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82 lines
2.5 KiB
Markdown
82 lines
2.5 KiB
Markdown
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# Dynamic Programming
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The book first goes into talking about the complexity of the Fibonacci algorithm
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```
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RecFibo(n):
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if n = 0
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return 0
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else if n = 1
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return 1
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else
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return RecFibo(n - 1) + RecFibo(n - 2)
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```
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It talks about how the complexity of this is exponential.
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"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)`"
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Now consider the memoized version of this algorithm...
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```
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MemFibo(n):
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if n = 0
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return 0
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else if n = 1
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return 1
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else
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if F[n] is undefined:
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F[n] <- MemFibo(n - 1) + MemFibo(n - 2)
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return F[n]
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```
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This actually makes the algorithm run in linear time!
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Dynamic programming makes use of this fact and just intentionally fills up an array with the values of $F$.
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```
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IterFibo(n):
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F[0] <- 0
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F[1] <- 1
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for i <- 2 to n
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F[i] <- F[i - 1] + F[i - 2]
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return F[n]
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```
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Here the linear complexity becomes super apparent!
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<u>Interesting snippet</u>
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"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?"
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Dynamic programming is essentially smarter recursion. It's about not repeating the same work.
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These algorithms are best developed in two distinct stages.
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(1) Formulate the problem recursively.
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(a) Specification: What problem are you trying to solve?
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(b) Solution: Why is the whole problem in terms of answers t smaller instances exactly the same problem?
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(2) Find solutions to your recurrence from the bottom up
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(a) Identify the subproblems
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(b) Choose a memoization data structure
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(c) Identify dependencies
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(d) Find a good evaluation order
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(e) Analyze space and running time
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(f) Write down the algorithm
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## Greedy Algorithms
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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.
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