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

Shortest paths Single-source shortest paths nonnegative edge weights Dijkstra's algorithm: O(E+ vlg n General

Paths in graphs Consider a digraph g=(v, E)with edge-weight function w:E→>R. The weight of path p=v1→ →>…→> vi is defined to be (D)=∑(n,1)

Dynamic programming Design technique, like divide-and-conquer Example: Longest Common Subsequence (LCs) Given two sequences x[l.. m] and yll.n], find a longest subsequence common to them both a' not the

Fixed -universe successor problem Goal: maintain a dynamic subset s of size n of the universe 0=10, 1,.,u-1 of size u subject to these operations INSERT(X∈U\\S): Add x to s DELETE(X E S): Remove x from S

ynamic order statistics OS-SELECT(i, S): returns the i th smallest element in the dynamic set S. OS-RANK(, S): returns the rank ofx E S in the sorted order of s s elements IDEA: Use a red-black tree for the set S, but keep subtree sizes in the nodes

Binary-search-tree sort 7∈ b Create an empty Bst for i=i to n do trEe-INSert(,AiD) Perform an inorder tree walk of t Example:

Symbol-table problem Symbol table T holding n records

How fast can we sort? All the sorting algorithms we have seen so far are comparison sorts: only use comparisons to determine the relative order of elements E. g, insertion sort, merge sort, quicksort heapsort The best worst-case running time that weve seen for comparison sorting is O(nIgn) Is o(nlgn) the best we can do?

The divide-and-conquer design paradigm 1. Divide the problem(instance) into subproblems 2. Conquer the subproblems by solving them recursively 3. Combine subproblem solutions