麻省理工学院：《算法导论》（英文版） Lecture 21 Prof charles e. leiserson

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

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

Welcome to Introduction to Algorithms, Fall 2001 Handouts 1. Course Information 2. Calendar 3. Registration (MIT students only) 4. References 5. Objectives and Outcomes 6. Diagnostic Survey

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

Recall from lecture 22 °Flow value:f=f(s,V) Cut: Any partition (S, T)of y such that s E S andt∈T Lemma. f=f(s, T) for any cut(S, T) Corollary. f(s, T) for any cut(S, T) Residual graph: The graph G=(v, ef) with strictly positive residual capacities c u, v) c(u,)-f(2y)>0

flow networks Definition. A flow network is a directed graph G=(, E)with two distinguished vertices:a source s and a sink t. Each edge(u, v)E E has a nonnegative capacity c(u, v). If(u, v) E, then c(u, v)=0 Example: c 2001 by Charles E Leiserson

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