chapter 8 readme

08. Recursion/README.md

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+# Chapter 8 Recursion and dynamic programming
+
+Questions p145, solutions p353
+
+You can know a problem is recursive when it can be built off of subproblems. If a program says: compute the nth, or the first n, or all..., it might be recursive.
+
+## How to approach
+
+* Bottom-up approach: solve the problem for a simple case, then for 2 elements, then 3, etc.
+* Top-down approach: think about how to divide the problem for case N into subproblems
+* Half-and-half approaches: divide the dataset in half, for example binary search, merge sort.
+
+## Recursive vs. iterative solutions
+
+Recursive algorithms are very space inefficient, each recursive call adds a new layer to the stack. If your algorithm recurses to a depth of n, it uses at least O(n) memory.
+
+It's better to implement a recursive algorithm iteratively. All recursive algorithms can be implemented this way, but sometimes its complex. Think how easy it would be, and discuss the tradeoffs with the interviewer.
+
+## Dynamic programming and memoization
+
+Dynamic programming is mostly just a matter of taking a recursive algorithm and finding the overlapping subproblems (that is, the repeated calls). You then cache those results for future recursive calls.
+
+> Fibonacci numbers
+
+```cpp
+int fibonacci(int i) {
+    if (i == 0) return 0;
+    if (i == 1) return 1;
+    return fibonacci(i-1) + fibonacci(i-2);
+}
+```
+
+Since the calls are branched out like in a tree, each node has 2 children, each children has 2 children, the runtime is roughlt O(2<sup>n</sup>).
+
+### Top-down, or memoization
+
+Each time we compute fib(i), we should just cache this result and use it later.
+
+```cpp
+int fibonacci(int n) {
+    return fibonacci(n, new int[n+1]);
+}
+
+int fibonacci(int i, int[] memo) {
+    if (i == 0 || i == 1) return i;
+    if (memo[i] == 0) {
+        memo[i] = fibonacci(i-1, memo) + fibonacci(i-2, memo);
+    }
+    return memo[i];
+}
+```
+
+The runtime now is O(n).
+
+### Bottom-up
+
+```cpp
+int fibonacci(int n) {
+    if (i == 0 || i == 1) return i;
+
+    int[] memo = new int[n+1];
+    memo[0] = 0;
+    memo[1] = 1;
+    for(int i = 2; i <= n; i++) {
+        memo[i] = memo[i-1] + memo[i-2];
+    }
+    return memo[n]
+}
+```
+
+But memo[i] are only used for memo[i+1], memo[i+2]. After that they're not used. We can get rid of the memo table and just store a few variables.
+
+```cpp
+int fibonacci(int n) {
+    if (i == 0) return 0;
+
+    int a = 0;
+    int b = 1;
+    for(int i = 2; i <= n; i++) {
+        int c = a + b;
+        a = b;
+        b = c;
+    }
+    return c;
+}
+```