Homework: 4.25, 4.28, 6.4, 6.6, 6.12, Read Chapter 6

Bellman-Ford and Dynamic Programming

Take an $N\times M$ grid, how many ways can the robot take to go from $(0,\,0)$ to $(m,\,n)$.

${n+m \choose m}$

Blueprint of Dynamic Programming:

1. Break up your problem into subproblems
2. What am I guessing when combining solutions from subproblems into solution for the main problem
3. Write a recursive solution
4. Convert to non recursive solution (fill in the table?)
5. Establish a topological order of filling the table

Coin Change Problem

Take coins $S_1,\,S_2,\,…,\,S_m$ each with some value. Given these coins and an infinite supply of each, use the smallest number of coins to get change for $N$.

$M(N) = \min_{1<i<M} M[(N-S_i)] + 1$

How many bits does it take to write down number $N$, $\log N$, so this is a pseudo-polynomial algorithm, or NP-incomplete. (Works for small $\log N$ however.)

Job Scheduling

Take the greedy Job Scheduling problem, but now assign weights to each job. How do we assign jobs that maximize weight?

1. Sort jobs by their finishing time
2. Define $\mathrm{OPT}(J)$ as the profit of the optimal selection of jobs $1\, …\,j$
3. $\mathrm{OPT}(N) = \mathrm{MAX}\{\mathrm{OPT}(N-1),\, w(N) + \mathrm{OPT}(P(N))\}$ where $w$ is the weight of some job $N$ and $P$ is the predecessor function of job $N$ (the previous job that will not overlap with job $N$)