cs340_09s_chapter10Part2 - CS 340 Similar to DP Outline...

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1 CS 340 Intro to Greedy Algorithms Outline • Review/continue of DP discuss / examples • Greedy algorithms – Similar to DP • not an actual algorithm, but a class of algorithms Two steps to dynamic programming • Formulate the answer as a recurrence relation or recursive algorithm. • Specify an order of evaluation for the recurrence so you always have what you need. Restaurant location problem • You work in the fast food business • Your company plans to open up new restaurants in IL along I-57 • Many towns along the highway, call them t 1 , t 2 , …, t n • Restaurants at t i has estimated annual profit p i • No two restaurants can be located within 10 miles of each other due to regulation • Your boss wants to maximize the total profit • You want a big bonus 10 miles Restaurant location problem • Could exhaustively search… – Produce all possible combinations – Eliminate the illegal combinations – Pick the one with the largest value – Runtime: 10 miles A DP algorithm • Suppose you’ve already found the optimal solution • It will either include t n or not include t n • Case 1: t n not included in optimal solution – Best solution same as best solution for t 1 , …, t n-1 • Case 2: t n included in optimal solution – Best solution is p n + best solution for t 1 , …, t j , where j < n is the largest index so that dist(t j , t n ) 10
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2 Recurrence formulation • Let S(i) be the total profit of the optimal solution when the first i towns are considered (not necessarily selected) – S(n) is the optimal solution to the complete problem S(n-1) S(j) + p n j , t n ) 10 S(n) = max S(i-1) S(j) + p i j , t i ) 10 S(i) = max Generalize Number of sub-problems: n . Boundary condition: S(0) = 0. Dependency: i i-1 j S Example 5 2 2 6 6 6 3 10 7 6 7 9 8 3 3 2 4 12 5 Distance (mi) Profit (100k) 6 7 9 9 10 S(i) S(i-1) S(j) + p i j , t i ) 10 S(i) = max 100 0 dummy 0 S(0) = 0 Starting over … Example 5 2 2 6 6 6 3 10 7 6 7 9 8 3 3 2 4 12 5 Distance (mi) Profit (100k) 6 7 9 9 10 12 12 14 26 26 S(i) S(i-1) S(j) + p i j , t i ) 10 S(i) = max 100 0 7 3 4 12 dummy Optimal: 26 Complexity • Time: O(nk), where k is the maximum number of towns that are within 10 miles to the left of any town – In the worst case, O(n 2 ) • Memory: Θ (n) Knapsack problem Three versions : 0-1 knapsack problem : take each item or leave it Fractional knapsack problem : items are divisible Unbounded knapsack problem : unlimited supplies of each item. Which one is easiest to solve? • Each item has a value and a weight • Objective : maximize value • Constraint : knapsack has a weight limitation Formal definition (0-1 problem) • Knapsack has weight limit W • Items labeled 1, 2, …, n (arbitrarily) • Items have weights w 1 , w 2 , …, w n – Assume all weights are integers – For practical reason, only consider
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cs340_09s_chapter10Part2 - CS 340 Similar to DP Outline...

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