[Algorithm] Median Maintenance algorithm implementation using TypeScript / JavaScript

Posted 2021-02-01 Answer1215

The median maintenance problem is a common programming challenge presented in software engineering job interviews.

In this lesson we cover an example of how this problem might be presented and what your chain of thought should be to tackle this problem efficiently.

Lets first refresh what is a median

• The median is the middle element in the sorted list
• Given a list of numbers

`
The median is the middle element in the sorted list.

Given
13, 23, 11, 16, 15, 10, 26

Sort them
10, 11, 13, 15, 16, 23, 26
            ↑
         Median

If we have an even number of elements we average

E.g.
10, 11, 13, 15, 16, 23, 26, 32
                /
             15.5

They way we solve the problem is by using two heaps (Low & High) to divide the array into tow parts.

 

                        Low                 |                      High

                     Max Heap          |                    Min Heap

Low part is a max heap, high part is a min heap.

`
(n/2 ± 1) smallest items in a low MaxHeap       (n/2 ± 1) biggest items in a high MinHeap

        peek => n/2th smallest                     peek => n/2th smallest
                                                   /
                                    MEDIAN!
`

If low part size is equals to high part size, then we get avg value, otherwise, we get from larger size heap.

function MedianMaintaince() {
  let lowMaxHeap = new Heap((b, a) => a - b);
  let highMinHeap = new Heap((a, b) => a - b);

  return {
    add(value) {
      // For the first element, we add to lowMaxHeap by default
      if (lowMaxHeap.size() === 0 || value < lowMaxHeap.peek()) {
        lowMaxHeap.add(value);
      } else {
        highMinHeap.add(value);
      }

      /**
       * Reblance:
       *
       * If low.size = 2; high.size = 4, then we move the root of high to the low part
       * so that low.size = 3, high.size = 3
       */
      let smallerHeap =
        lowMaxHeap.size() > highMinHeap.size() ? highMinHeap : lowMaxHeap;
      let biggerHeap = smallerHeap === lowMaxHeap ? highMinHeap : lowMaxHeap;
      if (biggerHeap.size() - smallerHeap.size() > 1) {
        smallerHeap.add(biggerHeap.extractRoot());
      }

      /**
       * If low.szie === high.size, extract root for both and calculate the average value
       */
      if (lowMaxHeap.size() === highMinHeap.size()) {
        return (lowMaxHeap.peek() + highMinHeap.peek()) / 2;
      } else {
        // get peak value from the bigger size of heap
        return lowMaxHeap.size() > highMinHeap.size()
          ? lowMaxHeap.peek()
          : highMinHeap.peek();
      }
    }
  };
}

const mm = new MedianMaintaince();
console.log(mm.add(4)); // 4
console.log(mm.add(2)); // 3
console.log(mm.add(5)); // 4
console.log(mm.add(3)); // 3.5

 

We have heap data structure:

function printArray(ary) {
  console.log(JSON.stringify(ary, null, 2));
}

function Heap(cmpFn = () => {}) {
  let data = [];
  return {
    data,
    // 2n+1
    leftInx(index) {
      return 2 * index + 1;
    },
    //2n + 2
    rightInx(index) {
      return 2 * index + 2;
    },
    // left: (n - 1) / 2, left index is always odd number
    // right: (n - 2) / 2, right index is always even number
    parentInx(index) {
      return index % 2 === 0 ? (index - 2) / 2 : (index - 1) / 2;
    },
    add(val) {
      this.data.push(val);
      this.siftUp(this.data.length - 1);
    },
    extractRoot() {
      if (this.data.length > 0) {
        const root = this.data[0];
        const last = this.data.pop();
        if (this.data.length > 0) {
          // move last element to the root
          this.data[0] = last;
          // move last elemment from top to bottom
          this.siftDown(0);
        }

        return root;
      }
    },
    siftUp(index) {
      // find parent index
      let parentInx = this.parentInx(index);
      // compare
      while (index > 0 && cmpFn(this.data[index], this.data[parentInx]) < 0) {
        //swap parent and current node value
        [this.data[index], this.data[parentInx]] = [
          this.data[parentInx],
          this.data[index]
        ];
        //swap index
        index = parentInx;
        //move to next parent
        parentInx = this.parentInx(index);
      }
    },
    siftDown(index) {
      const minIndex = (leftInx, rightInx) => {
        if (cmpFn(this.data[leftInx], this.data[rightInx]) <= 0) {
          return leftInx;
        } else {
          return rightInx;
        }
      };
      let min = minIndex(this.leftInx(index), this.rightInx(index));
      while (min >= 0 && cmpFn(this.data[index], this.data[min]) > 0) {
        [this.data[index], this.data[min]] = [this.data[min], this.data[index]];
        index = min;
        min = minIndex(this.leftInx(index), this.rightInx(index));
      }
    },
    peek() {
      return this.data[0];
    },
    print() {
      printArray(this.data);
    },
    size() {
      return this.data.length;
    }
  };
}