Add Jump Game.

Author: trekhleb

README.md

@@ -116,6 +116,7 @@ a set of rules that precisely define a sequence of operations.
 * **Uncategorized**  
   * `B` [Tower of Hanoi](src/algorithms/uncategorized/hanoi-tower)
   * `B` [Square Matrix Rotation](src/algorithms/uncategorized/square-matrix-rotation) - in-place algorithm
+  * `B` [Jump Game](src/algorithms/uncategorized/jump-game) - backtracking, dynamic programming (top-down + bottom-up) and greedy examples 
   * `A` [N-Queens Problem](src/algorithms/uncategorized/n-queens)
   * `A` [Knight's Tour](src/algorithms/uncategorized/knight-tour)
 
@@ -129,6 +130,7 @@ algorithm is an abstraction higher than a computer program.
   * `A` [Maximum Subarray](src/algorithms/sets/maximum-subarray)
   * `A` [Travelling Salesman Problem](src/algorithms/graph/travelling-salesman) - shortest possible route that visits each city and returns to the origin city
 * **Greedy** - choose the best option at the current time, without any consideration for the future
+  * `B` [Jump Game](src/algorithms/uncategorized/jump-game)
   * `A` [Unbound Knapsack Problem](src/algorithms/sets/knapsack-problem)
   * `A` [Dijkstra Algorithm](src/algorithms/graph/dijkstra) - finding shortest path to all graph vertices
   * `A` [Prim’s Algorithm](src/algorithms/graph/prim) - finding Minimum Spanning Tree (MST) for weighted undirected graph
@@ -142,10 +144,12 @@ algorithm is an abstraction higher than a computer program.
   * `B` [Quicksort](src/algorithms/sorting/quick-sort)
   * `B` [Tree Depth-First Search](src/algorithms/tree/depth-first-search) (DFS)
   * `B` [Graph Depth-First Search](src/algorithms/graph/depth-first-search) (DFS)
+  * `B` [Jump Game](src/algorithms/uncategorized/jump-game)
   * `A` [Permutations](src/algorithms/sets/permutations) (with and without repetitions)
   * `A` [Combinations](src/algorithms/sets/combinations) (with and without repetitions)
 * **Dynamic Programming** - build up a solution using previously found sub-solutions
   * `B` [Fibonacci Number](src/algorithms/math/fibonacci)
+  * `B` [Jump Game](src/algorithms/uncategorized/jump-game)
   * `A` [Levenshtein Distance](src/algorithms/string/levenshtein-distance) - minimum edit distance between two sequences
   * `A` [Longest Common Subsequence](src/algorithms/sets/longest-common-subsequence) (LCS)
   * `A` [Longest Common Substring](src/algorithms/string/longest-common-substring)
@@ -159,6 +163,7 @@ algorithm is an abstraction higher than a computer program.
 * **Backtracking** - similarly to brute force, try to generate all possible solutions, but each time you generate next solution you test
 if it satisfies all conditions, and only then continue generating subsequent solutions. Otherwise, backtrack, and go on a
 different path of finding a solution. Normally the DFS traversal of state-space is being used.
+  * `B` [Jump Game](src/algorithms/uncategorized/jump-game)
   * `A` [Hamiltonian Cycle](src/algorithms/graph/hamiltonian-cycle) - Visit every vertex exactly once
   * `A` [N-Queens Problem](src/algorithms/uncategorized/n-queens)
   * `A` [Knight's Tour](src/algorithms/uncategorized/knight-tour)

src/algorithms/uncategorized/jump-game/README.md

@@ -0,0 +1,128 @@
+# Jump Game
+
+## The Problem
+
+Given an array of non-negative integers, you are initially positioned at 
+the first index of the array. Each element in the array represents your maximum 
+jump length at that position.
+
+Determine if you are able to reach the last index.
+
+**Example #1**
+
+```
+Input: [2,3,1,1,4]
+Output: true
+Explanation: Jump 1 step from index 0 to 1, then 3 steps to the last index.
+```
+
+**Example #2**
+
+```
+Input: [3,2,1,0,4]
+Output: false
+Explanation: You will always arrive at index 3 no matter what. Its maximum
+             jump length is 0, which makes it impossible to reach the last index.
+```
+
+## Naming
+
+We call a position in the array a **"good index"** if starting at that position,
+we can reach the last index. Otherwise, that index is called a **"bad index"**.
+The problem then reduces to whether or not index 0 is a "good index".
+
+## Solutions
+
+### Approach 1: Backtracking
+
+This is the inefficient solution where we try every single jump pattern that 
+takes us from the first position to the last. We start from the first position 
+and jump to every index that is reachable. We repeat the process until last 
+index is reached. When stuck, backtrack.
+
+> See [backtrackingJumpGame.js](backtrackingJumpGame.js) file
+
+**Time complexity:**: `O(2^n)`.
+There are 2<sup>n</sup> (upper bound) ways of jumping from 
+the first position to the last, where `n` is the length of 
+array `nums`.
+
+**Auxiliary Space Complexity**: `O(n)`.
+Recursion requires additional memory for the stack frames.
+
+### Approach 2: Dynamic Programming Top-down
+
+Top-down Dynamic Programming can be thought of as optimized 
+backtracking. It relies on the observation that once we determine 
+that a certain index is good / bad, this result will never change.
+This means that we can store the result and not need to recompute
+it every time.
+
+Therefore, for each position in the array, we remember whether the 
+index is good or bad. Let's call this array memo and let its values
+be either one of: GOOD, BAD, UNKNOWN. This technique is 
+called memoization.
+
+> See [dpTopDownJumpGame.js](dpTopDownJumpGame.js) file
+
+**Time complexity:**: `O(n^2)`.
+For every element in the array, say `i`, we are looking at the 
+next `nums[i]` elements to its right aiming to find a GOOD
+index. `nums[i]` can be at most `n`, where `n` is the length 
+of array `nums`.
+
+**Auxiliary Space Complexity**: `O(2 * n) = O(n)`.
+First `n` originates from recursion. Second `n` comes from the 
+usage of the memo table.
+
+### Approach 3: Dynamic Programming Bottom-up
+
+Top-down to bottom-up conversion is done by eliminating recursion. 
+In practice, this achieves better performance as we no longer have the 
+method stack overhead and might even benefit from some caching. More 
+importantly, this step opens up possibilities for future optimization.
+The recursion is usually eliminated by trying to reverse the order of 
+the steps from the top-down approach.
+
+The observation to make here is that we only ever jump to the right.
+This means that if we start from the right of the array, every time 
+we will query a position to our right, that position has already be
+determined as being GOOD or BAD. This means we don't need to recurse
+anymore, as we will always hit the memo table.
+
+> See [dpBottomUpJumpGame.js](dpBottomUpJumpGame.js) file
+
+**Time complexity:**: `O(n^2)`.
+For every element in the array, say `i`, we are looking at the 
+next `nums[i]` elements to its right aiming to find a GOOD
+index. `nums[i]` can be at most `n`, where `n` is the length 
+of array `nums`.
+
+**Auxiliary Space Complexity**: `O(n)`.
+This comes from the usage of the memo table.
+ 
+### Approach 4: Greedy
+
+Once we have our code in the bottom-up state, we can make one final,
+important observation. From a given position, when we try to see if
+we can jump to a GOOD position, we only ever use one - the first one.
+In other words, the left-most one. If we keep track of this left-most
+GOOD position as a separate variable, we can avoid searching for it in
+the array. Not only that, but we can stop using the array altogether.
+
+> See [greedyJumpGame.js](greedyJumpGame.js) file
+
+**Time complexity:**: `O(n)`.
+We are doing a single pass through the `nums` array, hence `n` steps,
+where `n` is the length of array `nums`.
+
+**Auxiliary Space Complexity**: `O(1)`.
+We are not using any extra memory.
+
+## References
+
+- [Jump Game Fully Explained on LeetCode](https://leetcode.com/articles/jump-game/)
+- [Dynamic Programming vs Divide and Conquer](https://itnext.io/dynamic-programming-vs-divide-and-conquer-2fea680becbe)
+- [Dynamic Programming](https://en.wikipedia.org/wiki/Dynamic_programming)
+- [Memoization on Wikipedia](https://en.wikipedia.org/wiki/Memoization)
+- [Top-Down and Bottom-Up Design on Wikipedia](https://en.wikipedia.org/wiki/Top-down_and_bottom-up_design)