Disioint-set data structure (Union-Find) Problem: maintain a dynamic collection of pairwise-disjoint sets S=(S Each set S; has one element distinguished as the representative element, rep[sil lust support 3 operations

Negative-weight cycles Recall: If a graph G=(V, E) contains a negative weight cycle then some shortest paths may not exist

Graphs(review) Definition. a directed graph(digraph G=(, E)is an ordered pair consisting of a set y of vertices(singular: vertex) a sete c× of edges. In an undirected graphG=(V, E), the edge set e consists of unordered pairs of vertices In either case, we have El=O(v2).Moreover if G is connected, then E2v-l, which

How large should a hash table be? Goal: Make the table as small as possible but large enough so that it wont overflow(or otherwise become inefficient Problem: what if we don 't know the proper size

Computational geometry Algorithms for solving geometric problems in 2D and higher Fundamental objects: o point line segment

Balanced search trees Balanced search tree a search-tree data structure for which a height of o(g n)is guaranteed when implementing a dynamic set of n items

a weakness of hashing Problem: For any hash function h, a set of keys exists that can cause the average access time of a hash table to skyrocket An adversary can pick all keys from tkeU: h(k)=i for some slot i IDEA Choose the hash function at random independently of the keys

Order statistics Select the ith smallest of n elements(the element with rank i i=l: minimum, .i=n: marimum, i=L(n+1)/2]or[(n+1)/2 median Naive algorithm: Sort and index ith element Worst-case running time =o(n Ig n)+o(1 o(nIg n using merge sort or heapsort(not quicksort) c 2001 by Charles E Leiserson

Quicksort Proposed by C. A.R. Hoare in 1962 Divide-and-conquer algorithm Sorts“ in place”( like insertion sort, but not like merge sort Very practical(with tuning)

Solving recurrences The analysis of merge sort from Lecture I required us to solve a recurrence Recurrences are like solving integrals erential equations, etc o Learn a few tricks Lecture 3: Applications of recurrences