Lecture 11 • Shortest paths in graphs with positive weights 1 Algorithms-II : CS345A
- 2. Recap of lastlecture 2
- 3. Problem Definition Input: Adirected graph with and a source vertex Aim: • Compute for all • Compute for all 3
- 4. An example toget an insight into this problem 4 𝒔 G 𝒒 12 6 16 11 9 4 7 5
- 5. An example toget an insight into this problem Let us look at the neighborhood of 5 𝒔 𝒖 𝒙 𝒚 𝒛 𝒗 3 5 12 10 7 G
- 6. An example toget an insight into this problem The shortest path to vertex is edge (,). 6 𝒔 𝒖 𝒙 𝒚 𝒛 𝒗 3 5 12 10 7 G ¿ 𝟎 The nearest vertex of must be ’s neighbor.
- 7. Designing a greedyalgorithm for shortest paths What should be the greedy step such that “we can retrieve distance from in using distance from in ” ? 7 𝑮 𝑮 ′ instance of size Greedy step instance of size Opt() Opt() ?
- 8. We know thedistance to . Question: Can you remove vertex without affecting the distance from ? An example to get an insight into this problem 8 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 .
- 9. An example toget an insight into this problem 9 𝒙 𝒚 𝒛 5 12 10 7 G’ 3 2 8 21 𝒓 𝒕 +3 +3 +3 +3 𝒔 𝒗
- 10. An example toget an insight into this problem Theorem: For each , 10 𝒙 𝒚 𝒛 5 12 10 7 G’ 6 5 11 24 𝒓 𝒕 𝒔 𝒗
- 11. 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. 11 (,) + (,); Can you see some negative points of this algorithm ?
- 12. 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. 12
- 13. PROPERTIES OF ASHORTEST PATH 13
- 14. Optimal subpath property Considerany shortest path . = 51 Lemma 1: Every subpath of a shortest path is also a shortest path. 14 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 .
- 15. Exploiting the positiveweight 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. 15 𝒂 𝒈 𝒙 𝒚 𝒖 𝒔 ¿ 𝟎 ¿ 𝟎 ¿ 𝟎 ¿ 𝟎 ¿ 𝟎 ¿ 𝟎 (1)
- 16. More insights … Let: th nearest vertex from . = . Consider the shortest path . Lemma 2: must be of the form for some . 16 𝒔 𝒙 What picture captures all shortest paths from ? What can we say about ? must be for some . shortest
- 17. Complete picture ofall shortest paths ? Shortest paths tree 17 𝒔𝟏 𝒔𝟐 𝒔 𝒔𝟒 𝒔𝟓 𝒔𝟑 𝒔𝟕 𝒔𝟔
- 18. 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. 18 shortest
- 19. Suppose we havecomputed , ,…, . How to find out ? 19 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
- 20. Suppose we havecomputed , ,…, . How to find out ? For each is ?? 20 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 How to find ? Using Lemma 2
- 21. 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( , ) } } 21 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 ?
- 22. Dijkstra’s algorithm Dijkstra-algo(,) ; ; For = 0 to do { vertex from with minimum value of ; ; move from to ; For each with do { min( , ) } } 22 1 extract-min operation deg() Decrease-key operations for all ;
- 23. 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 ?