Data Structures And Algorithms

♨️ Data structures, algorithms and problem solving patterns using JavaScript.

Author

Aditya Hajare (Linkedin).

Current Status

WIP (Work In Progress)!

License

Open-sourced software licensed under the MIT license.

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Big O Shorthands

• Arithmatic operations are constant.
• Variable assignment is constant.
• Accessing elements in array (by index) or object (by key) is constant.
• In a loop, the complexity is the length of the loop times the complexity of whatever happens inside the loop.

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Auxiliary Space Complexity

• Refers to space required by the algorithm, not including space required by the inputs.
• Most primitive (boolean, numbers, undefined, null) are constant space.
• Strings require O(n) space where n is the string length.
• Reference types are generally O(n), when n is the length (for arrays) or the number of keys (for objects).

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Logarithmic Time/Space Complexity

• The logarithm of a number roughly measures the number of times we can divide that number by 2 before we get a value that is less than or equals to 1.
• Recursion sometimes involve logarithmic Space Complexity.

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Famous Problem Solving Patterns:

• Frequency Counter.
• Multiple Pointers.
• Sliding Window.
• Divide And Conquer.
• Dynamic Programming.
• Greedy Algorithms.
• Backtracking.
• Recursion.
• Brute Force.
• Branch & Bound.
• Linear Programming.
• Integer Programming.
• Neural Networks.
• Genetic Algorithms.
• Simulated Annealing.
• Memoization

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Frequency Counter

• This pattern uses objects or sets to collect values/frequencies of values.
• This can often avoid the need for nested loop or O(n^2) operations with arrays/strings.

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Multiple Pointers

• Creating pointers or values that corresponds to an index or position and move towards the beginning, end or middle based on a certain condition.
• Very efficient for solving problems with minimal space complexity as well.

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Sliding Window

• This pattern involves creating a window which can either be an array or number from one position to another.
• Depending on certain condition, the window either increases or closes (and a new window is created).
• Very useful for keeping track of a subset (data that is continuous in some way) of a data in an array/string etc.

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Divide And Conquer

• This pattern involves dividing a data set into smaller chunks and then repeating a process with subset of data.
• This pattern can tremendously decrease time complexity.

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Pure Recursion Tips

• For arrays, use methods like slice, spread operator and concat that makes copies of arrays so we don't end up mutating them.
• Strings are immutable!. So we will need to use methods like slice, substr or substring to make copies of string.
• To make copies of objects, use Object.assign, or the spread operator.

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Bubble Sort

• In this algorithm, the largest values bubble up to the top.

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Selection Sort

• Similar to bubble sort, but instead of first placing large values at sorted position, it places small values into sorted position.

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Insertion Sort

• Builds up the sort by gradually creating a larger left half which is always sorted.

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Merge Sort

• It's a combination of 2 things - Merging and Sorting.
• Exploits the fact that arrays of 0 or 1 element are always sorted.
• Works by decomposing an array into smaller arrays of 0 or 1 elements, then building up a newly sorted array.

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Quick Sort

• Like merge sort, exploits the fact that arrays of 0 or 1 elements are always sorted.
• Works by selecting 1 element (called pivot) and finding the index where pivot should end up in the sorted array.

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Radix Sort

• It works on list of numbers.
• It never makes comparison between elements.
• It exploits the fact that information about the size of a number is encoded in the number of digits.
• More digits means a bigger number!
• We can use it for any kind of data (e.g. strings) as long as we can convert and represent it in digits.
• In order to implement Radix Sort, it's helpful to build a few helper functions first:
  • getDigit(num, place): Returns the digit in num at the given place value.

  function getDigit(num, i) {
      return Math.floor(Math.abs(num) / Math.pow(10, i)) % 10;
  }

  • digitCount(num): Returns the number of digits in num.

  function digitCount(num) {
      if (num === 0) {
          return 1;
      }
      return Math.floor(Math.log10(Math.abs(num))) + 1;
  }

  • mostDigits(arr): Given an array of numbers, returns the number of digits in the largest numbers in list (using digitCount(num) above).

  function mostDigits(arr) {
      let maxDigits = 0;
      for (let i = 0; i < arr.length; i++) {
          maxDigits = Math.max(maxDigits, digitCount(arr[i]))
      }
      return maxDigits;
  }

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DFS PreOrder Psudo

• Visit the node.
• Traverse left.
• Traverse right.
• NOTE: Root is the first thing visited in PreOrder traversal.

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DFS PostOrder Psudo

• Traverse left.
• Traverse right.
• Visit the node.
• NOTE: Root is the last thing visited in PostOrder traversal.

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DFS InOrder Psudo

• Traverse left.
• Visit the node.
• Traverse right.
• NOTE: In order traversal gives us sorted data.

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Traversal Algorithms Usage

• BFS:
  • Good for traversing tree that is big in height (deep).
• DFS InOrder:
  • Mostly used on BST.
  • Gives data in sorted order.
• DFS PreOrder:
  • Can be used to export a tree structure so that it is easily reconstructed or copied.
  • We know root is at the beginning.
• When to use BFS vs. DFS:
  • That heavily depends on the structure of the search tree and the number and location of solutions (aka searched-for items).
    • If we know a solution is not far from the root of the tree, a breadth first search (BFS) might be better.
    • If the tree is very deep and solutions are rare, depth first search (DFS) might take an extremely long time, but BFS could be faster.
    • If the tree is very wide, a BFS might need too much memory, so it might be completely impractical.
    • If solutions are frequent but located deep in the tree, BFS could be impractical.
    • If the search tree is very deep we will need to restrict the search depth for depth first search (DFS), anyway (for example with iterative deepening).
    • Breadth-first search is less space-efficient than depth-first search because BFS keeps a priority queue of the entire frontier while DFS maintains a few pointers at each level.
    • If it is known that an answer will likely be found far into a tree, DFS is a better option than BFS. BFS is good to use when the depth of the tree can vary or if a single answer is needed—for example, the shortest path in a tree. If the entire tree should be traversed, DFS is a better option.
    • BFS always returns an optimal answer, but this is not guaranteed for DFS.
    • Depth-first search is used in topological sorting, scheduling problems, cycle detection in graphs, and solving puzzles with only one solution, such as a maze or a sudoku puzzle.
    • Other applications involve analyzing networks, for example, testing if a graph is bipartite. Depth-first search is often used as a subroutine in network flow algorithms such as the Ford-Fulkerson algorithm.
    • DFS is also used as a subroutine in matching algorithms in graph theory such as the Hopcroft–Karp algorithm.
    • Depth-first searches are used in mapping routes, scheduling, and finding spanning trees.
  • But these are just rules of thumb; we'll probably need to experiment.
    • Breadth First Search (BFS) algorithm, from its name "Breadth", discovers all the neighbours of a node through the out edges of the node then it discovers the unvisited neighbours of the previously mentioned neighbours through their out edges and so forth, till all the nodes reachable from the origional source are visited (we can continue and take another origional source if there are remaining unvisited nodes and so forth). That's why it can be used to find the shortest path (if there is any) from a node (origional source) to another node if the weights of the edges are uniform.
    • Depth First Search (DFS) algorithm, from its name "Depth", discovers the unvisited neighbours of the most recently discovered node x through its out edges. If there is no unvisited neighbour from the node x, the algorithm backtracks to discover the unvisited neighbours of the node (through its out edges) from which node x was discovered, and so forth, till all the nodes reachable from the origional source are visited (we can continue and take another origional source if there are remaining unvisited nodes and so forth).
    • Both BFS and DFS can be incomplete. For example if the branching factor of a node is infinite, or very big for the resources (memory) to support (e.g. when storing the nodes to be discovered next), then BFS is not complete even though the searched key can be at a distance of few edges from the origional source. This infinite branching factor can be because of infinite choices (neighbouring nodes) from a given node to discover. If the depth is infinite, or very big for the resources (memory) to support (e.g. when storing the nodes to be discovered next), then DFS is not complete even though the searched key can be the third neighbor of the origional source. This infinite depth can be because of a situation where there is, for every node the algorithm discovers, at least a new choice (neighbouring node) that is unvisited before.
    • Therefore, we can conclude when to use the BFS and DFS. Suppose we are dealing with a manageable limited branching factor and a manageable limited depth. If the searched node is shallow i.e. reachable after some edges from the origional source, then it is better to use BFS. On the other hand, if the searched node is deep i.e. reachable after a lot of edges from the origional source, then it is better to use DFS.
    • For example, in a social network if we want to search for people who have similar interests of a specific person, we can apply BFS from this person as an origional source, because mostly these people will be his direct friends or friends of friends i.e. one or two edges far. On the other hand, if we want to search for people who have completely different interests of a specific person, we can apply DFS from this person as an origional source, because mostly these people will be very far from him i.e. friend of friend of friend.... i.e. too many edges far.
    • Applications of BFS and DFS can vary also because of the mechanism of searching in each one. For example, we can use either BFS (assuming the branching factor is manageable) or DFS (assuming the depth is manageable) when we just want to check the reachability from one node to another having no information where that node can be. Also both of them can solve same tasks like topological sorting of a graph (if it has). BFS can be used to find the shortest path, with unit weight edges, from a node (origional source) to another. Whereas, DFS can be used to exhaust all the choices because of its nature of going in depth, like discovering the longest path between two nodes in an acyclic graph. Also DFS, can be used for cycle detection in a graph.
    • In the end if we have infinite depth and infinite branching factor, we can use Iterative Deepening Search (IDS).

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When to use Pre-Order, In-order or Post-Order?

• The traversal strategy the programmer selects depends on the specific needs of the algorithm being designed. The goal is speed, so pick the strategy that brings you the nodes you require the fastest.
• In Order:
  • If you know that the tree has an inherent sequence in the nodes, and you want to flatten the tree back into its original sequence, than an in-order traversal should be used. The tree would be flattened in the same way it was created. A pre-order or post-order traversal might not unwind the tree back into the sequence which was used to create it.
  • Used to get the values of the nodes in non-decreasing order in a BST.
• Post Order:
  • If you know you need to explore all the leaves before any nodes, you select post-order because you don't waste any time inspecting roots in search for leaves.
  • Used to delete a tree from leaf to root
• Pre Order:
  • If you know you need to explore the roots before inspecting any leaves, you pick pre-order because you will encounter all the roots before all of the leaves.
  • Used to create a copy of a tree. For example, if you want to create a replica of a tree, put the nodes in an array with a pre-order traversal. Then perform an Insert operation on a new tree for each value in the array. You will end up with a copy of your original tree.
  • If I wanted to simply print out the hierarchical format of the tree in a linear format, I'd probably use preorder traversal. Or in a TreeView component in a GUI application.

  - ROOT
      - A
          - B
          - C
      - D
          - E
          - F
              - G
  

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Dynamic Programming

• A method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions.
• When to use Dynamic Programming: Where there is optimal substructure present and overlapping subproblems.
• Overlapping Subproblem: A problem is said to have overlapping subproblems if it can be broken down into subproblems which are reused several times.
  • For e.g. Fibonacci Sequence: Every number after the first two is the sum of the two preceding ones.
  • Merge Sort DOES NOT have overlapping subproblems! For e.g.:
    • mergeSort([10, 24, 73 76]): DOES NOT have overlapping subproblems!
    • mergeSort([10, 24, 10, 24]): DOES have overlapping subproblems!
• Optimal Substructure: A problem is said to have Optimal Substructure if an optimal solution can be constructed from optimal solutions of its subproblems.

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Dynamic Programming - Memoization

• Storing the results of expensive function calls and returning the cached result when the same input occur again.

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