ADVANCED ALGORITHMSLecture 27: Course review

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ANNOUNCEMENTS

➤ HW 6 is due tomorrow

➤ Final exam — see practice finals and finals of 2016-17 ➤ Tuesday, 10:30 AM (same room)

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C I on

short questions1Sane everything offline 3 problems

EXAM

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➤ Much easier than HW — think simple

TOPICS REVIEW

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➤ Data structures ➤ what data is stored? ➤ what operations? time/space per operation ➤ common: tries for strings, binary search trees

➤ Divide and conquer ➤ analyze correctness via induction ➤ analyze running time via recursion — solving recurrences usually “easy” ➤ merge sort, multiplying integers (n^1.58.. time for n bit #s) ➤ selecting the k’th smallest number in unsorted list

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TOPICS REVIEW

�5

➤ Dynamic programming ➤ memorize answers to all sub-problems ➤ so only works if “number of sub-problems” is small ➤ subset-sum, “coin change”, max increasing subsequence

➤ Greedy algorithms ➤ loosely defined as algorithm that iteratively takes the “best decision at

the current state” ➤ sometimes finds optimal solution: minimum spanning tree ➤ usually does not; approximation algorithm — for covering

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TOPICS REVIEW

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➤ Local search ➤ start with a solution; try to go to a “better” one by “slight changes” ➤ how to change solution? ➤ works in some problems (convex optimization), 2-approximation for

matching

➤ Graph basics ➤ BFS/DFS — O(m+n) time — simplest procedures to explore graphs ➤ Dijkstra’s algorithm for shortest paths — works if all weights are +ve

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TOPICS REVIEW

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➤ Flows and cuts ➤ send as much “flow” as possible without violating capacities ➤ powerful framework: Ford/Fulkerson algorithm ➤ closely related to “cuts” ➤ definitions of flows and cuts; max flow = min cut

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Algorithms we knowfor min cut govia flows

TOPICS REVIEW

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➤ Probabilistic algorithms ➤ may have a “probability of success” ➤ expected running time ➤ analyze running time via defining appropriate “random variables” ➤ dealing with random variables

➤ express in terms of “simple” r.v.s ➤ linearity of expectation ➤ Markov, Chebychev inequalities ➤ sampling and how to analyze it

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expectation w h p

TOPICS REVIEW

�9

➤ Optimization formulations ➤ formulate different problems as optimization ➤ how to pick variables, their domains, … ➤ key: must make sure that formulation “captures” problem — any

solution to formulation gives valid solution to problem ➤ binary optimization is NP-hard — so we “relax” to linear programs ➤ linear programs can be solved in time poly(size of LP) ➤ LPs have nice geometry — feasible region is a “polytope” ➤ optimum solutions are always corner points

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TOPICS REVIEW

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➤ Optimization formulations ➤ relaxed LPs can be useful too ➤ sometimes all corner points are (luckily) integral! ➤ can “round” fractional solutions to integer ones

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