mirror of
https://github.com/Brandon-Rozek/website.git
synced 2024-12-24 01:03:06 +00:00
84 lines
2.5 KiB
Markdown
84 lines
2.5 KiB
Markdown
---
|
||
title: Dynamic Programming
|
||
---
|
||
|
||
The book first goes into talking about the complexity of the Fibonacci algorithm
|
||
|
||
```
|
||
RecFibo(n):
|
||
if n = 0
|
||
return 0
|
||
else if n = 1
|
||
return 1
|
||
else
|
||
return RecFibo(n - 1) + RecFibo(n - 2)
|
||
```
|
||
|
||
It talks about how the complexity of this is exponential.
|
||
|
||
"A single call to `RecFibo(n)` results in one recursive call to `RecFibo(n−1)`, two recursive calls to `RecFibo(n−2)`, three recursive calls to `RecFibo(n−3)`, five recursive calls to `RecFibo(n−4)`"
|
||
|
||
Now consider the memoized version of this algorithm...
|
||
|
||
```
|
||
MemFibo(n):
|
||
if n = 0
|
||
return 0
|
||
else if n = 1
|
||
return 1
|
||
else
|
||
if F[n] is undefined:
|
||
F[n] <- MemFibo(n - 1) + MemFibo(n - 2)
|
||
return F[n]
|
||
```
|
||
|
||
This actually makes the algorithm run in linear time!
|
||
|
||
|
||
|
||
Dynamic programming makes use of this fact and just intentionally fills up an array with the values of $F$.
|
||
|
||
```
|
||
IterFibo(n):
|
||
F[0] <- 0
|
||
F[1] <- 1
|
||
for i <- 2 to n
|
||
F[i] <- F[i - 1] + F[i - 2]
|
||
return F[n]
|
||
```
|
||
|
||
Here the linear complexity becomes super apparent!
|
||
|
||
<u>Interesting snippet</u>
|
||
|
||
"We had a very interesting gentleman in Washington named Wilson. He was secretary of Defense, and he actually had a pathological fear and hatred of the word *research*. I’m not using the term lightly; I’m using it precisely. His face would suffuse, he would turn red, and he would get violent if people used the term *research* in his presence. You can imagine how he felt, then, about the term *mathematical*.... I felt I had to do something to shield Wilson and the Air Force from the fact that I was really doing mathematics inside the RAND Corporation. What title, what name, could I choose?"
|
||
|
||
|
||
|
||
Dynamic programming is essentially smarter recursion. It's about not repeating the same work.
|
||
|
||
These algorithms are best developed in two distinct stages.
|
||
|
||
(1) Formulate the problem recursively.
|
||
|
||
(a) Specification: What problem are you trying to solve?
|
||
|
||
(b) Solution: Why is the whole problem in terms of answers t smaller instances exactly the same problem?
|
||
|
||
(2) Find solutions to your recurrence from the bottom up
|
||
|
||
(a) Identify the subproblems
|
||
|
||
(b) Choose a memoization data structure
|
||
|
||
(c) Identify dependencies
|
||
|
||
(d) Find a good evaluation order
|
||
|
||
(e) Analyze space and running time
|
||
|
||
(f) Write down the algorithm
|
||
|
||
## Greedy Algorithms
|
||
|
||
If we're lucky we can just make decisions directly instead of solving any recursive subproblems. The problem is that greedly algorithms almost never work.
|