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Dynamic Programming: Manhattan Tourist Problem Lecture 17

Dynamic Programming: Manhattan Tourist Problem Lecture 17. Manhattan Tourist Problem (MTP). Manhattan Tourist Problem (MTP). Imagine seeking a path (from source to sink) to travel (only eastward and southward) with the most number of attractions ( * ) in the Manhattan grid. Source. *. *. *.

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Dynamic Programming: Manhattan Tourist Problem Lecture 17

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  1. Dynamic Programming:Manhattan Tourist ProblemLecture 17

  2. ManhattanTouristProblem(MTP)

  3. Manhattan Tourist Problem (MTP) Imagine seeking a path (from source to sink) to travel (only eastward and southward) with the most number of attractions (*) in the Manhattan grid Source * * * * * * * * * * * * Sink

  4. Manhattan Tourist Problem (MTP) Imagine seeking a path (from source to sink) to travel (only eastward and southward) with the most number of attractions (*) in the Manhattan grid Source * * * * * * * * * * * * Sink

  5. Manhattan Tourist Problem: Formulation Goal: Find the longest path in a weighted grid. Input: A weighted grid G with two distinct vertices, one labeled “source” and the other labeled “sink” Output: A longest path in Gfrom “source” to “sink”

  6. MTP: An Example 0 1 2 3 4 j coordinate source 3 2 4 0 3 5 9 0 0 1 0 4 3 2 2 3 2 4 13 1 1 6 5 4 2 0 7 3 4 15 19 2 i coordinate 4 5 2 4 1 0 2 3 3 3 20 3 8 5 6 5 2 sink 1 3 2 23 4

  7. MTP: Greedy Algorithm Is Not Optimal 1 2 5 source 3 10 5 5 2 5 1 3 5 3 1 4 2 3 promising start, but leads to bad choices! 5 0 2 0 22 0 0 0 sink 18

  8. MTP: Simple Recursive Program MT(n,m) ifn=0 or m=0 returnMT(n,m) x  MT(n-1,m)+ length of the edge from (n- 1,m) to (n,m) y  MT(n,m-1)+ length of the edge from (n,m-1) to (n,m) returnmax{x,y}

  9. MTP: Simple Recursive Program MT(n,m) x  MT(n-1,m)+ length of the edge from (n- 1,m) to (n,m) y  MT(n,m-1)+ length of the edge from (n,m-1) to (n,m) return min{x,y} What’s wrong with this approach?

  10. MTP: Dynamic Programming j 0 1 source 1 0 1 S0,1= 1 i 5 1 5 S1,0= 5 • Calculate optimal path score for each vertex in the graph • Each vertex’s score is the maximum of the prior vertices score plus the weight of the respective edge in between

  11. MTP: Dynamic Programming (cont’d) j 0 1 2 source 1 2 0 1 3 S0,2 = 3 i 5 3 -5 1 5 4 S1,1= 4 3 2 8 S2,0 = 8

  12. MTP: Dynamic Programming (cont’d) j 0 1 2 3 source 1 2 5 0 1 3 8 S3,0 = 8 i 5 3 10 -5 1 1 5 4 13 S1,2 = 13 5 3 -5 2 8 9 S2,1 = 9 0 3 8 S3,0 = 8

  13. MTP: Dynamic Programming (cont’d) j 0 1 2 3 source 1 2 5 0 1 3 8 i 5 3 10 -5 -5 1 -5 1 5 4 13 8 S1,3 = 8 5 3 -3 3 -5 2 8 9 12 S2,2 = 12 0 0 0 3 8 9 S3,1 = 9 greedy alg. fails!

  14. MTP: Dynamic Programming (cont’d) j 0 1 2 3 source 1 2 5 0 1 3 8 i 5 3 10 -5 -5 1 -5 1 5 4 13 8 5 3 -3 2 3 3 -5 2 8 9 12 15 S2,3 = 15 0 0 -5 0 0 3 8 9 9 S3,2 = 9

  15. MTP: Dynamic Programming (cont’d) j 0 1 2 3 source 1 2 5 0 1 3 8 Done! i 5 3 10 -5 -5 1 -5 1 5 4 13 8 (showing all back-traces) 5 3 -3 2 3 3 -5 2 8 9 12 15 0 0 -5 1 0 0 0 3 8 9 9 16 S3,3 = 16

  16. si-1, j + weight of the edge between (i-1, j) and (i, j) si, j-1 + weight of the edge between (i, j-1) and (i, j) max si, j = MTP: Recurrence Computing the score for a point (i,j) by the recurrence relation: The running time is n x m for a n by mgrid (n = # of rows, m = # of columns)

  17. A2 A3 A1 B sA1 + weight of the edge (A1, B) sA2 + weight of the edge (A2, B) sA3 + weight of the edge (A3, B) max of sB = Manhattan Is Not A Perfect Grid What about diagonals? • The score at point B is given by:

  18. max of sy + weight of vertex (y, x) where y є Predecessors(x) sx = Manhattan Is Not A Perfect Grid (cont’d) Computing the score for point x is given by the recurrence relation: • Predecessors (x) – set of vertices that have edges leading to x • The running time for a graph G(V, E) (V is the set of all vertices and Eis the set of all edges) is O(E) since each edge is evaluated once

  19. Traveling in the Grid • The only hitch is that one must decide on the order in which visit the vertices • By the time the vertex x is analyzed, the values sy for all its predecessors y should be computed – otherwise we are in trouble. • We need to traverse the vertices in some order • Try to find such order for a directed cycle • ???

  20. DAG: Directed Acyclic Graph • Since Manhattan is not a perfect regular grid, we represent it as a DAG • DAG for Dressing in the morning problem

  21. Topological Ordering • A numbering of vertices of the graph is called topological ordering of the DAG if every edge of the DAG connects a vertex with a smaller label to a vertex with a larger label • In other words, if vertices are positioned on a line in an increasing order of labels then all edges go from left to right.

  22. Topological ordering • 2 different topological orderings of the DAG

  23. Longest Path in DAG Problem • Goal: Find a longest path between two vertices in a weighted DAG • Input: A weighted DAG G with source and sink vertices • Output: A longest path in G from source to sink

  24. max of sv= Longest Path in DAG: Dynamic Programming • Suppose vertex v has indegree 3 and predecessors {u1, u2, u3} • Longest path to v from source is: In General: sv = maxu (su + weight of edge from uto v) su1 + weight of edge fromu1to v su2 + weight of edge fromu2to v su3+ weight of edge fromu3to v

  25. Traversing the Manhattan Grid a) b) • 3 different strategies: • a) Column by column • b) Row by row • c) Along diagonals c)

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