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Problem Solving Dr. Andrew Wallace PhD BEng(hons) EurIng andrew.wallace@cs.umu.se

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Overview Breaking and entry How to get away? Which items to steal? Stash the loot

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Brute Force Simple Start at the beginning Keep working till.. You find a solution! Bang you head against a brick wall long enough, sooner or later the brick wall gives out

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Brute Force

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Searching a list Travelling salesman problem Deoxyribonucleic acid Heuristics!

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Brute Force It can work! Some times it’s the only solution! Problem: Inefficient Can become time consuming

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Optimal subproblems Optimal solution has optimal subproblem solutions S = {s 1, s 2, s 3, … s n } Optimal sub-structure Greedy algorithms Else Overlapping substructures Dynamic programming

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Greedy Algorithms Local optimum results in global optimum Problem: Shortest path from A to B Museum Safe House 10 12 9 6 10 12 3 8 2 15

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Greedy Algorithms Not all local optimal solutions can lead to global optimal solutions Can end up with the worse case If globally optimal can be fast

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Greedy Algorithms Set of candidate solutions Paths Selection function Which path to choose? End function Are we there yet?

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Greedy Algorithms A resource Set of agents who want to use the resource No overlaps in time allowed Agents (a)abcdefg Start (s) 1324675 Finish (f) 4566810 a, d, e

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Greedy Algorithms Earliest finishing time to maximise the number of participants Maximise the remaining time

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Greedy Algorithms O(n)

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Greedy Algorithms No back tracking Choices dependent on passed choices but not future choices

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Dynamic programming Overlapping sub problems Once calculated, save and reuse Dynamic Updates and changes things but not as in dynamic programming languages Programming Filling tables not computer programming

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Dynamic programming Items Weight Value But we have … Max weight!

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Dynamic programming Fibonacci sequence F n = F n-1 + F n-2 F 0 = 0, F 1 = 1 and F 2 = 1 F(6) F(4) F(5) F(3) F(4) F(3) F(1)F(2) F(0)F(1) F(2) F(0) F(1) F(2) F(0)F(1) F(2) F(0)F(1) F(2) F(1)F(0) F(1)

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Dynamic programming Items = {i 1, i 2, i 3 … i n } Weight = {w 1, w 2, w 3 … w n } Value = {v 1, v 2, v 3 … v n } W = max weight we can take

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Dynamic programming Optimal structure? w = {w 1, w 2 … w j } w = {w 1, w 2 … w j-1 } Overlapping subproblems Compare solution with item to solution without item

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Dynamic programming If i = 0 or w= 0 If w i > w If i > 0 or w >= w i

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Dynamic programming O(n)

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Divide and Conquer Recursion Break the problem reclusively into small problems Preferably evenly Solve the smaller problems Combine together to produce the overall solution

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Divide and Conquer T(n) = 2T(n/2) + (n) Master method

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Divide and Conquer

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Find 90 3345538290120150

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Divide and Conquer Find 90 Divide into two 3345538290120150

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Divide and Conquer Find 90 Divide into two And again! 90120150

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Divide and Conquer Find 90 Divide into two And again! 90

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Divide and Conquer

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Problem solving Top down greedy Bottom up dynamic

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Questions?

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0-1 Knapsack Problem A burglar breaks into a museum and finds “n” items Let v_i denote the value of ith item, and let w_i denote the weight of the ith.

0-1 Knapsack Problem A burglar breaks into a museum and finds “n” items Let v_i denote the value of ith item, and let w_i denote the weight of the ith.

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