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Dynamic Programming Techniques Lecture: Subset Sum & Shortest Path Problems

Discover the power of dynamic programming in solving subset sum and shortest path challenges. Join us for a session on optimal solutions and iterative problem-solving approaches. Learn how to schedule jobs efficiently and maximize task completion within resource constraints. Dive deep into the Subset Sum Problem and the Bellman-Ford algorithm for the Shortest Path Problem.

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Dynamic Programming Techniques Lecture: Subset Sum & Shortest Path Problems

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  1. Lecture 35 CSE 331 Nov 28, 2016

  2. Quiz 2 next Monday

  3. Comments on Feedback

  4. Official Feedback forms

  5. CS Ed week (Dec 5) We need volunteers! We need demos!

  6. When to use Dynamic Programming There are polynomially many sub-problems OPT(1), …, OPT(n) Richard Bellman Optimal solution can be computed from solutions to sub-problems OPT(j) = max {vj + OPT( p(j) ), OPT(j-1)} There is an ordering among sub-problem that allows for iterative solution OPT (j) only depends on OPT(j-1), …, OPT(1)

  7. Scheduling to min idle cycles n jobs, ith job takes wi cycles You have W cycles on the cloud What is the maximum number of jobs you can schedule?

  8. Subset sum problem n integers w1, w2, …, wn Input: bound W Output: • subset S of [n] such that • sum of wi for all i in S is at most W • w(S) is maximized

  9. Recursive formula OPT(j, W’) = max value out of w1,..,wj with bound W’ If wj > W’ OPT(j, W’) = OPT(j-1, W’) else OPT(j, W’) = max { OPT(j-1, W’), wj+ OPT(j-1,W’-wj) }

  10. Today’s agenda Dynamic Program for Subset Sum problem

  11. Shortest Path Problem Input: (Directed) Graph G=(V,E) and for every edge e has a cost ce (can be <0) t in V Output: Shortest path from every s to t Assume that G has no negative cycle 1 1 t s Shortest path has cost negative infinity 899 100 -1000

  12. May the Bellman force be with you

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