295. Find Median from Data Stream

At a Glance

Topic: Two Pointers
Pattern: Analyze Pattern
Difficulty: Hard
LeetCode: 295

Problem Statement

The median is the middle value in an ordered integer list. If the size of the list is even, there is no middle value, and the median is the mean of the two middle values.

For example, for arr = [2,3,4], the median is 3.
For example, for arr = [2,3], the median is (2 + 3) / 2 = 2.5.

Implement the MedianFinder class:

MedianFinder() initializes the MedianFinder object.
void addNum(int num) adds the integer num from the data stream to the data structure.
double findMedian() returns the median of all elements so far. Answers within 10-5 of the actual answer will be accepted.

Example 1:

Input ["MedianFinder", "addNum", "addNum", "findMedian", "addNum", "findMedian"] [[], [1], [2], [], [3], []] Output [null, null, null, 1.5, null, 2.0]

Explanation MedianFinder medianFinder = new MedianFinder(); medianFinder.addNum(1); // arr = [1] medianFinder.addNum(2); // arr = [1, 2] medianFinder.findMedian(); // return 1.5...

Approach & Solution Steps

Brute force — buffer all values, sort on every findMedian — O(n log n) per query.
Acceptable — balanced BST / order-statistic tree (policy-based or TreeMap multiset) — O(log n) insert + O(log n) median if supported.
Optimal — two heaps: max-heap for smaller half, min-heap for larger half, sizes differ by at most one — O(log n) addNum, O(1) findMedian for standard heap APIs.

Optimal JS Solution

class MinHeap {
	constructor() {
		this.values = [];
	}
	push(value) {
		this.values.push(value);
		this.bubbleUp(this.values.length - 1);
	}
	pop() {
		const top = this.values[0];
		const last = this.values.pop();
		if (this.values.length > 0) {
			this.values[0] = last;
			this.bubbleDown(0);
		}
		return top;
	}
	peek() {
		return this.values[0];
	}
	get size() {
		return this.values.length;
	}
	bubbleUp(index) {
		while (index > 0) {
			const parentIndex = Math.floor((index - 1) / 2);
			if (this.values[parentIndex] <= this.values[index]) {
				break;
			}
			[this.values[parentIndex], this.values[index]] = [
				this.values[index],
				this.values[parentIndex],
			];
			index = parentIndex;
		}
	}
	bubbleDown(index) {
		const length = this.values.length;
		while (true) {
			const leftIndex = index * 2 + 1;
			const rightIndex = index * 2 + 2;
			let smallest = index;
			if (
				leftIndex < length &&
				this.values[leftIndex] < this.values[smallest]
			) {
				smallest = leftIndex;
			}
			if (
				rightIndex < length &&
				this.values[rightIndex] < this.values[smallest]
			) {
				smallest = rightIndex;
			}
			if (smallest === index) {
				break;
			}
			[this.values[index], this.values[smallest]] = [
				this.values[smallest],
				this.values[index],
			];
			index = smallest;
		}
	}
}

class MaxHeap {
	constructor() {
		this.values = [];
	}
	push(value) {
		this.values.push(value);
		this.bubbleUp(this.values.length - 1);
	}
	pop() {
		const top = this.values[0];
		const last = this.values.pop();
		if (this.values.length > 0) {
			this.values[0] = last;
			this.bubbleDown(0);
		}
		return top;
	}
	peek() {
		return this.values[0];
	}
	get size() {
		return this.values.length;
	}
	bubbleUp(index) {
		while (index > 0) {
			const parentIndex = Math.floor((index - 1) / 2);
			if (this.values[parentIndex] >= this.values[index]) {
				break;
			}
			[this.values[parentIndex], this.values[index]] = [
				this.values[index],
				this.values[parentIndex],
			];
			index = parentIndex;
		}
	}
	bubbleDown(index) {
		const length = this.values.length;
		while (true) {
			const leftIndex = index * 2 + 1;
			const rightIndex = index * 2 + 2;
			let largest = index;
			if (
				leftIndex < length &&
				this.values[leftIndex] > this.values[largest]
			) {
				largest = leftIndex;
			}
			if (
				rightIndex < length &&
				this.values[rightIndex] > this.values[largest]
			) {
				largest = rightIndex;
			}
			if (largest === index) {
				break;
			}
			[this.values[index], this.values[largest]] = [
				this.values[largest],
				this.values[index],
			];
			index = largest;
		}
	}
}

class MedianFinder {
	constructor() {
		this.lowerMax = new MaxHeap();
		this.upperMin = new MinHeap();
	}
	addNum(value) {
		if (
			this.lowerMax.size === 0 ||
			value <= this.lowerMax.peek()
		) {
			this.lowerMax.push(value);
		} else {
			this.upperMin.push(value);
		}
		if (this.lowerMax.size > this.upperMin.size + 1) {
			this.upperMin.push(this.lowerMax.pop());
		} else if (this.upperMin.size > this.lowerMax.size + 1) {
			this.lowerMax.push(this.upperMin.pop());
		}
	}
	findMedian() {
		if (this.lowerMax.size > this.upperMin.size) {
			return this.lowerMax.peek();
		}
		if (this.upperMin.size > this.lowerMax.size) {
			return this.upperMin.peek();
		}
		return (this.lowerMax.peek() + this.upperMin.peek()) / 2;
	}
}

Edge Cases & Pitfalls

Always consider empty or null inputs.
Watch out for off-by-one index errors.