Recurrences
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Transcript of Recurrences
Recurrences / HW: 2.4Quiz: 2.1, 4.1, 4.2, 5.2, 7.3, 7.4Midterm: 8
given a recursive algorithm, state the recurrence
solve a recurrence, using Master theorem
given a recurrence and its solution, prove that the solution is correct
solve a problem using divide&conquer (study: merge sort, closest pair of points in R2)
Divide&Conquer
Basics
Quiz: 1.1, 1.2, 13.1Midterm: 7
group a bunch of function by their asymptotic growth
a = bloga b
(ab ) = abcc
ab+c = (ab)(ac)
Basics
Quiz: 1.1, 1.2, 13.1Midterm: 7
quicksort, heapsort, mergesort, counting sort, bucket sort, radix sort, sorting lower bound
heaps, B-trees
union find
basic geometry (do two lines intersect, is a point to the left of a line, ...)
Dynamic programming
HW: 5.1, 5.2, 5.3, 5.4, 6.3, 6.4, 7.1, 8.2, 10.1Midterm: 10, 11, 12Quiz: 6.1, 6.2, 7.1, 7.2
given a problem and the interpretation of the entries in the dynamic programming table, design the heart of the algorithm
Dynamic programming
Dynamic programming
Dynamic programming
Linear-time median
HW: 1.3, 3.1, 3.2, 5.5
employ the linear-time median algorithm
Greedy algorithms
HW: 4.1, 4.2, 4.3, 4.4, 6.2Midterm: 6Quiz: 5.1, 11.1, 11.2
does the greedy algorithm work for a given problem? find counterexample / prove it does
Linear programming
HW: 12.1, 12.2, 12.3Midterm:Quiz: 12.1, 12.2,
given a linear program, find its dual
given a problem, formulate it as a linear program (e.g., HW 12.1)
Linear programming
NP-completeness
HW:Midterm:Quiz: 10.3, 11.1, 11.2
is a given algorithm polynomial-time?
basic NP-complete problems:(3-SAT, Independent set, 3-Coloring, Clique,Subset-Sum, Hamilltonian path, Vertex Cover,Integer Linear Programming, Max-Cut)
Cook’s Theorem
NP-completeness
Graph algorithms
HW: 6.1Quiz 9.1, 9.2Midterm: 1, 2, 3, 5, 9
DFS, BFS, topological sort, connected components,minimum spanning tree (Kruskal, Prim), shortest paths (Dijkstra, Bellman-Ford, Warshall)maximum/maximal/perfect matching, augmenting paths, flows, residual networks, Ford-Fulkerson, maximum-weight matching, Max-Flow=Min-Cut theorem
Approximation algorithmsvertex cover 2 - approxset cover O(log n) - approxmetric TSP 1.5 - approxknapsack (1+) - approx
Reductions
HW: 8.1, 8.3, 9.1, 9.2, 9.3, 11.1 Quiz: 8.2, 8.3, 10.1
solve one problem using a black-box for another problem
Reductions
Reductions
Reductions
Reductions
Probability theory
HW: 2.1, 2.2 Quiz: 3.1, 3.2, 4.3, 5.3
random variable, expectationcoupon collector problem linearity of expectationLas Vegas / Monte Carlo
Score informationHW: total - two-lowest = score 1
rank a-b
Quiz: total - two-lowest = score 2
rank a-b
max220
330score 1
25 220
score 220 +
max330
75midterm
25 +