16.61 Aerospace Dynamics Spring 2003 Example Given: Catapult rotating at a constant rate(frictionless, in the horizontal plane) Find the eom of the particle as it leaves the

NUMERICAL SoL°To ·G川EN舟c。 MPLEX SET of0 YNAMICS 义(t)=斤(x,x) WHERE f()CouL0BE升MNLE升 R FUNCTI0N 工TCNB.工 MOss8 LE To ACTVALLY SoLVE FoR×(t)∈ ACTLY →0 EVELoP A则 UMERICAL SOLUTION. cA小NE0co0 ES TO HELP v5 D。TwsτN MATLAB BUT LET US CONSIDER THE BAsIcs

RIGI0 BoDY DYNAMICS ·T叭ocAP0 NENTS千0RtG1D80 y MoTIo 千 RANSLAT(0NAL F= M R升 TIONAL D ECOU PLE PRO vIED 工ND0FRTT0A AA0A工Nb0FTRA5L|N

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

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